OF THE LAPLACE TRANSFORMATION PROBLEMS IN€¦ · THE APPLICATION OF THE LAPLACE TRANSFORMATION TO...

T.P. 2732

THE APPLICATION OF THE LAPLACE TRANSFORMATIONTO FLOW PROBLEMS IN RESERVOIRS

A. F. VAN EVERDINGEN, SHELL OIL CO., HOUSTON, AND W. HURST, PETROLEUM

CONSULTANT, HOUSTON, MEMBERS AIME

ABSTRACT

For several years the authors have felt the need for a sourcefrom which reservoir engineers could obtain fundamentaltheory and data on the flow of fluids through permeable mediain the unsteady state. The data on the unsteady state flow arecomposed of solutions of the equation

O'P 1 oP oP-+--=-Or' r Or ot

Two sets of solutions of this equation are developed, namely,for "d~e constant terminal pressure case" and "the constantterminal rate case." In the constant terminal pressure case thepressure at the terminal boundary is lowered by unity at zerotime, kept constant thereafter, and the cumulative amount offluid flowing across the boundary is computed, as a functionof the time. In the constant terminal rate case a unit rateof production is made to flow across the terminal boundary(from time zero onward) and the ensuing pressure drop iscomputed as a function of the time. Considerable effort hasbeen made to compile complete tables from which curves canbe constructed for the constant terminal pressure and constantterminal rate cases, both for finite and infinite reservoirs.These curves can be employed to reproduce the elIect of anypressure or rate history encountered in practice.

Most of the information is obtained by the help of theLaplace transformations, which proved to be extremely helpfulfor analyzing the problems encountered in fluid flow. Theapplication of this method simplifies the more tedious mathe·matical analyses employed in the past. With the help of La·place transformations some original developments were obtained (and presented) which could not have been easilyforeseen by the earlier methods.

INTRODUCTIONThis paper represents a compilation of the work done over

the past few years on the flow of fluid in porous media. Itconcerns itself primarily with the transient conditions prevail·ing in oil reservoirs during the time they are produced. Thestudy is limited to conditions where the flow of fluid obeys the

Manuscript received at office of Petroleum Branch January 12 1949Paper pr...ented at the AIME Annual Meeting in San Francisco, 'Febru:

ary 13·17. 1949.1 Refereneee are eiven at end of paper.

diffusivity equation. Multiple·phase fluid flow has not beenconsidered.

A previous publication by Hurst' shows that when the pres·sure history of a reservoir is known, this information can beused to calculate the water influx, an essential term in thematerial balance equation. An example is offered in the literature by Old' in the study of dxe Jones Sand, Schuler Field,Arkansas. The present paper contains extensive tabulateddata (from which work curves can be constructed), which dataare derived hy a more rigorous treatment of the subject mat·ter than available in an earlier publication.' The application ofthis information will enable those concerned with the analysisof the behavior of a reservoir to obtain quantitatively correctexpressions for the amount of water that has flowed into thereservoirs, thereby satisfying all the terms that appear in thematerial balance equation. This work is likewise applicable tothe flow of fluid to a well whenever the flow conditions aresuch that the diffusivity equation is obeyed.

DIFFUSITY EQUATIONThe most commonly encountered flow system is radial flow

toward the well bore or field. The volume of fluid which flowsper unit of time through each unit area of sand is expressedby Darcy's equation as

K ,)Pv=---

II> Orwhere K is the permeability, II> the viscosity and oPlor thepressure gradient at the radial distance r. A material balanceon a concentric element AB, expresses the net fluid traversingthe surfaces A and B, which must equal the fluid lost fromwithin the element. Thus, if the density of the fluid is expressed by p, then the weight of fluid per unit time and perunit sand thickness, flowing past Surface A, the surface near·est ·the well bore, is given as

27rrp K oP = 2"K (pr OP)II> Or II> Or

The weight of fluid flowing past Surface B, an infinitesimaldistance ar, removed from Surface A, is expressed as

27rK oP 0 ( pr 0; )- [pr -.+ 6rJ

II> or or

December, 1949 PETROlEUM TRANSACTIONS, AIME 305

T.P. 2732 THE APPLICATION OF THE LAPLACE TRANSFORMATION TO FLOW PROBLEMSIN RESERVOIRS

(1lI·2)co ( O'P 1 oP ) 00 oPf e-" -- +- -- dt = J e-" -..:-...Jt

o Or' r Or 0 at

'I '2. '3" TIME

FIG. 1A - SEQUENCE CONSTANT TERMINAL PRESSURES.1B- SEQUENCE CONSTANT TERMINAL RATES.

implied by Eq. UI·I, the partial differential can be trans·formed to a total differential equation. This is performed bymultiplying each term in Eq. U4 by e-" and integrating withrespect to time between zero and infinity, as follows:

q(t}, RATE

Since P is a function of radius and time, the integration withrespect to time will automatically remove the ti~e functionand leave P a function of radius only. This reduces the leftside to a total differential with respect to r. namely,

t;I)

0' J ~ e-'" P dt ~00 O'p 0 dipl.)J e....' -- dt = = --- etc.

o or' Or' dr'

and Eq. Irr-2 becomes

d'PIP• 1 drIP. co dP-- +--- = f e....' -- dt

dr' r dr 0 dt

P, PRESSURE

(II.I)

(III.I)

K o(pr ~)= fr Op

p. or aTFrom the physical characteristics of fluids, it is known

that density is a function of pressure and that the density ofa fluid decreases with decreasing pressure due to the fact thatthe fluid expands. This trend expressed in exponential formis

I' = p.e.... (r.-P l ••••••• (II.2)where P is less than P., and c the compressibility of the fluid.If we substitute Eq. II·2 in Eq. II.I, the dilfusivity equationcan be expressed using density as a function of radius andtime! or

( 0'1' +!.- 01') ~ = 01' (IIoS)Or' r or fp.c aT

For liquids which are only slightly compressible, Eq. II·2simplifies to I' e= P. [1- c (P. - P) ] which further modifiesEq. 11·3 to give

(O'P + _1 ap ) _K __ aP

Furthermore, if theOr' r or fltc aT'

radius of the well or field, R., is referred to as a unitradius, then the relation simplifies to

a"P -L I oP _ oP-or' . -; or - at ..... {n-4}

where t = KT!fp.cR.' and r now expresses the distance as amultiple of R., the unit radius. The units appearing in thispaper are always used in connection with Darcy's equation, sothat the permeability K must be expressed in darcys, thetime T in seconds, the porosity f as a fraction, the viscosity FJ.

in centipoises. the compressibility c as volume per volumeper atmosphere, and the radius R. in centimeters.

The difference between these two terms, namely,

o( pr~)2rK or .

- -- or,p. or

is equal to the weight of fluid 105t by the element AB, orOp

- 2..fr -- oraTwhere f is the porosity of the formation.This relation gives tl:e equation of continuity for the radialsystem, namely,

LAPLACE TRANSFORMATIONIn all publications, the treatment of the diffusivity equation

has been essentially the orthodox application of the FourierBessel series. This paper presents a new approach to thesolution of problems encountered in the study of flowing fluids,namely, the Laplace transformation, since it was recognizedthat Laplace transformations offer a useful tool for solvingdifficult problems in less time than by the use of Fourier.Bessel series. Also, original developments have been obtainedwhich are not easily foreseen by the orthodox methods.

If PI') is a pressure at a point in the sand and a functionof time, then its Laplace transformation is expressed by theinfinite integral

where the constant p in this relationship is referred to as theoperator. If we treat the diffusivity equation by the process

306 PETROLEUM TRANSACTIONS, AIME December, 1949

A. F. VAN EVERDINGEN AND W. HURST T.P. 2732

A

y

y

The next step in the development is to reproduce the boundary condition at the well bore or field radius, r = I, as aLaplace transformation and introduce this in the general solution for Eq. II1-S to give an explicit relation

PI,I = f l , ••)

By inverting the term on the right by the Mellin's inversionformula, or other methods, we obtain the solution for thecumulative pressure drop as an explicit function of radiusand time.

ENGINEERING CONCEPTSBefore applying the Laplace transformation to develop the

necessary work-curves, there are some fundamental engineering concepts to be considered that will allow the interpretation of these curves. Two cases are of paramount importancein making reservoir studies, namely, the constant terminalpressure case and the constant terminal rate case. If we knowthe explicit solution for the first case, we can reproduce anyvariable pressure history "at the terminal boundary to determine the cumulative influx of fluid. Likewise, if the rate offluid influx varies, the constant terminal rate case can he usedto calculate the total pressure drop. The constant terminalpressure and the constant terminal rate case are not independent of one another, as knowing the operational form ofone, the other can he determined, as will be shown later.

Comtant Terminal Pressure CaseThe constant terminal pressure case is defined as follows:

At time zero the pressure at all points in the formation is constant and equal to unity, and when the well or reservoir isopened, the pressure at the well or reservoir boundary, r = I,immediately drops to zero and remains zero for the durationof the production history.

If we treat the constant terminal pressure case symbolically,the solution of the problem at any radius and time is givenby P = PI•. t). The rate of fluid influx per unit sand thicknessunder these conditions is given by Darcy's equation

qCT) = 2rK (r ap) ..... (IV.I),. ar r = I

If we wish to determine the cumulative influx of fluid inabsolute time T, and having expressed time in the diffusivityequation as t =KTjf/'CR.', then

T 2...K £,.cR: t ( ap )QlTl = J qCT) dT = --x-

K- J -- dt

o ,. 0 Or r =1= 2"£cR.' Q'tl ••••• (IV-2)

(meg)

i!: PLANE

B

Furthermore, if we consider that Pltl is a cumulative pressuredrop, and that initially the pressure in the reservoir is everywhere constant 50 that the cumulative pressure drop P, ...,=O,the integration of the right hand side of the equation becomes

j e...• dP dt =e...• PIC) Ico + p j e-11' PICI dto dt 0 0

co

= poI e-Pt Pu > dt

As this term is also a Laplace transform, Eq. 111-2 can be written as a total differential equation, or

d'Pc,1 + I dPc,)_--- - -- - pPI,1

dr' r dr

CL.----_f----............---------_-+;:rl-_+..:.(.,..::..L.:O:.:)~ xo

Dr-----i~---__f..,

FIG. 2 - CONTOUR INTEGRATION IN ESTABLISHING THE CONSTANTTERMINAL RATE CASE FOR INFINITE EXTENT.

i!:PLANE

-t-if--:H----+-1-+-+-1-+-t--f-f.--f-l--+-x(~ ,0)

where

Q(t) = / (ap) dt (IV-3)

o ar r =1In brief, knowing the general solution implied by Eq. IV·S,

which expresses the integration in dimensionless time, t, of thepressure gradient at radius unity for a pressure drop of oneatmosphere, the cumulative influx into the well hore or into theoil-bearing portion of the field can be determined by Eq. IV-2.Furthermore, for any pressure drop, L\.P, Eq. IV-2 expressesthe cumulative influx as

Q(T) = 2".fcRb• l:l.P Q(cl • • • • • (IV4)per unit sand thickness.*

FIG. 3 - CONTOUR INTEGRATION IN ESTABLISHING THE CONSTANTTERMINAL RATE CASE FOR LIMITED RESERVOIR.

• The set of symbols now introduced and the symbols reported InHurst'a1 earlier paper on water-drive are related as follQW1I:

tG(Cl' 8jR') = QUI and G(Cl' ejR') = J Q(tl dt where

Cl' SIR' = t



(IV-H)

When an oil reservoir and the adjoining water·bearing formations are contained between two parallel and sealing fault.ing planes, the How of Iluid is essentially parallel to theseplanes and is "linear," The constant terminal pressure casecan also be applied to this case. The basic equation for linearflow is given by

a'p aPax' =at (IV.S)

where now t = KT/fpc and x is the absolute distance meas.ured from the plane of inHux extending out into the waterbe4I"ing sand. If we assume the same boundary conditions asin radial flow, with P =P(x, t) as the solution, then byDarcy's law, the rate of fluid influx across the original water·oil contact per unit of cross-sectional area is expressed by

q,,'l = K (~) (IV-6),. ax X=o

The total fluid influx is given by

'! K fl"C t(ap)Q(TI = J q""dT=-.-- J -- dto ,. K 0 OX x=o

=f C Qltl • : • • • • (IV.7)

where Qltl is the generalized solution for linear flow and isequal to

Q", = j (~) dt (IV-B)o ax x=o

Therefore, for any over·all pressure drop liP, Eq. IV·7 gives

Q<Tl = fcAP Q", (IV·9)per unit of cross-sectional area.

Comtant Terminal Rate Case

In the.constant terminal rate case it is likewise assumed thatinitially the pressure everywhere in the formation is constanthut that from the time zero onward the fluid is withdrawnfrom the well bore or reservoir boundary at a unit rate. Thepressure drop is given by P =p(.,t\> and at the boundary ofthe field, where r = 1, (aP/ar) •., = -1. The minus signis introduced because the gradient for the pressure drop rela·tive to the radius of the well or reservoir is negative. If thecumulative pressure drop is expressed as liP, then

e:,.p = q", p(.,,, (IV·lO)

where q", is a constant relating the cumulative pressure dropwith the pressure change for a unit rate of production. Byapplying Darcy's equation for the rate of fluid flowing intothe well or reservoir per unit sand thickness

q'T' = -2rK ( oL\P ) =-2rK q(t) (ap(.,tl ),. or =I,. or r =I

h · h . lifi q('f,1' Thw Ie simp es to q(1l =--. erefore, for any constant2,..K

rate of· production the cumulative pressure drop at the fieldradius is given by

L\P = q(T)I' Pu

)2..K

Similarly, for the constant rate of production in linear {low,the cumulative pressure drop is expressed by

where q,n is the rate of water encroachment per unit area ofcross-section, and p,,) is the cumulative pressure drop at thesand face per unit rate of production.

