CopYright 1931 by E. ]. Brill,Leiden, Netherlands.

All rights reserved, including the rightto translate or to reproduce this paper

or parts thereof in any fo~

PRESSURE BUILD-UP IN WELLS

BY

D. R. HORNER'"

Section II, Preprint 7

SynopsisThe report presents a method of analysis of the

pressure build-up curve obtained from a closed-inwell by plotting the bottom hole pressure against the

logarithm of to t &, where .& is the closed-in time

and to is the past producing life of the well.In all cases it proves to be possible to determine

the permeability of the formation from the slope ofthis curve.

Mehods are also given for extrapolating therecorded pressures to infinite dosed-in time for thecases of

I-a new well far· from. al.lY reservoir boundaryII-a new well close to a fault, but far from any

other boundaryIII-a well in a fiti'ite reservoir.

Several illustrath'e examples are discussed.

ResumeLe rapport presente une methode d'analyse de la

courbe de remontee de pression dans un puits fermeobtenue en representant la pression de fond en

fonction du logarithme de tot .&, a etant Ie temps

ecouIe apres la fermeture et to Ia duree de la vieproductive passee dupuits.

II en ressort qu'il est possible dans tous les casde determiner Ia permeabilite de Ia formation apar.tir de l'inclinaison de <;ette courbe.

On indique aussi des methodes pour l'exfrapolationdes pressions enregistrees pour une periode defermeture infinie dans les cas ci-apres:

I-un nouveau puits eloigne des fimites du reser­voir;

II-un nouveau puits se trouvant aproximite d'unefaille, mais loin de loute autre limite;

III-un puits situe dans un reservoir limite.

• N.V. De Bataafsche Petroleum Maatschappij, The Hague.

Plusieurs exemples donnes -a titre d'i\Iustration sontdiscutes.

Acknowledgements

The author wishes to state that he was not con­cerned with the development of the theories hereinpresented. They were evolved over a considerableperiod by the reservoir engineering section of theproduction department of the head office of theBataafsche Petroleum Maatschappij in The Hague.

Among others who worked on the matters hereinpresented should be mentioned Messrs B. P. Boots,F. Brons, D. N. Dietz, M. Jakobs and W. R. vanWijk**.

Contents

PART I-THEORY

I, I-Basic Equations and Assumptions • . • •• 2I, 2-Basic: Solution to .the Equation of Flow . ., 2I, 3-~r~ssure Build:up in a Single Well in an In-

finite ReserVOir • . • . . • . • • • •• 2a) Constant Production Rate before Closing in 2b) Variable Production Rate bsfore Closing in 3c) Interpretation of Build-up Equations V; VI,

VII • . . • • • • . • • . • . •• 4d) Conditions of Applicability. ." . • •.•• 4

I, 4-Influence of a Fault in ali Otherwise InfiniteReservoir .......:...... 4

I, s-Well at the Centre ofa Finite Circular Reservoir' 6

PART II-APPLICATI.ONS

II, I-Introduction ••••.••.•• 8II, 2-Examples of. the Pressure Build-up in a New

Well .•.•.••.••.•••• , 8II, 3-Examples of the Pressure Build-up in a Well

near a Linear Barrier. Fault . . • • • •. II

II, 4-Example of the Pressure Build-up in a Wellin a Finite Reservoir 15

n, S-Comments • . • . • • • . . • • . ., 16Bibliography •.•. " • • • • . • • • • •• 16

•• Now professor of Physics a.t the Landbouwhogeschool,Wageningen.

25

PROCEEnINGS THIRD WORLD PETROLEUM CONGRESS

PART I-THEORY

26

I-Basic Equations and AssumptionsThe mathematical study of the sub-surface flow

of reservoir fluid requires that certain simplifyingassumptions be made as to the nature of the porousmedium and the fluids which it contains. In effectthe only practicable method at present available re~quires such sweeping simplifications in order toobtain a solution at all, that the solution so obtainedrequires considerable testing in practice in order todetermine its usefulness and its limitations;

Thus it is general practice to develop the flowequations assuming the reservoir to be homogeneoushorizontal and of uniform thickness throughout. Th~fluid is assumed to obey d'Arcy·s· law and to bepresent in one phase only. Furthermore. it is assumedthat the compressibility and the absolute vis.cosityof the fluid remain sensibly constant over the rangeof temperaturt::and pressure variation encountered inthe formation and that the density of the fluid obeysan exponential type law

p = Poe·clP.·p) •••••••••••• (I)

where P is the density at some pressure P.Po is the density at some standard pressure