Superposition Theorem

With these fundamental relationships available, it remainsto be shown how the constant pressure case can be interpretedfor variahle terminal pressures, or in the constant rate case,for variable rates. The linearity of the dilfusivity equation al·lows the application of the superposition theorem as a se·quence of constant terminal pressures or constant rates insuch a fashion that it reproduces the pressure or productionhistory at the boundary, r =1. This is essentially Duhaznel'sprinciple, for which reference can be made to transient electriccircuit theory in texts py Karman and Biot,' and Bush" It hasbeen applied t oilie flow of fluids hy Muskat,' Schilthuis andHurst,' in employing the variable rate case in calculating thepressure drop in the East Texas Field.'

The physical significance can best be realized by an appli.cation. Fig. I·A shows the pressure decline in the well boreor a field that has been flowing and for which we wish to ob·tain the amount of fluid produced. As shown, the pressurehistory is reproduced as a series of pressure plateaus whichrepresent a sequence of constant terminal pressures. Therefore,by the application of Eq. IV4, the cumulative fluid producedin time t by the pressure drop e:,.P0' operative since zero time,is expressed by Q<Tl =2..fcR: L\P. QU)' If we next consider

r-01tJ3o..------r-------..,....-----.

20b------l--.1'....----/

IOI---/---/-+"L------+------I

O~I----~5------I~O:--------J

(IV.12) FIG. 4 - RADIAL FLOW, CONSTANT TERMINAL PRESSURE CASE, INFIN·ITE RESERVOIR, CUMUtATIVE PRODUCTION VS. TIME.

308 PETROlEUM TRANSACTIONS. AIME December, 1949


ASYMTOTIC VAI.UE O.ElZ!>

"R • 1.5

(Y-I)

(IV·16)

(IV·17)

_ CIO -e'P

' I(Xl 1P (p) == f e-P

' I dt == - 1 =-o p p

o

If the increments are infinitesimal, or the smooth curve reb.·tionship applies, Eq. IY·IS becomes

t dq(t')lI.P =q,.) Pm + f -- PH"') dt'

o dt'

If ql.) = 0, Eq. IV·16 can also be expressed ast

lI.P =of q(.') P'".,,) dt'

In applying the Laplace transformation, there are certainfundamental operations that must be clarified. It has beenstated that if P ICI is a pressure drop, the transformation forPIt) is given by Eq. III-I, as

FUNDAMENTAL CONSIDERATIONS

where P'w is the derivative of Pl') with respect to t.

Since Eqs. IY·13 and lV·IS are of such simple algebraicfOrIDs, they are most practical to use with production historyin making reservoir studies. In applying the pressure or rateplateaus as shown in Fig. 1, it must be realized that the timeinterval for each plateau should be taken as small as possible,50 as to reproduce within engineering accuracy the trend ofthe curves. Naturally, if an exact interpretation is desired, Eqs.IV.14 and JV·16 apply.

_ CIO

PIP) == of e....• Pm dt

To visualize more concretely the meaniilg of this equation, ifthe unit pressure drop at the boundary in the constant termi·nal pressure case is employed in Eq. Ill.I, its transform is

given by

The Laplace transformations of many transcendental functionshave been developed and are available in tables, the most com·plete of which is thc tract by Campbell and Foster.' It is there·fore often possible after solving a total differential such asEq. ill-3 to refer to a set of tables and transforms and deter-

mine the inverse of Pl.) or PH)' It is frequently necessary to

simplify PIP) before an inversion can be made. However, Mel.lin's inversion formula is always applicable, which requiresanalytical treatment whenever used.

There are two possible simplifications for PIP) when timeis small or time is large. This is evident from Eq. m·3, wherep can be interpreted by the operational calculus as the operator dldt. Therefore, if we consider this symbolic relation,then if t is large, p must be lImall, or inversely, if t is small,

p will be large. To understand this, if P,P) is expressed by aninvolved Bessel relationship, lhe substitution for p as a small

or large value will simplify PCP) to give PIt) for the corre·sponding times.

Mellin's inversion formula is given on page 71 of Carslawand Jaeger:'

(IV.IS)QIT) =2,..fcR.' [LlP.QII) + t\.P,Qu.•,) +

lI.P,Q".<,,) + lI.P,Q[ •.•,) + ... ]

FIG. 5 - RADIAL FLOW, CONSTANT TERMINAL PRESSURE CASE,CUMULATIVE PRODUCTION VS. TIME FOR LIMITED RESERVOIRS.

when t > t,. To reproduce the smooth curve relationship ofFig. I-A, these pressure plateaus can be taken as infinitesim·ally small, which give the summation of Eq. IY·I3 by theintegral

t Oll.PQ(T) == 2ricR.' f -- Ql•.•·) dt' . • . (IV·H)

a at'By considering variable rates of fluid production, such as

shown in Fig. I·B, and reproducing these rates as a serles ofconstant rale plateaus, then by Eq. IV·ll the pressure drop inthe well bore in time t, for the initial rate q. is lI.P. = q.PI.).At time t" the comparable increment for constant rate is ex·pressed as q, - q., and the effect of this increment rate onthe corresponding increment of pressure drop is lI.P, =(q. - q.) PI'· .... Again by superimposing all of these effects,the determination for the cumulative pressure drop is ex·pressed by

lI.P == ql.) Pl') + [q,(t,) - qt.)] Pl.· ••) + [q"(.,) - q(.,)]

p("',) + [q(,,)-q(,:/] PI ....) + (IV.IS)

the pressure drop t\.p.. which occurs in time t" and treat thisas a separate entity, but take cognizance of its time of inception t.. then the cumulative fluid produced by this incrementof pressure drop is Q(t) = 2".fcR.' t\.P, Q("'I)' By superimposing all these effects of pressure changes, the total influxin time t is expressed as

3.ot---+---+----h~,.L--_r_--_1_--_l

rr=-Q(-t)--r----r----,r--~..__,,,.._-_r_--..,

35f----l----f---+

1.51-__-if-__-:=::!==---r--+__-=:A:;SY;.:"'::..:T..:OT.:.:'::.C....:V.:::J\L:r;:U::E....:I::..:.500=L/

2.0-1---+--+-7-+---+---+---+---1

O.OO~----;,'=.O,--------:2-:-..0;;----3,fO;;----4-='.O~----=5!-::.O;----6-=!.O

December, 1949 PETROLEUM TRANSACTIONS, AIME 309


r-PRESSURE DROP IN ATMOSPHERES-PCtl.

1.80.....--,,-.......""'T"---r---r---r--"""T--..-----.

2.00J.---f--H--\t-\\-i--1t--1t--I---;

2.101--+---+11--+-\

2:3()I.-........,,12!::--~16,.----;!2:!;-0--::;2;l;4---;:;;~--;f;;-.....l.-:h---i40'

TIME(t)FIG. 6 - RADIAL FLOW, CONSTANT TERMINAl RATE CASE, PRESSURE

DROP VS. TIME. P(t) VS. t

2.201---I----!--\-+--+--\--+-\

where P ) is the transform PCP) Where this report is con(X

corned with pressure drops, the above can be written as

'Y+iex. J.901---++-t\-1t.--+--f---+--f---+--l1 Xt, At, -

PI,,) -pCt,) = - f (e -e ) P d A. (Y.2)2ri (X)

')'-i ex.

The integration is in the complex plane). =x + iy, along aline parallel to the y·axis, extending from minus to positiveinfinity, and a distance 'Y removed from the origin, so that allpoles are to the left of this line, Fig. 2. The reader who has acomprehensive understanding of contour integrals will recognize that this integral is equal to the integration around a~-circle of infinite radius extending to the left of the iinex = 'Y, and includes integration along the "cuts," which joinsthe poles to the semi-circle. Since the integration along thesemi-circle in the second and third quadrant is zero for radiusinfinity and t>O, this leaves the integration along the "cuts"and the poles, where the latter, as expressed in Eq. Y·2, arethe residuals.

= P PCP) - PWo)dPlt) - •

or the transform of --=p PCP) - PWolt provided e-9' P Ct)dt

Theorem A - If PCP) is the transform of Pc 'll then

I0

00 00

+ P of e-9 l Pl') dt

Certain fundamental relationships in the Laplace transformations are found useful:'·

approaches zero as time approaches infinity.

00

Theorem B - The transform of of P (,') dt' is expressed by

00 t -e-9' t

f e-9t f Pet') dt'dt =-- fPc.') dt'o 0 p 0

p

or the t~ansform of the integration P Ct') with respect to t'_ t

from zero to t is P cpJP, if e-9t f P (t') dt' is zero for timeo

infinity.

Theorem C - The transform for e±ct p,,) is equal to00 00 _ _

of e-9' e±ct Pct) dt = of e-CP.C)t PCt) dt = P CP;C)

if p - c is positive.

TheoremD-Ifp,(p) is the transform of P,(t), and P,(p)is the transform of P,(tlt then the product of these two transforms is the transform of the integral

tof PUt') P,Ct.,·) dt'

This integral is comparable to the integrals developed by thesuperimposition theorem, and of appreciable use in thispaper.

CONSTANT TERMINAL PRESSURE AND

CONSTA1'I"T TERMINAL RATE CASES,

INFINITE MEDIDM

The analytics for the constant terminal pressure and ratecases have been developed for limited reservoirs'" when theexterior boundary is considered closed or the production ratethrough this boundary is fixed. In determining the volume ofwater encroached into the oil.bearing portion of reservoirs.few cases have been encountered which indicated that thesands in which the oil occurs are of limited extent. For themost part, the data show that the influx behaves as if thewater-bearing parts of the formations are of infinite extent,because within the productive life of oil reservoirs, the rate ofwater encroachment does not reflect the influence of an ex·terior boundary. In other words, whether or not the water sandis of limited extent, the rate of water encroachment is such asif supplied by an infinite medium.



( _o~) =-1 at all times.or r=l

A reference to a text on Bessel functions, such as Karmanand Biot: pp. 61.63, shows that the general solution for Eq.III·3 is given hy

To fulfill the second boundary condition for unit rate ofproduction, namely (oPlor)r-1 = -1, the transform forunity gives

(VI.l)

(VI.2)

where I. (rvp) and K.(rYp) are modified Bessel func.tions of the first and second kind, respectively, and of zeroorder. A and B are two constants which satisfy a second order

differential equation. Since P(r,p) is the transform of thepressure drop at a point in the formation, and hecause at apoint not yet affected hy production the absolute pressure

equals the initial pressure, it is required that P(r,p) shouldapproach zero as r becomes large. As shown in Karman and

Biot: I.(rVp) hecomes increasingly large and K.(r'/p)

approaches zero as the argument (r'/p) increases. There·fore, to obey the initial condition, the constant A must equalzero and (VI·I) becomes

Computing the water inflWl for an infinite reservoir with the

help of Fourier-Bessel expansions, an exterior boundary can

be assumed so far removed from the field radius that the production for a considerable time will reflect the infinite ca~e.

Unfortunately, the poor convergence of these expansions inval.

idates this approach. An alternative method consists of using

increasing values for exterior radius, evaluating the water in.

flux for each radius separately, and then drawing ~he envelope

of these curves, which gives the infinite case, Fig. 5. In such

a procedure, each of the branch curves reflects a water reser.

voir of limited extent. Inasmuch as the drawing of an envelope

does 110t give a high degree of acuracy, the solutions for the

constant terminal pressure and constant terminal rate cases

for an infinite medium are presented here, with values forQ(C) and Pit) calculated directly.

The constant terminal pressure case was first developed by

Nicholson" by the application of Green's function to an instan.

taneous circular source in an infinite medium. GoldsteinU pre.

sented this solution by the operational method, and Smith"

employed Cai"slaw's contour method in its development. Cars.

law and Jaeger..·.. later gave the explicit treatment of the

constant terminal pressure case by the application of the La.place transformation. The derivation of the constant terminal

rate case is not given in the literature, and its developmentis presented here.

...

(VI-3)

--1----4--16.0

by Eq. V·l. The differentiation of the modined Bessel func·tion of the second kind, Watson's Bessel Functions," W.B.F.,p. 79, gives K.' (z) =- K, (z). Therefore, differentiation Eq.

The Constant Rate Case

As already discussed, the boundary conditions for the con

stant rate case in an infinite medium are that (1) the pres·

sure drop p(•. lI is zero initially at every point in the forma.

tion, and (2) at the radius of the field (r =1) we have

r-: PRESSURE DROP

11,......xtl::lnO!:- -..:3i!-_-..:jS~_..!8i!_!_¥=:....,.--_r~_r_--,...-__r-r_-----T"':;r_-r_-..,.._,~ u

.-l._--R-200I

3..~-----f_-----1I--_+A-~~,.£..--+15.2--+--___.,I__+_---__,"__~=---+_-_+__16.S

I3.6.~----1----+ ~~-k-$------!15r-~f---+--+---#~---=====-~(-6

3.41-------I..'--:l~-+_7~jl-+_----_f14.8 R-'OO'-+---1U

5.21-----..P+--+-If---=l--f--- R.wo+l;--t--79f=--t-:::::===--I

F-:±===F=hlfi1jro---l14.4-~;.q...---j---j-----t--+--+---J5.8

I _~--R-300

2.8~(L--.l.~--L---L--l.-..L.T..----L.l..L.-..c....L.---I-.L....s~---___:~-_:_--_:___'5.6IXlOs 3 l5 8 IXIO 3 3 !l 8

tFIG. 7 - RADIAL FLOW, CONSTANT TERMINAL RATE CASE. CUMULATIVE PRESSURE DROP VS. TIME PIt) VS. t



and ..ince

VI·2, with re~pect to r at r = 1, gives

( .at ) =-BVp K.( 'vp)or r=l.

p'rK, (vp)To determine the inverse of Eq. VI-4 in order to establish

the pressure drop at radius unity, we can resort to the sim.plification that for small times the operator p is large. Since

(VI·12)

(VI.l3)

(VI·14)

(VI.lS)

(log2-"Y)

p

ro

zK.,(z) ~ - [log:2 + 'Y]

K,(z) ~ liz .. ..Therefore. Eq. VI·4 becomes

- -logpp u." = --- +

2p

o.

f-- e·o7' ./

/ V /'

YlP 17/

1/

r=E='t.'~~, ..~

:to' "",

17 r7 ~.p

.J? I....

:j::j::Ij

V

Ol ,ro' ro'

Eq. V·I. Therefore, the pressure drop at the boundary of thefield when t is large is given by

1= -- [log 4t - "f]

21

= - [log t + 0.80907]2

The inversion for the first term on the right is given by Camp.bell and Foster, Eq. 892, and the inverse of the second term by

o

where "f is Euler's constant 0.57722, and the logarithmic termconsists of natural logarithms. When z is small

The solution given by Eq. VI-ISis the solution of the con·tinuous point source problem for large time t. The relationshiphas been applied to the flow of fluids by Bruce," Elkins," andothers, and is particularly applicable for study of interferencebetween flowing wells.

The point source solution originally developed by Lord Kel·vin and discussed in Carslaw" can be expressed as

III

FIG. 8 - CONSTANT RATE OF PRODUCTION IN THE STOCK TANK,ADJUSTING FOR THE UNLOADING OF FLUID IN THE ANNULUS. pet)VERSUS t where;= c!2ncR.', AND c is the VOLUME OF FLUID UN·LOADED FROM THE ANNULUS. CORRECTED TO RESERVOIR CONDI.

TlONS, PER ATMOSPHERE BOTTOM·HOLE PRESSURE DROP. PER UNITSAND THICKNESS.

r-"'"I

Pl•. ., =.-! sex> e-' dn =..!. { -Ei (_.! ) ~ . (VI.16)2 it n 2 4t

often referred to as the logarithmic integral or the Ei·func·tion. Its values are given in Tables of Sine, Cosine, and Expo.nential Integrals, Volumes I and II, Federal Works Agency,W.P.A., City of New York. For large values of the time,t,

1Eq. VI·16 reduces to P" .., =2 [log 4t-')'] which is Eq.

VI-IS, and this relation is accurate for values of t>100.

(VI-4)

(VI·S)

(VI-6)

(VI.S)

(VI·H)

is given in Campbell and

( -=-)u'"1 ex> oJ

+- (_1)' 2: ----- [ ~ m-' + 2: m-']2 ,.. r! (n+r) ! m-' mA

1 n·l ( Z )_.., (n-r-l)!+ - 2: (-1)' -2 ,.. 2 r!

K.,(z) = ~;zfor z large, W.B.F., p. 202, then

1PU.,I = -,,;

pThe inversion for this transformFoster, Eq. 516, as

d' :PI" ---- = pP,,1dx'

for which the general solution is the expression

(-~~ )r=1 = - ;

the constant B = IIp'/1 K, (Vp). Therefore, the transformfor the pressure drop for the constant rate case in an infinitemedium is given by

2PUI = -- t'/1 (VI-7)

v-;;In brief, Eq. VI.7 states that when t = K T/fpeR.' is small,which can be caused by the boundary radius for the field, R.,being large, the pressure drop for the unit rate of productionapproximates the condition for linear flow.

To justify this conclusion, the treatment of the linear flowequation, Eq. IV-S, by the Laplace transformation gives

P,x." = Ae-' vp + Be"V> (VI·9)By repeating the reasoning already employed in this develop.ment, the transform for the pressure drop at x = 0 gives

P(OVPf = lip'!'which is identical with (VI.6) with p the operator of t =KT/fl'c.

The second simplification for the transform (VI-4) is toconsider p small, which is equivalent to considering time, t,large. The expansions for K. (z) and K, (z) are given in Cars.law and Jaeger,' p. 248.

K.,(z) = - I.(z) { log~ + 'Y ~ +( ~ )'2 :l

(1 ) ( )' ( 1 1 ) ( )' (VI.lO)

+ 1+ 2 ~ + 1+"2+"3 ~(2!)' (3!)' +

zK.(z)=-(-I)'·'I.(z) {log2+'Y~



~.{S K.( v'A )2 ir

-ut, -- e ) 1(. (u e' r) du

(VI·24)

(VI-23)

and

ruI, (u) Y. (u) - J. (u) Y. (u) =

-i,..negative real "cut" is expressed by A = u· e

~t. At. -1 1 (e -e ) 1(.(V A r) d A

2". 0:>

By the recurrence formula given in W.B.F., p. 772

~'i' K,(v'~

" . /2= -1 0:> (e-ut'_e-U 1,) 1(. (u e-1,.. r) du

-;-J -j,r/2 -ir/2u' e K,(u e )

Using Eq. VI·IS, yields the relationship-u't, -u't

Io:>(e -e ') [Y1 (u) I.(ur)-J,(u) Y.(ur)] du

-;-J u' [1,'(u) + Y,'(u)](VI·20)

The integration along Paths DO and DC is the sum of therelations VI·I9 and VI·20, or

p(" '1) - PI', ,,) =2 0:> (e-u't'_eu't,) [Y,(u} l.(ur} -l,(u) Y.(ur}] du-f ----~~~~_:_::_-_...:...--r O u'[1,'(u) + Y,'(u)]Initially, that is Ilt time zero, the cumulative pressure drop at

any point in the formation is zero, p(,. ,••) =O. Hence, thepressure drop since zero time equals:

-u't2 o:>(l-e ) [1,(u) Y.(ur) -Y.(u) l.(url] du

p(,.., =-;J u'[1,'(u) + Y.'(u)]

(VI·21)which is the explicit solution of the constant terminal rate casefor an infinite medium.

To determine the cumulative pressure drop for a unit rateof production at the well bore or field radius, (where r == 1)then Eg. VI·21 changes to

-u't2 0:> (1 -e ) [1.(u) Y.{u) - Y.{u) I.(ll)] du

pu• t) =-;of u' [J,'(u) + Y,'(u)](VI·22)

Equation VI·22 simplilies to-u't

4 co (I - e ) du

p(,) =,...of u' [J,'(u) + Y1'(u)]

Constant Terminal Pressure CaseAs already shown, the transform of the pressure drop in

Likewise, the integration along the under portion of the

- ,..i -+ -~ I.(z) +i Y.(z) }

2

1. (z)

i,..±

2I. (z e ) ==

111':!:-

21(.(z e ) =

-u't,1 00 (eria) ----i-r----ir----

By this development it is evident that the point source solu·tion doe3 not apply at a boundary for the determination of thepressure drop when t is small. However, when the radius, R..is small, such as a well radius, even small values of the abso·lute time, T will give large values of the dimensionless time t,and the point source solution is applicable. On the otherhand, in considering the pressure drop at the periphery of afield (in which case R. can have Il large numerical value) thevalue of t can be easily less than 100 even for large values ofabsolute time, T. Therefore, for intermediate times, the rig·orous solution of the constant rate case must be used, whichwe will now proceed to obtain.

To develop the explicit solution for the constant terminalrate case, it is necessary to invert the Laplace transform, Eq.VI4, by the Mellin's inversion formula. The path of integra·tion for this transform is described by the "cut" along thenegative real axis, }'ig. 2, which give3 a single valued functionon each side of the "cut." That is to say that Path AB re·quired by Eq. V-2 is equal to the Path AD and CB, both ofwhich are described by a semi-eircle of radiu3 infinity. Since'Its integration is zero in the second and third quadrant, thisJeaves the integration along l'aths DO and UC equal to All.The integration on the upper portion of the "cut" can he ob-

tained by making ~ == u' e +i~ which yieldsAt, At. -

1 _~(e -e )1(.(Y A r)J dA ==

2ri a

u' e' K, ( u e' ) (VI.17)Carslaw and Jaeger' (page 249) shows that modified Bassel

i1l'±-

2functions of the /irst and second kind of arguments z ecan be expressed by the regular Bessel functions as complexvalues, as follows:

(VI.I9)where the imaginary term has been dropped.

The substitution of the corresponding values for

iTr/2 ir/2K, (u e r) and K, (u e ) from Eq. VI·IS in Eq. VI-17gives the integration along the upper portion of the negativereal "cut" as

-ut t1 _utt:1 CD (e -e ) [Y,(u) J.Cur) -I,(u) Y.(ur) ] du_f ;-;:-;,-:-;-;--;--:::-;;-:;-::-;;- ..:.-_

u' [J,'(u) + Y,'(u) ]

I".±-

I,(z e ') = ± i I,(z) • • • • . (VI·IS)

(VI·25)

aninfinitemediumisP(,.•,=B K,('Ipr). In the constantterminal pressure case it is assumed that at all times the pressUre drop at r =1 will be unity, which is expressed as atransform by Eg. V·I

PU••l =1/pBy solving for the constant B at r == 1 in the above formula,

we find B =lip K,(vp), so that the transform for thepressure at any point in the reservoir is expressed by

- 1(.(Vp r)PI'." =_..:-....:.........:...

p1(.(Vp)The comparable solution of VI·25 for a cumulative pressuredrop can be developed as before by considering the paths ofFig. 2, with a pole at the origin, to give the solution

r-"2 [I,(z) + i Y,(z) ]

i".±-

2K,(z e ) =

and



(IV·3)

TABLE I - Radial Flow, Comtant Terminal Pressure~ Constant Terminal Rate ClUes Jar Infinite

Reservoirs

p(l'. t:L) -P(r. tl) =u't u'Es2 co(e- '-e- ) [J.(u) Y.(ur) - Y.(u) J.(ur)] du

-;J u'[J:(u) + Y.'(u)] (VI·26)

If we are interesterl in the cumulative Buid influx at the fieldradius, r =I, then the relationship Eq. IV·3 applies, or

t (oP)Q(t) = J -- dto or r=l

The determination of the transform of the gradient of thepressure drop at the field's edge follows from Eq. VI·25,

(oPI •••)) _ K.(Yp )

or r=l p'I'K.(Vp)

since K.'(z) =-K,(z). Since the pressure drop Pl', Cj corre·aponds to the difference between the initial and actual pres'sure, the transform of the gradient of the actual pressure at

r = I is given by

( oP) _ (-OP",.))or r=1 or r=l

or

(oP ) K...c1 (c....Y.;:,..P..;..)_

or r=l p'fJ K.( Vp>which corresponds to the integrand of Eq. IV-3. Further, from

the definition given by Theorem H, namely, that if P(p) is thet

transform of P(Ch then the transform of f PCC') dt' is given byo

P(p)/p and the Laplace transform for Q(tl is expressed by

1.0(10)"~.O "1.0(10)-'1~6 ..2.0 "2.6 u3.0 "4.0 "6.0 u

0.0 "1.0 u8.0 u8.0 u1.01.82.02.03.04.08.00.07.08.08.01.0Cl0)'1.8 "2.0 u2.5 u3_0 u

4.0 ..6.0 ..0.0 ..7.0 u8.0 ..0.0 u1.0(10)11.5 •2.0 ..2.6 •3.0 •4.0 •8.0 ue.o ..7.0 u8.0 u0.0 ..1.0(10)'

0.1120.27S0.1040.6200.5OG0.5800.7$0O.80S1.0201.1401.261I.~

1.4501.6702.0322.4422.83S3.m3.m4.5415.1485.740U148.11517.417O.oeoI.22tCIO)11.4M ..1.881 "2.088 ..2.482 ..2.1150 ..3.228 ..3.500 ..3.1l42 ..4.301 "6.080 ..7.m ..9.120 ,~

10.&3 "13.48 "18.24 "J8.~ Ie

21.150 Ie

24..23 j~

20.17 ..2G.31 ..

p(U

0.1120.2200.3150.3750.4240.4500.150:10.6M0.0100.8600.7020.136o.m0.8020.127UI201.101l.tau1.2761.31121.4301.600U801.504U611.820!.OSO2.0012.1412.282.2.3&32.4782.6602.1152.1122.7232.0213.0043.1133.2533.40$3.518U083.5843.7503.8183.1150

1.8(10)'2.0 ..2.6 u3.0 "4.0 ..5.0 ..0.0 ..7.0 ..S.O "G.O "1.0(10)'1.5 ..2.0 oW

2.6 ..3.0 "4.0 M

8.0 ..0.0 ..7.0 "8.0 ..0.0 ..1.0(10)'1.5 "2.0 ..2.5 ..3.0 ..4.0 ..5.0 ..0.0 M

7.0 ..8.0 ..8.0 ..1.0(10)'1.5 M

2.0 ..2.5 M

3.0 ..4.0 u6.0 oW

8.0 ..7.0 ..8.0 ..0.0 ..1.0Cl0l'

4.138(10)'UI5 ..8.480 ..1.890 ..0.757 ..

11.88 "13.05 ..15.QU ..18.00 "IG.QU ..21.1l8 ..

3.140(10)'4.070 ..4.QU4 ..5.801 ..7.534 ..0.542 M

11.03 ..12.80 ..14.33 ..16.8$ ..17.88 ..2.538(10)'3.308 M

4.080 M

4.817 ..8.207 ..7.0Q9 ..9.113 ..

10.51 ..11.80 ..13.28 ..14.112 ..2.120(10)2.781 ..3.m ..4.1ld4 ..5.313 ..8.544 ..7.781 •8.1l88 ..

10.18 ..11.14 ..12.62 ..

(VI.29)

(VI·30)

(VI·27)

(VI·28)

1.17CI0)"1.85 ..1.82 ..2.20 ..3.02 ..3.15 "~.47 ,.5.10 ..6.8Q ..8.&3 "7.28 "1.08(10)"1.42 ..

l.5(10)1l2.0 ..2.5 ..3.0 ..4.0 ..5.0 ..6.0 ..7.0 'I.8.0 C4

G.O ..1.0(10)0:1.6 ..2.0 "

1.828(10)'2.3G8 ..2.G81 u3.011 ..4.&10 ..5.889 "8.788 ..7.818 If

8.M8 ..G.Oll ..