(conveniently taken as the original reser­voir pressure Po)

and c is the compressibility (assumed constant).Then if we consider one well drilled into such a

reservoir and assume furthermore that the flow tothe well is radial (which implies either an infinitereservoir or a finite circular reservoir with the wellat its centre) it may be shown that the equation offlow (r) * is

~'p Ii'p _ fCfl-i\pi\r' + rilr - k 3t (II)

where r is the distance from the centre line of thewell in centimetres,

t is the time in seconds,p is the reservoir pressure in atmospheres at

distance r and time t,f is the formation porosity expressed as a

fraction of the bulk volume,k is the formation permeability in darcies,p. is the fluid viscosity in centipoises

and c is the fluid compressibility measured involumes per volume per atmosphere.

Of these basic assumptions it would seem that byfar the most critical is the one which requires thepresence of only a single phase of reservoir fluid inthat both 'the compressibility and the permeability ~re

• References given at end of paper.

\'e~y sensitive to changes in pressure below the bubblepomt. However, although the theory is developed forthe case where pressures are everywhere above thebubble point, the equations often seem to fit whenthis condition does not hold.

2-Basic Solution to the Equation of FlowThere are a number of exact solutions of equation

1.1 (for various boundary conditions) gi"en in theliterature. but these suffer from the disadvantagethat they involve complicated integrals and Besselfunc!ions which make them very unwieldy for cal­culation purposes. \Ve therefore make use of the so­called "point-source" solution

qfl- .( r2ffl-C)tp = Po +41tkh El - 4kt ... (III)

where Po' is the initial reservoir pressure in atmos­pheres.

h is tHe formation thickness in centimetresand q is a COllstOllt rate of production of the well

expressed as cubic centimetres of sub­surface volume (under original conditions)per second.

This equation III is an exact solution of equationII for .the following boundary conditious:

(1) external boundary at infinity at constantpressure Po

and (ii) internal boundary (i.e. the well radius)vanishing and with a constant flow ratt qacross it (i.e. for a mathematical sink in aninfinite reservoir).

Thus the only error introduced b)' using equationIII as the basic solution for the case of an infinitereservoir is in considering the well radius as infinitelysmall. This error is considered to be negligible forthe applications of this report.

3-Pressure Build-up in a Single Well in anInfinite Reservoir

a) Constant Production Rate beforeClosing in

Consider a single well in an infinite reservoir

t Values of the Ei-function, defined b)' the equation

J'ooe·uEi(-x) -- - dxu u

are available. from references (2) and (3).This funcllon is -00 for x zero aild increases monotonic­

ally to zero as x goes' from zero to +00. As x approacheszero, Ei (-x) -In x -+ .5772.... and so for small values ofx. say for x smaller than about .01. we may write withclose approximation

Ei(-x) l':Ilnx + .5772 ..••

SECTION II, l'REPRINT 7

"iVr.

Fig. I. Illustration of accuratt method for correcting for\'ariable production rate.

application, and will normally be more precise thanis warranted by the other inaccuracies which are un­avoidably present-in fact .only rarely has it everproved of value to apply this elaborate method. In­stead, the equation V is usually modified by simplyintroducing a so-called corrected time t., and writ,ing

q !L tc+& (VII)Pw=Po- 41tkh In~ .....

where' q is calculated from the last established pro­duction rate before closing in the well; t. is ob­tained by dividing the total cumulative p~u!=tionof the well by the last established production rate.

in: However, such conditions do not normally obtain,and so some correction must be applied to takeaccount of the varying rates at which a well will havepwduced during its hist.ory. Two methods of cor­rection may be used, one of which may be said toenable a theoretically.prec:ise solution to be obtained(at least in principle) while the other is nothing buta ·good working approximation.

To illustrate the precise meth6d we suppose thatthe pr.oduction history of th·~ well was as shownby the broken line in f;gure I. To this we ap­proximate by a series of steps (as shown) and thenmodify the eq'!ation.v to read

_ -'-~~ In t o+&Pw - Po 41tkh?qo to+&-t

1

+I to+&-t1 + In to+&-t2

ql nto+&-tl q. to+8o-tato+8-ta l (VI+qa In &. ~,....... )

where the t's and the q's are as indicated in fig. I,

and are so chosen that they represent the same totalproduction as the well actually made.