)G.D.S "1.804(101'2.108 ..2.S01 u

3.1DO "4.071 ..5.032 ..5.084 ..8.028 ..1.866 oW

8.701 ..o.m"1.429(10r1.880 u2.328 ..2.171 "3.M5 ..4.510 II

8.388 ..8.220 ..1.OM "7.000 ..8.747 ..1.288(10)'1.097 ..2.103 "2.505 ..3.2GO ..4.081 ..4.868 ..lii.643 u8.414 ..7.183 ,.7.048 ..

TABLE 1-COnlirwed

1.5(10)'2.0 ..2.5 ..3.0 ..4.0 ..5.0 II

8.0 ..7.0 u8.0 ..U.O ..1.0(10)11.5 "2.0 ..2.8 "3.0 ut.O ..5.0 ..8.0 ..7.0 ..8.0 ..0.0 ..1.0(10)'l.5 u2.0 ..2.5 "3.0 "4.0 ..5.0 II

5.0 "7.0 "8.0 u9.0 ..1.0(10)"1.6 '12.0 ..2.5 "3.0 "4.0 u4.0 u8.0 "7.0 ..8.0 u9.0 u1.0(10)11

With respect to the transform 'O"H there is the simplificationthat for time small. p is large, or Eq. VI·27 reduces 10

K,( Yp)QI.) = ._-:.:~.:..-.:....-

pi/, K.( Vp)

The application of the Mellin's inversion formula 10 Eq. VI·27follows the paths shown in Fig. 2, giving

-u't4 ex> (I-e ) du

Q(tl =rlof u' [J:(u) + Y:(u) ]

'0(0) = IIp'l'and the inversion is as hefore

2QUl =--- t'l'

Y-;-which is identical to the linear flow case. For all other valuesof the time, Eq. VI·28 must be solved numerically.

Relation Between Q(p) and pep)It is evident from the work that has already gone before.

that the Laplace transformation and the superimposition the·orem offer a basis for interchanging the constant terminalpressure to the constant terminal rate case, and vice versa. Inany reservoir study the essential interest is the analyses ofthe flow either at the well hore or the field boundary. Thepurpose of this work is to determine the relationship betweenQ (t), the constant terminal pressure case, and P (tl. the con·stant terminal rate case, which explicitly refer to the boundaryr =1. Therefore, if we conceive of the influx of fluid into a


A. f. VAN EVERDINGEN AND W. HURST T.P. 2732

TABLE II - Comtant Terminal Pressure Ca.seRadial Flow. Limited Reservoirs

R _ 1.6 R _ 2.0 R _ 2.5 R = SoDlit = 2.8899 CI, = 1.3606 lit = 0.8663 lit = 0.6%56... = '.S452 lit = 4.U58 lit = 3.0876 ... = 2.3041

t Q lI) t QCt) t Q(t) t Q(l)

-----5.0(10)-' 0.275 5.0(10)-> 0.278 1.0(IW' 0. 4081 3.0(10)-< 0.7558.0 M 0.304 7.6 II 0.3~ 1.5 M 0.509. t.O W 0.8957.0 M 0.330 1.0(10)-' 0.404 2.0 M 0.899 5.0 .. 1.a238.0 .. 0.304 1.25 .. 0.448 2.5 M 0.881 5.0 .. 1.1'39.0 .. 0.318 1.SO •• 0.507 3.0 .. 0.758 7.0 .. 1.2581.0(10)-1 0.398 1.75 .. 0.553 3.8 .. O.ll2lI 8.0 .. 1.31l31.1 M 0.41' 2.00 .. 0.597 4.0 .. o.m '.0 .. 1.4481.2 .. 0.431 2.25 .. 0.838 4.6 II 0.962 1.00 l.81l31.3 M 0.4~ 2.50 .. 0.878 8.0 .. 1.024 1.25 1.1911.4 .. 0.~1 2.78 .. 0.715 &.5 I' 1.083 1.50 1.9971.8 .. 0.414 3.00 .. 0.751 5.0 .. 1.140 1.15 2.1841.8 .. 0.484 3.25 II 0.785 8.8 .. 1.198 2.00 2.&l31.7 M 0.4f7 3.50 .. 0.817 7.0 M 1.248 2.25 2.5071.8 .. 0.507 3.75 .. 0.848 1.5 .. 1.z:n 2.50 2.848l.t .. 0.817 4.00 .. 0.817 8.0 .. 1.148 2.15 2.7722.0 .. 0.&28 4.25 .. 0.908 8.8 .. 1.398 3.00 2.8852.1 M 0.833 4.50 .. 0.832 9.0 .. 1.440 3.25 2.9902.2 M 0.041 4.78 M 0.988 9.8 .. l.484 3.50 8.0842.3 M 0.848 5.00 .. 0.183 1.0 1.525 3.75 8.1102.4 .. 0.884 5.50 .. 1.028 1.1 1.505 4.00 a.2412.' M O.sst 8.00 .. 1.070 1.2 1.571 4.25 a.8112.8 .. 0.868 8.50 .. 1.108 1.3 1.747 4.50 8.1812.8 .. 0.814 1.00 M 1.143 1.4 1.811 4.75 a.4393.0 .. 0.682 1.50 .. 1.174 1.8 1.810 5.00 8.m3.2 .. 0.883 8.00 .. 1.203 1.8 1.924 8.50 8.8813.4 .. 0.894 t.OO M 1.253 1.7 1.118 5.00 8.6558.5 M 0.899 1.00 1.298 1.8 2.022 8.50 3.7173.' .. 0.803 1.1 1.330 2.0 2.106 7.00 3.7574.0 .. '0.505 1.2 1.388 2.2 2.178 7.80 3.l!OD4.8 .. 0.813 1.3 l.882 2.4 2.241 8.00 3.8438.0 .. 0.817 1.4 1.402 2.8 2.294 9.00 3.~

5.0 M 0.821 l.5 1.432 2.8 2.340 10.OJ 3.11281.0 .. 0.823 1.7 1.«4 3.0 :1.380 11.00 3.9818.0 .. 0.824 1.8 1.453 3.4 2.«4 12.00 8.m

2.0 1.468 3.8 2.491 1UlO 1.8852.5 1.481 4.2 2.&28 15.00 Ut38.0 1.498 4.8 2.851 18.00 3.9974.0 1.499 8.0 2.570 20.00 3.m5.0 l.500 8.0 2.599 22.00 3.m

1.0 2.813 24.00 4.0008.0 2.81tt.O 2.822 I10.0 2.824

TABLE IT - Continued

B _U R _ 4.0 R _ 4.6 It _ 6.0 It = 6.0

... = 0.4861 ... = 0.8986 ... = 0.3295 ... = 0.2823 lit = 0.2182

... = 1.8374 ... = 1.5267 ... = 1.3061 ... = 1.1S92 ... = 0.9025

t Q(,) t Q(,) t Q(l) t Q(t) t Q Ct )

- -1.00 1.511 2.00 2.W! 2.5 2.835 3.0 3.195 8.0 5.1481.20 1.781 2.20 2.598 3.0 3.196 3.5 3.542 6.5 5.4401.40 1.940 2.40 2.148 3.8 3.537 4.0 3.875 7.0 5.724l.50 2.111 2.80 2.893 4.0 3.859 4.5 4.113 7.8 8.0021.80 2.273 2.80 3.034 4.5 4.185 5.0 4.4g~ 8.0 8.2732.00 2.427 3.00 3.170 5.0 UM 5.5 4.792 8.5 8.5372.20 2.674 3.25 3.334 5.6 4.727 6.0 5.074 9.0 5.7952.40 2.715 3.50 3.493 8.0 4.984 6.6 5.345 9.6 7.0472.50 2.849 3.76 3.845 8.5 5.231 7.0 $.IlOS 10.0 7.~3

2.80 2.978 4.00 3.792 7.0 5.484 7.5 6.854 10.5 7.5333.00 3.098 4.25 3.932 7.5 6.884 8.0 8.094 11 7.7873.25 3.242 4.50 4.088 8.0 5.892 8.5 8.325 12 8.2203.50 3.379 4.76 4.198 8.5 6.089 9.0 8.547 13 8.6513.16 3.507 6.00 4.323 9.0 8.218 9.5 8.780 14 9.0834.00 3.828 5.80 4.580 9.5 6.453 10 8.985 15 9.4584.26 3.742 5.00 4.779 10 8.821 11 7.350 18 9.8294.50 3.850 8.50 4.982 11 6.930 12 7.708 17 10.194.15 3.951 7.00 6.169 12 7.208 13 8.035 18 10.538.00 4.047 7.50 6.343 13 7.457 14 8.339 19 .0.656.50 4.222 8.00 6.504 14 7.880 15 8.820 20 11.188.00 4.378 8.80 5.653 15 7.880 16 8.879 22 11.745.80 4.518 9.00 8.790 15 8.060 18 9.338 24 12.287.00 4.839 9.80 5.917 18 8.385 20 9.731 25 12.507.80 4.749 10 8.035 20 8.511 22 10.07 31 13.748.00 4.845 11 8.248 22 8.809 24 10.35 35 14.408.50 4.!l32 12 8.425 24 8.958 26 10.89 39 1U39.00 5.009 13 8.5SO 25 9.097 28 10.80 51 15.059.80 5.078 14 8.712 28 9.200 30 10.98 80 15.58

10.00 8.138 15 5.825 30 9.283 34 11.25 70 15.9111 5.241 18 8.922 34 9.404 38 11.48 80 17.1412 5.321 17 7.004 38 9.481 42 11.81 eo 17.2713 8.385 18 7.078 42 9.532 46 11.7. 100 17.38l4 5.435 20 7.189 46 9.565 50 11.79 110 lUI15 5.475 22 7.272 50 9.586 50 11.91 120 17.4518 5.806 24 7.332 60 9.612 70 11.96 130 11.4617 6.531 26 7.377 70 9.821 60 11.98 140 .7.4818 6.851 30 7.434 80 9.823 90 11.99 150 17.4920 5.579 34 7.454 90 9.624 100 12.00 160 IU925 5.811 38 1.481 100 9.625 120 12.0 180 IUO30 8.821 42 7.490 200 17.5035 5.824 \6 7.4" 220 17.8040 5.825 50 7.497

well or field 85 a constant rate problem, then the actual cumu·lative fluid produced as a·function of the cumulative pressuredrop is expre3~ed by the superposition relationship in Eq.IV·14 as

t d6PQ(T) = 2...fcR.' of ---;w- Quo,'. dt' (IV-H)

when 6P is the cumulative pressure drop at the well boreaffected by producing the well at constant rate which is established by

6P = Q(\t')2::ct) (IY·U)

The substitution of Eq. IY-l1 in IV·14 givesQ _ q(T) f/'CR.' ft d p u') Q

(T) - K 0 ~ u·,', dt'

Since the rate is constant. Q(T,=q(T' x T, and as t=KT/f.ucR.'this relation becomes

t dPc.')t = J~ Qc.-") dt' • • • • (VI-31)

To express Eq. VI·31 in transformation form. the tran3formfor t is I/p', Campbell and Foster, Eq. 408.1. The trllIl9form

for PO) at r = 1 is P(Ph and it follows from Theorem A thatdP(t) -

the transform of --- is pPCp) as the cumulative pressuredt

drop P(t) for constant rate is zero at time zero. Finally fromTheorem D. the transform for the integration of the form Eq.VI-31 is equ81 to the product of the transforms for each of thetwo terms in the integrand, or

TABLE IT - Continued

R 7_0 R _ 8.0 It _ 9.0 R _10.0

... = 0.1167 ... = 0.1476 ... = 0.1264 ... ::::: 0.1104

... = 0.7634 ... = 0.8438 ... = 0.6140 ... =0.4919

t Q(I) t Q(U t Q lI) t Q(,)--------------9.00 6.861 9 6.851 10 7.417 15 9.9859.80 7. i27 10 7.398 15 9.945 20 12.32

10 7.389 11 7.920 20 12.26 22 13.2211 7.902 12 8.431 22 13.13 24 14.0912 8.397 .3 8.930 24 .3.98 25 14.9513 8.876 14 9.418 26 14.79 28 16.7814 '.341 15 9.895 28 15.89 30 16.5915 g.791 16 10.361 30 15.35 32 17.3616 10.23 17 10.82 32 '7.10 34 18.1617 10.85 18 11.26 34 17.82 36 18.9118 11.05 19 11.70 36 18.82 38 19.8519 11.45 20 12.13 36 19.19 40 20.3720 11.85 22 12.95 40 19.85 42 21.0722 12.58 24 13.14 42 20.48 44 21.1824 13.27 26 14.80 H 21.09 .8 22.4226 13.92 28 15.23 46 21.89 48 23.0728 14.53 30 15.92 48 22.28 50 23.7130 15.11 34 17.22 80 22.82 82 24.33S5 16.39 38 18.41 82 23.36 84 24.t.40 17.49 40 18.97 84 23.89 58 25.53.5 16.43 45 20.28 50 24.39 58 28.1150 19.24 80 21.42 18 24.88 60 25.8780 20.51 55 22 ••6 50 25.36 65 26.0270 2'.45 60 23.40 85 26.48 70 29.2980 22 ••3 70 24.98 70 27.62 75 3O.4ggo 22.83 80 26.26 75 28.46 80 3Ul

100 23.00 90 27 .26 SO 2g.38 85 32.87120 23.47 100 28.11 85 30.•8 90 33.86140 23.71 120 29.31 go 30.g3 95 34.50180 23.85 140 30.08 . g6 31.53 100 36.46ISO 23.92 160 30.88 100 32.27 120 38.81200 23.96 ISO 30.U 120 34.39 IH 40.89500 24.00 200 31.12 140 35.92 160 42.15

240 31.34 160 37.04 180 44.212SO 31.43 180 37.85 200 45.38320 31.47 ZOO 38.U 240 46.95360 31.49 240 39.17 280 47.94400 31.50 280 39.16 320 48.54500 31.80 320 39.77 380 48.tl

300 39.88 400 49.14400 39.94 440 49.28440 39.97 4SO 49.36480 39.16



(VI-32).& du t

Q Ct, == t J ---,:-::----:-:--:-:-:-:- == --::~:-- ----=& 0 u(log u - 0.11593]' [0.11593 -log &J

(VI.34)

Substitution of this relation in Eq. VI-3S gives&'t (1- e-) dn CXl e- 1 dn

f ='Y+ f - dn- f-o n &'t n .ftn

and since the second term on the right is the Ei·function already discussed in the earlier part of this work, Eq. VI-37reduces to

(VI-37)

(VI.39)1

P (I) = - ['Y-Ei(-a't) + log &'t]& 2

The integration for P Itl close to the origin is expressed by

-u't40 &=0.02 (1 - e ) duP .Ct) =.. J . (VI-.35)

u ... 0 ul[J,'(u) + Y,'(u)]

For u equal to or less than 0.02, J,{U) =0, and Y,(u) ==2/... u so that Eq. VI.35 reduces to

-u't& (l-e )

P It) == r du (VI-36)" o' u

If we let n =u't

Further,

"'t(I- e-n ) dll 1 (1- e-lI ) dnf = J---

o nOli

-n1 . .1t (1- e )

P (t)=- J dn& 2 o' n

Since Euler's constant 'Y is equal to

1 (l-e-) dn CXl e-'Y= J -J- dn

• n

1pi = pPIPl Qu,

Evidence of this identity can be confirmed by substitutingEqs. VI-4 and VI·27 in Eq. VI·32. In brief, Eq. VI·32 is therelationship between constant terminal pressure and constantterminal rate cases. If the Laplace transformation for one isknown, the transform for the other is established. This inter·change can only take place in the transformations and thefinal solution must be by inversion.

Computation 01 Pit) and Q",

To plot PItl and Qlt' as work-curves, it is necessary to de·termine numerically the value for the integrals shown in Eqs.VI·24 and VI.28. In treating the infinite integrals for PItI andQUit the only difficult part is in establishing the integrals forsmall values of u. For larger values of u the integrands converge fairly rapidly, and Simpson's rule for numerical integration has proved sufficiently accurate.

To determine the integration for Qlt' in the region of theorigin, Eq. VI-28 can he expressed as

-u't4 .& (1- e ) duQ"m = "..oJ ul[J"{u) +Y.'(u)] (VI-33)

where the value for & is taken such that 1- e-u't ::: u't.

which is troe fgr u't equal to or less than 0.02, or" = YO.02/tand the simplification for Eq. VI·33 becomes

4t" duQ,Ilt) =".. J u[J:(u) + Y:(u)]

For u less than 0.02, I.(u) == I, and

2 u 2Y.{Ii) ::: - { log - + 'Y ~ == - { log u - 0.11593 ~

... 2 ...

AB the logarithmic term is most predominant in the denominator for small values of u, this equation simplifies to

TABLE III - Comtant Terminal Rate Case Radial Flow - Limited Reservoirs

H, 1.5 R _ 2.0 R _2.6 R _ 3.0 R _ 8.5 R _ CR - C.5

P, == 8.3225 P, == 3.1955 p, = 2.1584 p, == 1.8358 p, =1.3218 p, = 1.1120 fl, =0.9609fll = 11.92C fl, == 8.8118 p, = 4.2280 fJ. ::; 3.t787 (J, = 2.6526 p, = 2.1342 fl, = 1.8356

t Pltl

t P Ctlt PC,) t PIll t P Ctl t P

ltl t P ltl

5.0(10)" 0.251 2.2(10)-1 0.«3 C.0(10)-I O.aM 5.2(10)-1 0.ll27 1.0 0.802 1.5 0.927 2.0 1.0238.0 u 0.288 2.4 u 0.459 C.2 u 0.578 5.4 u 0.&8 1.1 0.830 1.8 0.948 2.1 1.0401.0(10)-1 0.322 2.8 II 0.476 C.C U 0.187 8.5 U 0.14$ 1.2 0.857 1.7 0.988 2.2 I.Ose1.2 u 0.385 2.8 U 0.492 C.8 u 0.598 8.0 U 0.e82 t.3 0.882 1.8 0.988 2.' t.0721.4 u 0.387 3.0 u 0.807 4.8 U 0.808 8.5 .. 0.883 I.C 0.906 1.0 1.007 2.4 1.0871.8 u 0.420 3.2 .. 0.522 6.0 .. 0.618 7.0 U 0.703 1.6 0.929 2.0 1.1>26 2.8 1.1021.8 u 0.452 3.C U 0.&38 6.2 .. 0.528 7.5 .. 0.721 1.8 0.9St 2.2 I.OS9 2.8 1.1162.0 .. 0.4IU 3.8 u 0.651 5.C .. 0.&8 8.0 II 0.740 1.7 0.973 2.4 1.092 2.7 1.1302.2 u 0.at6 3.8 .. O.aM 5.6 .. 0.647 8.5 U 0.788 1.8 0.9U 2.8 1.123 2.8 1.1CC2.' .. 0.M8 4.0 .. 0.579 5.8 U 0.667 9.0 U 0.776 1.9 1.014 2.8 1.IM 2.9 I.ISS2.5 .. O.UO ,(.2 .. 0.593 5.0 .. 0.566 9.5 .. 0.791 2.0 1.034 3.0 1.184 3.0 1.17l2.8 " 0.6IZ 4.4 u 0.807 6.5 ,. 0.1188 1.0 0.8>6 2.25 1.083 3.8 1.258 3.2 1.1973.0 .. 0.1l« 4.5 U 0.521 7.0 H 0.710 1.2 0.866 2.80 1.130 4.0 1.32C 3.C 1.2223.5 .. 0.724 4.8 u 0.&34 7.3 u O.73t t.f 0.920 2.78 1.176 U 1.392 3.8 1.246C.O .. 0.1ll4 5.0 u 0.648 8.0 .. 0.752 1.8 0.973 3.0 1.221 8.0 1.4OO 3.8 1.269C.5 .. o.ssc 5.0 II 0.715 8.5 .. 0.772 2.0 1.075 4.0 UOI U 1.527 4.0 t.2925.0 " 0.1lM 7.0 .. 0.782 9.0 u 0.792 3.0 1.328 6.0 U79 6.0 1.894 4.3 1.3495.3 u 1.0« 8.0 U O.84D D.5 .. 0.8IZ 4.0 1.578 6.0 1.787 U 1.6OO 5.0 1.4036.0 u 1.124 9.0 U 0.915 t.O 0.832 5.0 1.828 7.0 1.727 8.5 1.457

1.0 0.982 2.0 1.215 8.0 1.861 8.0 UIO2.0 1.649 3.0 1.893 9.0 1.994 7.0 1.8153.0 2.31B 4.0 U77 10.0 2.IZ7 8.0 1.7198.0 3.649 3.0 2.368 9.0 1.823

10.0 1.92'111.0 2.03112.0 2.135

I!J.O 2.239H.O 2.34315.0 2.CC7



o= AI. (v'pR) - BK. (v'pR) . . • . (VII-2)since it is shown in W.B.F., p. 79, that K,,' (z) = - K. (z), andI: (z) = I. (z). The solutions for A and B from these twosimultaneous algebraic expressions are

A=K.(v'pR)/p[K.(v'pR) !.(v'p)+K.(v'p) I,(v'pR)]and

B=I,( v'p R)/p[K.( v'p R) 1.( v'p)+K.( v'p) 1.( yIp Rl]

By substituting these constants in Eq. VI-I, the general solution for the transform of the pressure drop is expressed by

- [K.(v'p R) r.(vp r) +I.(v'p Rl K.(Vpr)]Pc •. p> =-=-...::..:.-..:...--::....-:..:..-=_:.-_.:....:..--::.-....:....-::.;..-.....:....:.::..::.

p[K.(vpR) I.(v'p) +I.(VpR) K.(vP}J(VII-3)

To find Q(t) the cumulative fluid produced for unit pressure drop, then the transform for the pressure gradient atr = 1 is obtained as follows:

there exists a restriction such that no fluid can flow past this

barrier 50 that at that point ( OP) R = O.Or r=

The general solution of Eq. VI.I still applies, but to fulfillthe boundary conditions it is necessary to re-determine valuesfor constants A and B. The transformation of the boundarycondition at r = 1 is expressed as

(VII-I)1 - -

- = AL( \' p ) + BK,,( v' p )p

and at r = R the condition is

No Fluid Flow Across E:tterior Boundary

The values for the integrands for Eqs. VI-24 and VI-28have been calculated from Bessel Tables for or greater than0.02 as given in W.RF., pp. 666-697. The calculations havebeen somewhat simplified by using the square of the modulusof

IH.c'> (u) 1=IJ. (u) +i Y.(u) I and IH.(1) (u) l=jJ.Cu) +i Y,(u) Iwhich are the Bessel functions of the third kind or the Hankelfunctions.

Table I shows the calculated values for Qlt) and PIt) tothree significant figures, starting at t = 0.01, the point wherelinear flow and radial flow start deviating. PIt) is calculatedonly to t = 1,000 since beyond this range the point sourcesolution of Eq. VI-15 applies. The values for Qlt) are givenup to t = 10".

The reader may reproduce these data as he sees fit: Fig. 4is an illustrative plot for QI'» and Fig. 7 is a semi-logarithmicrelationship for P (I).

LIMITED RESERVOIRS

As already mentioned, the solutions for limited reservoirsof radial symmetry have been developed by the Fourier-Besseltype of expansion.I

•I•n Their introduction here is not only to

show how the solutions may be arrived at by the Laplacetransformation, hut also to furnish data for PIll and QIllcurves when such ca::es are encountered in practice.

The first example considered is the constant terminal pressure case for radiallIow of limited extent. The boundary conditions are such that at the well bore or field's edge, r =1,the cumulative pressure drop is unity, and at some distanceremoved from this boundary at a point in the reservoir r = R.

_(O:).~ [I,(Vp_R) K.(v'~ -K.(V!R) II(V~)]

-0 pIP[K.( v' p R) 1.( v' p) +1.('1 p R) K.( Vp )]

where the negative sign is introduced in order to make Q(t)

TABLE III - Continued

R 5 R 6.0 R 7.0 R 8.0 R 9.0 R - 10fl. =0.8472 P. = 0.6864 fl. = 0.5782 P. = 0.4999 P, = 0.4406 P, = 0.3940P, = 1.6112 P. = 1.2963 P. = 1.0860 P. = 0.9352 P. = 0.8216 P. = 0.7333

t Pit> t Pit> t p(,> t PIt) t PI'> t Pc,>

3.0 1.167 4.0 1.275 6.0 1.436 8.0 1.558 10.0 1.lJ51 12.0 1.7323.1 1.180 4.5 1.322 6.5 U70 8.5 1.582 10.5 l.G73 12.6 1.7503.2 1.1D2 6.0 1.384 7.0 1.501 9.0 1.807 11.0 1.8G3 13.0 •.7883.3 1.204 5.5 U04 7.6 l.631 9.5 1.831 11.6 1.713 .3.6 1.7843.4 1.215 8.0 1.441 8.0 1.659 10.0 1.863 12.0 1.732 14.0 1.8013.5 1.227 0.6 1.477 8.6 l.68& 10.5 1.575 12.6 1.750 14.5 1.8173.8 1.238 7.0 1.511 9.0 1.813 11.0 1.897 13.0 1.788 15.0 1.8323.7 1.249 7.6 1.5« 9.5 1.838 1l.G 1.717 13.5 1.78& 15.5 1.8473.8 1.259 8.0 1.878 10.0 1.863 12.0 1.737 14.0 1.803 18.0 1.8823.9 1.270 8.6 1.507 11.0 1.711 12.5 1.757 14.5 1.819 17 .0 1.8904.0 1.28t 9.0 1.838 12.0 1.757 13.0 1.778 15.0 1.835 18.0 1.9174.2 1.301 9.6 1.088 13.0 1.801 13.5 1.795 15.5 1.861 19.0 1.9434.4 1.321 10.0 1.898 14.0 1.845 14.0 I.S13 18.0 1.8&7 20.0 1.9884.8 1.340 11.0 1.757 15.0 1.888 14.5 1.831 17.0 •.897 22.0 2.0174.8 1.350 12.0 1.815 15.0 1.G31 15.0 1.849 18.0 1.D28 24.0 2.0835.0 1.378 13.0 1.873 17.0 l.914 17.0 1.919 19.0 1.9055 28.0 2.1085.6 1.424 14.0 1.931 18.0 2.018 19.0 1.988 20.0 1.983 28.0 2.1518.0 1.489 15.0 1.988 19. 2.~8 21.0 2.051 22.0 2.037 30.0 2.1946.5 1.513 10.0 2.045 20.q 2.100 23.0 2.118 24.0 2.090 32.0 2.2387.0 1.558 17.0 2.103 22.Q 2.184 25.0 2.180 28.0 2.142 U.O 2.2787.5 1.698 18.0 2.150 24.00 2.287 30.0 2.340 28.0 2.t93 30.0 2.319S.O 1.841 19.0 2.217 28.0 2.351 35.0 2.499 30.0 2.244 38.0 2.3609.0 1.725 20.0 2.214 28.0 2.434 40.0 2.lJ58 34.0 2.345 40.0 2.401

10.0 1.808 25.0 2.sao 30.0 2.617 45.0 2.817 38.0 2.440 50.0 2.80411.0 1.892 30.0 2.846 40.0 2.496 00.0 2.80612.0 U75 45.0 2.621 70.0 3.00813.0 2.~9 50.0 2.14814.0 2.14215.0 2.225

December, 1949 PETROLEUM TRANSACT/ONS, A/ME 317


pos1tlve. Theorem B shows that the integration with respectto time introduces an additional operator p in the denomi.nator to give

QIP' = [I, ( vp R) K,( v'p] -K,(vp R) I,(vp )]

p"l'[K, ( vp R) 1.( vp ) +1.( vp R) K.( vp )](Vll4)

In order to apply Mellin's inversion formula, the first con·sideration is the roots of the denominator of this equation

which indicate the pole~. Since the modified Bessel functionsfor positive real arguments are either increasing or decreasing, the bracketed term in the denominator does not indicateany poles for positive real values for p. At the origin of theplane of Fig. 2 a pole exists and this pole we shall have toinvestigate first. Thus, the substitution of small and realvalues for z (Eqs. VJ·12 and VI·13) in Eq. VII4, gives

(R'-I)QIP) = 2p

~

TABLE IV - Comtant Terminal Rate CaJe Radial Flow

Preuure at Exterior Radiu$ Comtant

B = L5

~LOB = 2.11 R = 3.0 B = 3.5

A, = a.4~2~ >., = 1.7140 >., = 1J!426 A, = 0.9696 A, = 0.7862>.. = 9.5207 A. = 4.8021 >.., = 3.2285 A. = 2.4372 A. = 1.9624

t Pu > t Pit) t Pit) t Pc I) t pc.)