However, this equation VI is rather laborious In

which was completed and first brought into produc­tion at time zero, and which subsequently producedat a constant rate q until time to, when it was closedin. Then, ignoring the effects of the after-produc­tion *, the well pressure p" at time to +& (i.e. &after closing in) may be obtained by superimposinr.two solutions of the form of equation III and thenwriting r" for r thus

~ (If .

q!L E" rw !LC ')'Pw =Po+ 41tkh 1 - 4 k (to+&)

E'( rw1f!Lc\l- 1 - 4U.) ~ .,'.,,'. (IV)

where p" == pressure. at well bore in atmospheresand r" = well radius in centimetres.Now for small values of its argument the Ei­

function may be accurately approximated by a loga­rithmic function. (See footnote to page 2.) If thisapproximatiC?n be in fact made in equation IV abovewe derive the basic build-up equation for a singlewell, in an infinite reservoir as

q!L t o+&Pw = Po--khln --. , ..... (V)

4.. &The error involved in this approximation will be

very small after a comparatively short time. It will,25r tf!Lc

indeed, have dropped to Y-i 90 as soon as&> : '

a condition which will usually be satisfied within amatter of seconds after closing in the well. As wehave introduced errors into the calculations (by com­pletely ignoring the after-production) which mayaffect the validity of our equations for a period ofan hour or more after closing the well in, it is clearthat both the approximation to the exact solution bythe Ei-function and the approximation to the Ei­function by the In-funCtion are entirely justif.iable.

Thus in the case of a well which has produced uni­forml)' since completion at a rate q from an infinitereservoir we may expect the bottom hole pressure tobuild up in accordance with equation V.

b) Variable Production Rate beforeClosing in

In the pre\'ious paragraph we derived a: build-upequation V for a well which produces uniformly atrate q from time zero to time to and is then closed

• When a well is dosed in at the surface, production fromthe formation does not cease immediately. Instead, there issome as yet undetermined period of time during which theformation produces fluid into the well bore (at a decreasingrate) thereby building up the pressure within the well. Itit this quantity of. fluid-the volume which is produced bythe formation into the well after dosing in at the surface­which is herein termed the "after-production".

il

~~:"

H~ASVA£"Mtr ,?Il,..... / ,A,qct,q(KI~TINS I'IAT~: ...

"

II

rt.I "

27

PROCEEDINGS THIRD WORLD PETROLEUlo1 CONGRESS

28

c) Interpretation of Build-up Equa­t ion S V, VI, VII

It is immediately evident that if we plot p. against

In tot & (using t. for to as necessary) or against

to+& to+&-tl .qo In tO+&_t

1+ QlInto+&-tll + etc. If we" are

using the accurate correction method, we mayexpect the points to fall on a straight line, at leastafter the effects of the after-production have dis­appeared. If indeed it does prove possible to drawsuch a straight line, two deductions can be madeimmediately. They are:(i) by extrapolating the line to

In to+& =0&

to+& In 1o+&-t1(or qo In 10+&-1";. +Ql to+&-t

2+ .....

etc. = 0 in the ac.:urate case), which is equiv­alent to extrapolating to & infinite, we mayread P. = p. which is of course equal to P. thefully built up pressure of the well. This vafue is,of course, identical with the initial pressure, aswe are at prescnt only considering an infinitercservoir.

(ii) The gradient of this line is equal to 4;:h (or to

4~h in the accurate case) and so Imowing

values for q, p. and h it is possible to determinea value for the permeability k measured in situ.This "alue of k has the advantage that it is amean value for the whole well drainage area.

d) Condi ti ons 0 f Appl icab il ity

The theory detailed above is, however, only ap­plicable strictly to an infinite reservoir, which is atheoretical cOllception which does not exist in fact.Thus the above method can only be expected to be ap­plicable in the case of a well which has not yetproduced sufficient fluid materially to ha\"e dimin­ished the overall static re~e.rvoir pressure, i.e. a newweIl in which the effects of the reservoir boundaryhave not yet become apparent.

4-Influence of a Fault in an Otherwise InfiniteReservoir

The problem of a well producing from a, pointdistant "au em from a linear barrier fault may besimply solved by the method of images. This meansthat instead of considering one well Q producingfrom a semi-infinite reservoir bounded by the linear

fault (fig. 2 a) we may consider two similar wellsQ and Q' producing from an infinite reservoir (fig.2 b), where the fault has now been removed and theweIl Q' is inserted" in such a manner as to have ane!fect equivalent to that of the fault. This require­ment is simply that Q' be identical to Q and at themirror image of Q in the fault plane, i.e. at a distance2a from Q. Thus the pressure" in Q at time to + &(& after closing in) is

qp. ~. rw2fp.c""