5.0(10)-1 0.230 2.0(10)-1 0.424 3.0(10)-1 0.502 5.0(10)-1 0.617 5.0(10)-1 0.1201.1 .. 0.2((l 2.2 .. 0."1 3.5 .. 0.1535 1.1 .. O.S(() 8.0 .. o.eea1.0 .. 0.249 2.4 .. 0.457 4.0 .. 0.1iM 1.0 .. 0.142 7.0 .. 0.7057.0 .. 0.2flG 2.5 .. 0.472 4.5 .. 0.691 7.0 .. 0.702 8.0 .. 0.7418.0 .. 0.282 2.8 .. 0.486 5.0 .. 0.618 8.0 .. 0.738 9.0 .. 0.774~.O .. ·0.2ll2 3.0 .. 0.498 1.1 .. 0.838 ~.O .. 0.770 1.0 0.S>41.0(10)-1 0.307 3.5 .. 0.527 6.0 .. 0.159 1.0 O.ns 1.2 0.8581.2 .. 0.828 4.0 .. 0.552 7.0 .. 0.691 1.2 0.850 U 0.11041.4 .. 0.144 4.5 .. 0.573 8.0 .. 0.728 U 0.892 1.8 0.1451.6 .. 0.140 5.0 .. 0.691 9.0 .. 0.755 1.6 0.1l27 1.8 0.9811.1 .. 0.167 5.5 .. O.IOS 1.0 0.771 1.8 0.155 2.0 1.0122.0 .. 0.175 6.0 .. 0.119 1.2 0.115 2.~ 0.980 2.2 t.0412.2 .. 0.881 6.5 .. 0.830 U 0.842 2.2 1.000 2.4 1.0512.4 .. O.JIlI 7.0 .. 0.839 1.5 0.801 2.4 1.011 2.5 1.0172.6 .. O.IBO 7.5 .. 0.647 1.8 0.8711 2.1 1.030 2.8 1.10112.1 .. 0.8n 1.0 .. 0.854 2.0 0.887 2.8 1.042 3.0 1.1211.0 .. O.UG 8.5 .. O.IGO 2.2 0.8115 1.0 1.051 3.6 1.158I .... 0.400 9.0 .. 0.1&1 lU O.BOO 3.5 1.059 4.0 l·m4.0 .. 0.402 9.5 .. 0.849 2.8 0.905 4.0 1.080 5.0 1.254.1 .. 0.404 1.0 0.1173 2.8 0.90S 4.5 1.017 11.0 1.2#1.0 .. 0.405 1.2 0.582 3.0 0.910 5.0 1.0Il1 7.0 1.242e.o .. 0.405 U 0.188 3.1 0.~13 1.1 l.em 8.0 1.2477.0 .. 0.405 1.1 O.IBO 4.0 0.~1I 8.0 1.096 9.0 1.2aO1.0 .. 0.405 1.8 0.1182 4.6 0.9111 11.1 1.097 10.0 1.251

2.0 0.582 1.0 0.9111 7.0 1.01l7 12.0 1.2522.1 O.eva 1.1 0.9111 5.0 1.0Il8 14.0 1.2633.0 0.893 8.0 0.9111 10.0 1.01l9 18.0 1.253

TABLE IV - Continued

R = 4.0 R = 6.0 R = 8.0 R = 10 £=15A, = 0.5870 >., = 0.4205 A, = 0.8090 A, = 0.2448 >., = 0.1518>.., = 1.6450 >.., = 1.0059 >.., = 0.7285 >.., = 0.6726 >.., = 0.8745

t pc.) t PCI) t PCI) t PCI) t pc.)

1.0 0.802 4.0 1.276 7.0 1.499 10.0 1.161 20.0 1.veo1.2 0.867 4.6 1.320 7.6 1.527 12.0 1.730 22.0 2.0031.4 0.901 5.0 1.381 8.0 1.654 14.0 1.798 24.0 2.0431.8 0.947 6.6 1.398 8.6 1.680 18.0 1.856 28.0 2.0801.8 0.985 8.0 1.432 9.0 1.604 18.0 1.907 28.0 2.114'!.O 1.020 8.5 1.4e2 ~.8 1.827 20.0 1.952 30.0 2.1482.2 1.052 7.0 UBO 10.0 1.1148 25.0 2.043 35.0 2.2182.4 1.080 7.6 1.518 12.0 l.72t 30.0 2.111 ((l.0 2.2792.6 1.108 8.0 1.539 14.0 1.785 35.0 2.160 45.0 2.3322.8 1.130 8.6 1.551 16.0 1.837 fO.O 2.197 60.0 2.3793.0 1.152 9.0 1.580 18.0 1.879 45.0 2.224 60.0 2.4553.4 1.190 10.0 1.816 20.0 1.914 50.0 2.245 70.0 2.5133.8 1.222 12.0 1.587 22.0 U43 55.0 2.260 80.0 2.1684.5 1.258 14.0 1.704 24.0 1.987 60.0 2.271 90.0 2.5825.0 1.290 111.0 1.730 28.0 1.985 55.0 2.279 10.0(10)1 2.8195.5 1.309 18.0 1.749 28.0 2.002 70.0 2.286 12.0 .. 2.1668.0 1.325 20.0 1.7112 30.0 2.016 76.0 2.290 H.O .. 2.8777.0 1.347 22.0 1.771 35.0 2.0fO SO.O 2.293 18.0 .. 2.88118.0 1.381 24.0 1.777 ((l.0 2.055 BO.O 2.297 18.0 4f 2.8979.0 1.370 25.0 1.781 45.0 2.064 10.0(10)1 2.300 20.0 .. 2.701

10.0 1.376 28.0 1.784 50.0 2.070 11.0 .. 2.301 22.0 .4 2.70t12.0 1.382 30.0 1.787 60.0 2.078 12.0 ,( 2.302 24.0 .. 2.705H.O 1.386 36.0 1.789 70.0 2.078 13.0 .. 2.302 28.0 .. 2.70718.0 1.386 ((l.0 1.791 80.0 2.079 It.O .. 2.302 28.0 .. 2.707

LIS.O 1.386 50.0 1.792 18.0 .. 2.303 30.0 .. 2.708



and by the application of Mellin's inversion formula appliedat the origin, then

An investigation of the integration along the negative real"cut" both for the upper and lower portions, Fig. 2, revea1lthat Eq. VII4 is an eYen function for which the integrationalong the paths is zero. However, poles are indicated alongthe negative real axis and these residuals together with Eq.VII·S make up the solution for the constant terminal pressurecase for the limited radial system. The residuals are estab·

[J,(a.R) Y.(a.) - Y,(a.R) I.(a.)] = 0 (VII·7)and the poles are represented on the negative real axisby A. = - a.', Fig. 3. The residuals of Eq. VII-6 are the seriesexpansion

(VII-6)

lished by the Mellin's inversion formula by letting A =u'eir ;then by Eqs. VI-IS

At-e Q(A)d A=2~ f

A" A" etc.-u't_~fe [1.(uR) Y,(u) -Y.(uR) ll(u)] du

ri u'[J,(uR) Y.(u) -YI(uRl J.(u)]as, a:. etc.

where a" as, and a. are the roots of

(VII·S)_1_ f/t..!- (R'-I) R'-I2ri 2 --A-- dA = -2-

A=.

TABLE IV -Continueda _ 20 B_ 25 R =80 a _ 40 R =50As = 0.1208 As = 0.09648 A, = 0.08032 A, = 0.05019 A, = 0.04813As =0.2788 A, = G.2:l23 >., =0.1849 A, =0.1384 A, =0.1106

t l'(t) i I'll) t PC,) i I'll) i I'll)

30.0 2 ••t8 110.0 2.18ll 70.0 2.MI 12.0(10)1 2.813 20.0(10)1 3.08485.0 2.219 M.O 2.434 SO.O 2.815 H.O U 2.888 22.0 " 3.11140.0 2.282 SO.O 2.478 ;0.0 2.872 15.0 U 2.~ 24.0 u 3.18441.0 2.838 85.0 2.114 10.0(10)1 2.723 18.0 u 3.011 28.0 u 3.19310.0 2.888 70.0 2.&60 12.0 U 2.812 20.0 .. 3.003 28.0 u 3.22910.0 2.f75 75.0 2.583 14.0 U 2.88& 22.0 U 3.109 30.0 " 3.28370.0 2.841 SO.O 2.814 15.0 u 2.0lI0 2t.0 .. 3.152 35.0 u 3.33980.0 2.tlO9 85.0 2.843 •S.I U 2.985 25.0 .. 3.191 40.0 " 3.40690.0 2.858 90.0 2.871 17.0 U 2.979 28.0 .. 3.228 45.0 .. 3.48110.0(10)' 2.107 95.0 2.897 17.5 U 2.m 30.0 .. 3.259 10.0 .. 3.11210.1 U 2.728 10.0(10)1 2.721 18.0 U a.DOe U.O .. 3.831 85.0 .u a.55811.0 u 2.7f7 12.0 U 2.Sl7 20.0 •• a.084 40.0 u a.891 eo.O u 3.195n.1 U 2.7&4 14.0 U 2.878 25.0 U a.115O ·45.0 .. a.44O 85.0 U a.83012.0 U 2.181 15.0 u 2.935 30.0 .. a.219 60.0 CI 3.482 '70.0 •• 3.GS112.1 U 2.m 18.0 U 2.984 35.0 U 3.289 M.O U 3.515 75.0 u a.88813.0 U 2.810 20.0 U 3.024 40.0 .. UOO eo.O U 3.845 SO.O U 3.713la.& .. 2.823 22.0 U 3.057 45.0 U U32 85.0 U 3.888 85.0 u 3.73514.0 .. 2.835 24.0 U 3.085 150.0 U 3.351 70.0 u 3.888 go.O u 3.7M14.1 .. 2.848 25.0 U 3.107 GO.O u 3.375 SO.O .. a.819 g6.0 u a.77115.0 U 2.857 28.0 U 3.125 70.0 u a.887 go.o u a.S40 10.0(10)' 3.78718.0 .. 2.875 30.0 u 3.142 SO.O u a.a~ 10.G(l0)' U611 12.0 .. 3.83318.0 U 2.go5 35.0·" a.171 90.0 u 3.397 12.0 U 3.1172 14.0 u 3.88220.0 II 2.1129 40.0 u 3.189 10.0(10)' a.399 14.0 " 3.581 15.0 " 3.88124.0 " 2.g58 46.0 u 3.200 12.0 u 3.401 15.0 .. 3.585 18.0 u 3.89228.0 U 2.g7S 60.0 u· 3.207 14.0 " 3.401 18.0 .. a.m 20.0 u 3.;0030.0 .. 2.9ao eo.O .. 3.2U 20.0 u 3.188 22.0 " 3.go440.0 u 2.m 70.0 .. 3.217 25.0 It 3.889 24.0 " 3.;07150.0 •• 2.996 ao.O u 3.218 25.0 " 3.m

90.0 If 3.2U 28.0 " 3.UO


n =60 R =10 R = 80 B =90 R =100

tl Pit) t I'll) t pc.) t P ltl t 1'Ct)

3.0(10)' 3.257 S.0(10)' 3.612 8.0(10)' 3.eoa 8.0(10)' 3.747 1.0(10)' 3.SS9t.O U 3.401 5.0 U 3.eoa 7.0 U a.sso 9.0 " 3.B05 1.2 u 3.8495.0 U 3.612 7.0 U 3.SSO 8.0 .. 3.747 1.0(10)' 3.SS8 1.t: It 4.0255.0 .. a.802 8.0 " 3.748 9.0 Ie 3.ll3S 1.2 u 3.U4U 1.5 U 4.0927.0 ,. 3.575 9.0 U 3.803 10.0 U 3.SS7 1.3 u 3.U88 1.8 .. 4.11508.0 U a.nu 10.0 U a.884 12.0 " a.945 1.4 U 4.025 2.0 .. 4.200U.O u a.7UZ 12.0 If 3.US7 14.0 " 4.01U 1.6 .. 4.058 2.5 U 4.303

10.0 " a.832 14.0 II 4.003 HLO ., 4.061 1.8 .. Uff 3.0 " 4.:17912.0 It 3.908 18.0 " 4.084 15.0 u 4.0BO 2.0 .. UUZ 3.a n 4.43414.0 " 3.959 18.0 U 4.095 18.0 " 4.lao 2.5 " 4.285 4.0 U 4.47815.0 " 3.905 20.0 II 4.127 20.0 " 4.171 3.0 u 4,34U 4..5 u 4.51018.0 u 4.023 25.0 u 4.181 25.0 f. 4.248 3.6 .. 4.3Uf 5.0 If 4.53420.0 U 4.043 30.0 u 4.211 30.0 " 4.297 4.0 If 4.425 5.5 If 4.85225.0 .. 4.071 35.0 •• 4.228 305.0 If 4.328 4.5 n 4.448 8.0 " 4.56530.0 .. 4.084 40.0 .. 4.2:17 4O:G ,. <1.847 &.0 " U&4 0.3 If 4.57935.0 .. 4.090 405.0 It 4.242 45.0 .. 4.3eo 5.0 " t.482 7.0 U 4.88340.0 I' 4.m ~.O It 4.245 ro.D U 4.358 7.0 II 4.491 7.3 If 4.58845.0 u 4.093 611.0 U 4.247 eo.O .. 4.375 8.0 .. 4.496 8.0 .. 4.59380.0 Ie 4.094 60.0 u 4.247 70.0 .. 4.380 g.o u 4.498 9.0 .. 4.6U855.0 II 4.094 65.0 It 4.248 80.0 .. 4.381 10.0 u 4.4UU 10.0 .. 4.1101

70.0 " 4.248 ;0.0 u 4.382 11.0 " U90 12.1 If 4.110475.0 .. 4.248 10.0(10)' 4.382 12.0 u 4.1500 15.0 .. 4.110680.0 u 4.248 11.0 .. 4.382 14.0 (l 4.1500



andJ:(z) = -J.(z)

which are recurrence formulae for both first and eecond kindof Bessel functions, W.B.F., p. 45 and p. 66, then by the identities of Eqs. VII·7 and VI-23, the relation VII-B reduces to

CD e-Ga't J.' (a.R)- 2 %

a"a, a.' [J:(a.) -J,'(aoR)]etc.

This is essentially the solution developed in an earlier work,'but Eq. VII·IO is more rapidly convergent than the solutionpreviously reported.

The values of Q(t) for the constant terminal pressure casefor a limited reservoir have been calculated from Eq. VII-lOfor R = 1.5 to 10 and are tabulated in Table 2. A reproductionof a portion of these data is given in Fig. 5. As Eq. VII-IO israpidly convergent for t greater than a given value, only two

(VII.IO)

-all'te J.'(a.R)

a.' [J.' (a.) - J,') a.R)]

CD%a"a,

etc.

Therefore, the solution for Q(t) is expressed by

R'-IQ(C) =----2

2

(VII·9)

• (VII-B)

J.'(z) =J.(z) -J.(z)/z

-Ga'te [J.(a.R) Y.(a.) - Y.(a.R) JI(a.)]

da.'lim-[J.(uR) Y.(u) - Y.(uR) J.(u)]

duu~

since

CD-2 %

a"a.etc.


B =200 R =SOO R =400 R =500 R =eoo,PCII t PUI t P cu t P

Ctl t PUI~

U(10)' 4.11e1 e.o(IO)' 4.754 1.5(10)4 1.212 2.0(10)' 1.3M 4.0(0)4 6.7032.0 .. UOS 8.0 " 4.lIa8 2.0 " 1.166 2.1 Ie 5.468 4.8 " 8.71122.1'" 4.117 10.0 " 1.010 a.o .. 1.616 a.o .. 8.569 &.0 .. 1.814a.o .. H08 12.0 " 5.101 4.0 " 1.68ll a.1 " &.838 8.0 " 8.004••8'" H8& 14.0 " 5.177 5.0 " 1.781 4.0 " &.102 7.0 .. I.m••O~ .. 4.682 18.0 " 1.242 !.O " 1.1146 4.1 " &.75Q 8.0 .. 8.041'.0"'''' 4.833 18.0 .. 5.2eO 7.0 .. 1.880 5.0 .. 5.810 0.0 .. 8.548.0 .. 4.754 20.0 ., 5.~8 8.0 " I.v.zo 8.0 " 5.8a4 10.0 " Ua97.0 .. 4.1129 24.0 " 5.429 9.0 " 1.942 7.0 " &.DeD 12.0 " 802101.0 " 4.8a4 28.0 .. 5.401 10.0 " 1.957 8.0 " 8.013 14.0 " 8.2S'Z'.0 " 4.H9 30.0 .. 5.117 11.0 .. I.M7 9.0 " 8.051 18.0 .. 8.299

10.0 " 4.998 40.0 " 5.aoe 12.0 " 6.976 10.0 .. 8.088 18.0 .. 8.12a12.0 " 6.072 &0.0 " 1.M2 12.1 " 1.977 12.0 " 8.151 20.0 " U41114.0 " 3.m SO.O" 5.878 13.0 .. 5.98> 1•.0 " 5.154 21.0 .. U7.18.0 " 1.171 70.0 " I. lIDO 14.0 " 1.983 18.0 " 8.183 30.0 " 8.18718.0 " &.203 80.0 4f 5.806 18.0 " 1.988 18.0 " 8.103 3&.0 " 8.19220.0 " &.227 90.0 " 8.700 18.0 u I.m 20.0 .. 8.202 40.0 c. U9S21.0 .. 6.284 10.0(10)' 1.702 20.0 II 1.991 25.0 .. 8.211 SO.O .. 8.a9710.0 " 6.282 12.0 u 5.703 24.0 " 1.991 30.0 .. 8.213 80.0 " 8.1.738.0 " 1.2DO 14.0 .. 5.704 28.0 " 1.991 a5.0 .. 8.21440.0 " 1.294 11.0 .. 5.704 40.0 .. 1.214


R = 700 R = 800 R = 900 R = 1000 R = 1200

t P Ctl t PCt) t P Ctl,

PC'I t Pcu

6.0(10)' 5.814 7.0(10)' 5.983 8.0(10)' 1.040 1.0(10)1 6.UI 2.0(10)1 8.11078.0 .. 1.90S 8.0 " 8.049 SI.O .. 8.108 1.2 .. 8.252 3.0 .. 6.7047.0 .. U82 0.0 " 6.108 10.0 .. 6.181 1.4 " 8.329 4.0 " 6.8338.0 " 1.048 10.0 " 6.180 12.0 .. 8.251 1.8 u 8.395 5.0 " UI89.0 .. 8.105 12.0 II 8.249 14.0 .. 6.327 1.8 " 6.432 8.0 u 8.976

10.0 " 8.118 14.0 .. 8.322 18.0 .. 8.392 2.0 u 8.503 7.0 u 7.01312.0 .. 8.239 16.0 .. 8.382 18.0 " 6.447 2.5 II 8.805 8.0 " 7.03814.0 .. 6.305 18.0 .. 8.432 20.0 .. 6.49. 3.0 u 8.881 9.0 u 7.058IS.O" 6.357 20.0 .. 6 .•74 25.0 " 8.187 3.6 II 1.738 10.0 .. 7.08718.0 .. 1.308 25.0 .. 8.8&1 30.0 .. 6.852 4.0 u 8.781 12.0 I' 7.08>20.0 Ie 1.430 30.0 " 8.6DD 40.0 .. 6.729 4.1 .. 8.813 14.0 .. 7.08&25.0 " 8.484 35.0 " 8.830 45.0 .. 8.751 5.0 I' &.837 16.0 .. 7.08110.0 .. 1.614 fO.O u 6.850 &0.0 II 8.788 6.6 u 8.8&4 18.0 " 7.08036.0 " 8.830 45.0 .. 1.833 15.0 II 8.777 6.0 'I 8.868 19.0 " 7.08940.0 " 8.140 60.0 II 8.171 80.0 " 8.785 7.0 It 1.885 20.0 II 7.09046.0 .. 1.545 M.O Ie 6.878 70.0 .. 8.794 8.0 u 8.SD5 21.0 .. 7.09080.0 .. 8.1148 eo.O u 8.879 SO.O u 6.798 9.0 .. 8.901 22.0 .. 7.09080.0 " 8.150 70.0 .. 8.882 GO.D U 6.800 10.0 .. 8.90~ 23.0 .. 7.09070.0 • 8.511 so.o .. 1.884 10.0(10)1 6.8>1 12.0 .. 6.907 2~.0 .. 7.00J80.0 .. 8.511 100.0 .. 1.884 H.O .. 6.007

18.0 .. 8.908



R'logR

(R'-I)'

(VII·13'

r3R'-4R'log R-2R'-Il4(R'-I)'

- 1 R'P I ...l =pi (R'-I)p-70

R (R'-r')log-- +

r 2(R'-I)

(R' + 1) 1 2- 4(R'-I)'~ +7 (R'-l) (VII-I2)

This equation now indicates both a single and double pole atthe origin, and it can be shown from tables or by applyingCauchy's theorem to the Mellin's formula that the inversion ofEq. VII·I2 is

P I ... l = (R~I) [~ + t JR'

----logr(R'-I)

which holds when the time, t, is large.As in the preceding case, there are poles along the negative

real axis, Fig. 3, and the residuals are determined as before

by letting A =u' lr, and Eqs. VI·IS give

terms of the expansion are necessary to give the accuracyneeded in the calculations.

Likewise from the foregoing work it can be easily shownthat the transform of the pressure drop at any point in theformation in a limited reservoir for the constant terminal ratecase, is expressed by

[K,(vpR) I.(vpr) +I,(VpR) K..(Vpr)]P,•.P) = .:0- ""'-

p'I'[I,(VpR) K,(V Pl-K,(VpR) I, vp)](Vil.Il)

An examination of the denominator of Eq. VII-Il indicatesthat there are no roots for positive values of p. However, adouble pole exists at p =O. This can be determined by ex·panding K..(z) and K,(z) to second degree expansions forsmall values of z and third degree expansions for I.(z) and,I, (z), and substituting in Eq. VII·ll. It is found for smallvalues of p, Eq. VII-ll reduces to

TABLE IV -Continued

R :=: 1400 R :=: 1600 R :=: lS00 R :=: 2000 R:=: 2200

t Pit) t PIt) t PI') t PIt} t P Ul

2.0(10)1 8.lID7 2.8(10)' 8.619 3.0(10l' 8.710 4.0(10)1 8.854 6.0(10)1 8.tee2.6 " 8.819 8.0 " 8.710 4.0 " 8.854 8.0" UM 8.8 " 7.0131.0 " 8.7ug 3.8 " 8.787 8.0 u 8.tee 8.0 " 7.088 8.0 u 7.0671.8 u 8.788 4.0 u 8.883 8.0 u 7.084 7.0 u 7.132 8.8 u 7.01174.0 u 8.849 8.0 u 8.tG2 7.0 u 7.120 8.0 " 7.1ge 7.0 u 7.1118.0 u 8.880 e.o U 7.00 S.O u 7.188 9.0 u 7.26\ 7.8 u 7.11l7'.0 " 7.028 7.0 u 7.114 9.0 U 7.238 10.0 u 7.:gS S.O u 7. log7.0 U 7.082 8.0 u 7.187 10.0 u 7.2SO 12.0 u 7.374 8.8 u 7.%n1.0 " 7.\23 9.0 " 7.210 18.0 " 7.407 14.0 u 7.431 9.0 u 7.2M'.0 u 7.IM 10.0 .. 7.244 20.0 u 7.489 10.0 .. 7.474 10.0 " 7.307

10.0 .. 7.177 18.0 .. 7.334 30.0 u 7.4llll 18.0 .. 7.800 12.0 _t 7.m15.0 u 7.~ 20.0 n 7.304 40.0 .. 7.4~5 20.0 u 7.1530 18.0 ~ 7.1la720.0 fC 7.241 25.0 .. 7.373 &0.0 fC 7.498 26.0 .. 7.860 20.0 II 7.67921.0 .. 7.243 30.0 .. 7.375 81.0 " 7.4~6 30.0 u 7.884 25.0 u 7.83110.0 I' 7.244 36.0 .. 7.377 62.0 u 7.4~6 38.0 .. 7.093 30.0 .. 7.116111.0 .. 7.244 40.0 If 7.378 13.0 u 7.408 40.0 tt 7.597 35.0 II 7.87732.0 It 7.244 42.0 " 7.378 504.0 u 7.496 60.0 " 7.800 fO.O tt 7.88033.0 .. 7.244 H.O .. 7.378 88.0 .. 7.496 00.0 " 7.001 60.0 .. 7.893

6-4.0 If 7.001 60.0 tl 7.89570.0 .. 7.8geSO.O .. 7.596


R :=: 2400 R :=: 2600 R = 2S00 R = 3000

t Pit) t Pit) t Pit) t PCtl

8.0(10)1 7.057 7.0(10)1 7.134 8.0(10l' 7.21.'1 1.0(10)' 7.3127.0 " 7.134 8.0 " 7.201 0.0 If 7.2l1D 1.2 " 7.4038.0 " 7.200 0.0 u 7.269 10.0 .. 7.312 1.4 I~ 7.4SO9.0 " 7.259 10.0 .. 7.312 12.0 u 7.403 J.G I' 7.546

I10.0 II 7.310 12.0 u 7.401 10.0 .. 7.542 1.8 u 7.00212.0 .. 7.a~S 14.0 .. 7.478 20.0 .. 7.844 2.0 .. 7.~1 I18.0 .. 7.625 10.0 .. 7.1535 24.0 .. 7.71~ 2.4 If 7.732 I

20.0 II 7.811 18.0 u 7.588 28.0 If 7.775 2.8 II 7.794 I

24.0 .. 7.168 20.0 .. 7.1531 30.0 u 7.797 3.0 " 7.820 !28.0 " 7.706 2••0 " 7.099 35.0 .. 7.840 3.6 .. 7.871 I30.0 " 7.720 28.0 .. 7.740 40.0 If 7.870 4.0 u U0835.0 II 7.746 30.0 .. 7.765 lID.O " 7.905 4.5 II 7.938

I40.0 II 7.7l1D 36.0 .. 7.799 lID.O .. 7.1122 5.0 u UM50.0 II 7.776 40.0 .. 7.821 70.0 .. 7.1130 8.0 II 7.979GO.G II 7.7SO 60.0 II 7.1lt6 SO.O .. 7.934 7.0 II 7.119270.0 II 7.782 80.0 II 7.866 90.0 u 7.938 8.0 .. 7.999SO.O ~ 7.783 70.0 " 7.800 10.OCI0)' 7.937 g.O It 8.002

I90.0 II 7.783 80.0 .. 7.11e2 12.0 .. 7.937 10.0 .. 8.00493.0 u 7.793 90.0 u 7.883 13.0 .. 7.937 12.0 If 8.006

10.0{lO)' 7.833 15.0 " 8.005 i



(VII.20)00 e->..'t J:(>..R)

P (OJ =log R - 2 1:n=l >..'[J,'(>'.) - J.'(>..R)]

where >.. is the root established from

J.(>.,.) Y.p.•R) - Y.(>.,.) J.(h.R) = 0 (VII.2I)Point source:

-I/oo't2 00 e J.(I/oo)PC') =logR-- ::& (VII 22)

R' n=l 1/00' J,' (J1.R) -

where the root p.,. is determined from J.(Il.R) =O. W.B.F.,p. 748. Table 4 is the summary of the calculated pet) em·ploying Eq. V1I.20 for R =1.5 to 50, the cylinder sourcesolution, which applies for small as well as large times. Thedata given for R = 60 to 3,000 are calculated from the pointsource solution Eq. V1I·22. Plots of these data are given inFig. 7.

When developing the solution hy meal15 of the Laplacetransformation, it is assumed that the exterior boundary r = R,

P(R,p) =0, which fixes the pressure at the exterior boundaryas constant. Since the ahove-quoted references contain com·plete details, the final solutions are only quoted here forcompleteness' sake.

Cylindrical source:

SPECIAL PROBLEMSThe work that has gone before shows the facility of the

Laplace transformation in deriving analytical solutions. Notyet shown is the versatility of the Laplace transformation inarriving at solutions which are not easily foreseen by the ortho·dox methods. One such solution derived here has shown to heof value in the analysis of flow tests.

When making flow tests on a well, it is often noticed thatthe production rates, as measured by the fluid accumulatingin the stock tanks, are practically constant. Since it isdesired to obtain the relation hetween flowing hottom holepressure and the rate of production from the formation, it isnecessary to correct the rate of production as measured in theflow tanks for the amount of oil obtained from the annulusbetween casing and tubing. To arrive at the solution for thisproblem, we use the basic equation for the constant terminalrate case given by Eq. IV·H, where q(T) is the constant rate offluid produced at the stock tank corrected to reservoir condi·ditions, but Pm is a pseudo pressure drop which is adjustedmathematically for the unloading of the fluid from the annulusto give the pressure drop occurring in the formation.

It is assumed that the unloading of the annulus is directlyreflected by the change in bottom hole pressure as exerted bya hydrostatic head of oil column in the casing. Therefore, therate of unloading of the annulus qA,(T), expressed in cc. persecond corrected to reservoir conditions, is equal to

dAPqA(T) = C----;rr- (VIII.l)

where C is the volume of fluid unloaded from the annulusper atmosphere bottom hole pressure drop per unit sand thick·ness. The rate of fluid produced from the formation is thengiven by q'T) - qA(T)' As the hottom hole pressure is continuoously changing, the prohlem becomes one of a variable rate.The substitution of the form of Eq. IV·n in the superpositiontheorem, Eq. IV·16, gives

(Vll·19

e -P2 t [J, (fJ.R) Y. (fJ.r) - Y, (fJ.R) J. (fJ.)]

. dPo' lim.- [J,(uR) Y,(u) -J,(u) Y,(uR)]du

u~fJ. (Vll·I6)

f

002::&

P,,{J.. etc.

1

2ri>." >.., etc.

-u't=.!.-fe [J,(uR) Y.(ur) - Y,(uR) 1.(ur)] du (VII.I4),..i u'[l,(uR) Y,(u) -l,(u) Y,(uR)){J" {J" etc.

where fJ" fJ" etc., are roots of[J,(fJ.R) Y,(fJ.) -J,CP.) Y,(fJ.R)] = 0 . (VII·I5)

with >.. =-fJ.'. The residuals at the poles in Eq. VII·I4 givethe series

By the recurrence formulae Eqs. VII·9, the identity VII·IS,and Eq. VI·23. this series simplifies to

-P'te • J,(fJ.R) [1,(,11.) Y.(P.r) - Y,(fJ.) J.C.6.r)]

P.[J,'CfJ.R) - l,'(P.)]

00.. ::&

fJ.. P.. etc.(VII·17)

Therefore, the sum of all residuals, Eqs. VII·13 and Vll·I7 isthe solution for the cumulative pressure drop at any point inthe formation for the constant terminal rate case in a limitedreservoir, or

__2_(~+t)_~10 r- (3R'-4R'logR-2R'-I)p(•.,)- (R'-I) 4 (R'-I) g 4CR'-I)'

00 cPo't J.(.6.R) [J,(P.) Y.(P.r) - Y,(P.)J.(p.r)]+r::&

fJ., (I, fJ.[J,'(fJ.R) -J,'(P.)](VII·IS)

which is essentially the solution given by Muskat: now de·veloped by the Laplace Transformation. Finally. for the cumu·lative pressure drop for a unit rate of production at the wellbore, r =1, this relation simplifies to

= 2 ('!'+t) _ (3R'-4R'logR-2R'-1)p(.) (H' - 1) 4 4(R' - I)'

00 e-P.'t J,'(P.R)

+2::& 'rJ' J' ]p•• fJ, ,II.. I (p.R) - ,(13, )The calculations for the constant tenninal rate case for a

reservoir of limited radial extent have been determined fromEq. VII·19. The summary data for R =1.5 to 10 are given inTable 3. An illustrative graph is shown in Fig. 6. The effectof the limited reservoir is quite pronounced as it is shownthat producing the reservoir at a unit rate increases the pres·sure drop at the well bore much faster than if the reservoirwere infinite, as the constant withdrawal of fluid is reflectedvery soon in the productive life by the constant rate of dropin pressure with time.

Pressure Fixed at Exterior Boundary

As a variation on the condition that ( dP = 0 ) wedr r=R

may assume that the pressure at r =R is constant. In effect,this assumption helps to explain approximately the pressurehistory of Bowing a well at a constant rate when, upon open·ing, the bottom hole pressure drops very rapidly and thenlevels out to be:ome constant with time. The case has heendeveloped by Hurbt' using a cylinder source and by Muskat'using a point 50urce solution.

322 PETROlEUM TRANSACTIONS, AIME December, 1949


1£ tAP =-- f [qlT', - qAIT"] P'It.•·, dt'

2rK 0

and from Eq. VIII·I

transform of P'I., as pli;., and the transform of AP as AP,

80 that the transform for dAP/ dt is pAP, then it followsthat

AP =...!:... j[qIT" - C dAP ] P'e •.••' dt' • (VIII·2)2rK 0 dT'

Since T = fi<CR: t/l(, and the unit rate of production at theqlTl1'

surface corrected to reservoir conditions is qlt, = - K' Eq.- 2r

VIII·2 becomes

(VIII·9)

(VIII.IO)

AP=

and it will be recognized from Campbell and Foster, Eq. 920.1,

that the integrand is the transform for K" (Vp). Further,the integration with respect to time follows from Theorem B,Chapter V, so that the transform of Eq. VIII·7 is the relation

- K.,(Vp)PIP) = . . . . . . (vm-8)

p

The same result can be gleaned from Eq. VIll-6 since for t

large, p is small and K, (vp) = 1/vp. Substitution ofthis approximation in Eq. VIIl-6 yields Eq. VIll.a. Therefore,

introducing the expression for PIP) in Eq. VIII·5 gives

q K.,( v'p)

p [1 + C p K.,(Vp)]to obtain values for Pith the cumulative pressure drop for unitrate of production in the stock tank which automatically takescognizance of the unloading of the annulus.

The inverse of the form of VIII-iO by the Mellin's inversionformula can be determined by the path described in Fig. 2.The analytical determination is identical with the constantterminal rate case given in Section VI. Therefore, the cumulative pressure drop in the well bore, for a unit rate of production at the surface, corrected for the unloading of the fluidin the casing, is the relation

-u't00 (I-e .) J.(u) du

Pit) =of ------r--------r----

u[ (1 + u'C 2 Y.(u)· + (u·C2"J.(u»']

(VIll-1l)Fig. 8 presents a plot of the computed values for PIll cor-

p [1 +C p K.,(Vp)]for which it is necessary only to find the inverse of

(VIll.s)

• (VIII·5)

. • (VIIl4)- q - ~ -

AP= [--C PAP]P P tp,

p

AP=----

[1 +C p·Pe.)]

t[ - dAP ]AP= f qlt')-C-- p'e •••·) dt'o dt'

where C= C/2rf~R.'.

Eq. VIll·3 presents a unique situation and we are confronted with ~etermination of AP, the actual pressure drop,appearing both in the integrand and to the left side of theequation. The Laplace transformation offers a means of soly·ing for AP which, by orthodox methods, would be difficultto accomplish.

It will be recognized that Theorem D, from Chapter V, isapplicable. Therefore, if Eq. VIII-3 can be changed to a Laplace transformation, AP can be solved explicitly. If weexpress the transform of the constant rate qltl as q/p, the

and on solution gives

. • • • (VI.I6)

Since q = qml'/211"K, then the term ------ in Eq.

[1 + C p·PI.J.VIII-5 can be interpreted as the transform 'of the pseudo pres·sure drop for the unit rate of production at the stock tank.

No mention has been made as to what value can be substi-

tuted for Pl.). If we wish to apply the cylinder source, Eq.VI4 applies, namely,

K,,(Vp)Pl., = . (VIII-6)

p'" K,(\lp)

However, from the previous discussion it has been shownthat for wells, t is usually large since the well radius is small,and the point source solution of Lord Kelvin's applies, namely,

I 00 e......

PIt! = 2 1/ 4 / -;;- du

the Ei.function. Therefore, to apply this expression in Eq.VIII.5, it is necessary to obtain the Laplace transform of thepoint source solution of Eq. VI-I6. By an interchange ofvariables, this equation becomes

I t e-1/U

Pet! =- f - dt ••••• (VIII-7)2 0 t

responding to C from 1,000 to 75,000. It can be observed thatthe greater the unloading from the cBsing, the smaller theactual pressure drop is in a formation due to the reduced rateof fluid produced from the sand. For large times, however, allcurves become identified with the point source solution whichis the envelope of these curves. After a sufficient length oftime, the change in bottom hole pressure is 80 slow that therate of production from the formation is essentially that produced by the well, and the point source solution applies.

ACKNOWLEDGMENTSThe authors wish to thank the Management of the Shell Oil

Co., for permission to prepare and present this paper forpublication. It is hoped that this information, once availableto the industry, will further the analysis and understandingof the behavior of oil reservoirs.

The authors acknowledge the help of H. Rainbow of theShell Oil Co., whose suggestions on analytic developmentwere most helpful, and of Miss L. Patterson, who contributedthe greatest amount of these calculations with untiring effort.

REFERENCES1. "Water Infiux into a Reservoir and Its Application to the

Equation of Volumetric Balance," William Hurst, Trans.,AIME,I943.



2. "Analysis of Reservoir Performance," R. E. Old, Trans.,AIME, 1943.

3. "Unsteady Flow of Fluids in Oil Reservoirs," WilliamHurst, Physics, January, 1934.

4. "The Flow of Compressible Fluids Through Porous Media and Some Problems in Heat ConductioU;" M. Muskat,Physic:J, March, 1934.

5. Mathematical Methods in EngineerinG, Karman and Biot,p. 403, McGraw-Hill, 1940.

6. Operational Circuit Analysi:s. Vannevar Bush, Chapter V,John Wiley and Sons, 1929.

7. "Variations in Reservoir Pressure in the East Texas Field,"R. J. Schilthuis and W. Hurst, Trans., AIM£, 1935.

B. "Fourier Integrals for Practical Applications," G. A.Campbell and R. M. Foster, American Telephone andTelegraph Company.

9. Operational Methods in Applied Mathematics, H. S. Canlaw and J. C. Jaeger, Oxford Univ. Press, 1941. (Chap·ter IV).

10. Ibidu11l. p. 5 to 7.

11. "A Problem in the Theory of Heat Conduction," J. W.Nicho1sen, p. 226, Proc. Royl Soc., 1921.

12.. "Some Two-Dimensional Diffusion Problems with Circu.lar Symmetry," S. Goldstein, p. 51, Froc. London Math.Soc. (2), Vol XXXIV, 1932.

13. "Heat Flow in an Infinite Solid Bounded Intemally by aCylinder," L. P. Smith, p. 441, !. App. Physics, 8, 1937.

14. "Some Two-Dimensional Problems in Conduction of Heatwith Circular Symmetry," H. S. Carslaw and J. C. Jaeger,p. 361, Proc. London Math. Soc. (2), Vol. XlVI.

15. "Heat Flow in the Region Bounded Intemally by a Cir.cular Cylinder," J. C. Jaeger, p. 223, Proc. Royal Soc.,Edinb. A, 61, 1942.

16. A Treatise on the Theory of Bessel Functions, G. W.Watson, Cambridge Univ. Press, 1944.

17. Modern Arwlrsis, E. T. Whittaker and G. W. Watson,Cambridge Univ. Press, 1944.

18. The Conduction of Heat, H. S. Carslaw, pp. 149-153,MacMillan and Company, 1921.

19. "Pressure Prediction for Oil Reservoirs," W. A. Bruce,Trans., AWE, 1943.

20. "Reservoir Performance and Well Spacing," Lincoln F.Elldns, Oil and Cas !ourrwl, Nov. 16, 1946, API, 1946.

21. Conduction of Heat in Solids, H. S. CarsIa'" and J. C.Jaeger, Oxford at the Clarendon Press, 1947.

Note: This book came to our notice only after the text ofthis paper was prepared and for that reason refer·ences to its contents are incomplete. The carefulreader will observe that, for instance, equation yj.21in this paper is similar to equation (16), p. 283when k and a_reo given unit values; also that"Limited Reservoirs" cOntains equations quite similar to those appearing in Section 126, "The HollowCylinder," of Carslaw and Jaeger's book. * * *


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