Pw = Po + 41;kh~ E1(- 4 k (1o+&) I.( rwllfp.c) . (" "a2fp.c)

-El - 4k &, + E1 -k(1o+&)

.( a2ftLc ~-E1 -k"& l~ ..: ..... (VIII)

where the first two Ei-functions, unchanged fromt~uation IV, represent the effect of the well Q. andthe last two Ei-functions are the contribution of theimage well Q'. As before we may substitute theIn-function for the Ei-function in the first two termsto give the build-up equation

_ qp. ~ to+& . a2fp.c "\Pw - Po- 41; kh i In-&--El(-k(to+&)'

+Ei (- a~fr)~ (IX)

a bF)g. 2. Application of the method of images to the case of

the linear barrier fault

Now the distance "au will normally be large enoughto make the last term sensibly zero and the oth~r Ei­function sensibly constant until & becomes quitelarge. This means that for the first part of the build­up the curve will be approximate1)'

qll- I to+8o ." atf!L~ iP... =Po - 41;kh In-&--Et(- k10 Ii'" (X)

that is the first part is a straight line of norma!

SECTION II, PREPRINT 7

(4;:h) slope when plotted against In to;8o.Extrapolation of this line, however, would give a

final pressure below the true value (Po)' However,as & becomes very large the arguments of both theEi-functions in equation IX become small and theymay then be approximated to by In-functions. Thusfor very large & equation IX becomes

qp. t o+8oPw=Po-2nkh In.-8o- ...... (XI)

that is the last part of the build-up curve is a straight

line of slope2~~h~i.e.twice the normal value) when

plotted against In tot & ; this part of the curve mayt +8o

be correctly extrapolated to p. '== Po at In T = o.

Oearly there will be a range of transition values

of In tot8o where the line of the fomi of equation X

merges into the line of the form of equation XI. Itis also theoretically possible to determine -the quantity

a I ~p. C (and hence a) by fitting the build-up curve

to the exact equation IX and thus to determine thedistance of a fault from a well (although not, ofcourse, its orientation).

P/"Im/

'D

(/D

tJ fhwr.rJ ! ! I ! i i In, t ~ .~

f • ISD mYtl~)' bhrin,f"s" fUel#/" 0.11)'--j, ::

so~ J L,f .. 1D0mO

/' . 20 %c = lO~ YDI/.w/l1lm

-h. /Som

Po .. ZOOI1/m. //0 • 10 ""'.1'$ I /8 • Ism

I .I'. .IA -£fh /In .t;/ I-EifiJZ1J5) "'El(--lliJe.J/ /• -IX

II /i

I vIi

I ~

I /I

~I

/'V

/V

• 7 IS S ., " &

"In t.+&&

Fig. 3. Illustration of the theoretical case of a linear barrier fault.

I

29

PROCEEDINGS THIRD WORLD PETROLEUM CONGRESS

30

A calculated plot of equation IX is included (fig.3) to illustrate the form of the build-up which canbe expected.

S-Well at the Centre of a Finite CircularReservoir

E.'Cact solutions to the flow equation II are avail­able in the literature for the case of a single well inthe centre of a finite circular reservoir. A solution (4)which onlv errs insofar as it treats the well as amathematiCal sink in a finite reservoir (i.e. is asaccurate in this case as is the Ei-solution for an in­finite reservoir) is

[where r. -'extcm"J reservoir radius (ems) and thex. (n = I, 2, .:. co) are the roots of Jl(X) = 0],hut this solution .is clearly too complicated to be ofother than thtoretical interest.

Accordingly an approximate solution is derivedby modifying the equation III. Taking r. to be theradius of the finite resen'oir boundary, we may saythat the pressure drop caused by a well in an infiniteres':f\'oir is less than the pressure drop caused by anid<:ntical well, in a finite reservoir by an amount;!ependent on the quantity of fluid which (in theinfinite case) has flowed in across a circle of radiusr•. The niethod then is to calculate the volume offluid which in the infinite case has crossed the ex­ternal boundary r. it any time t. Then we may saythat this quantity of fluid (say Q.), had it beenproduced from the finite reservoir, would have ca~sed

an additional average pressure drop throughout thefinite reservoir equal to

Q.hydrocarbon-filled reservoir pore volume

X I . .. and this quantity may be cal-compresslblhty

culated to be

. r.1 fILe)..-9.L~Ei(_r.lflLe)+~e-4kT(4'1tkh l 4 kt ' T.lflL e ~

Thus the pressure history of a circular field withone uniformly producing well at its centre may be

obtained by includin'g this correction term in equationIII to give

qlL 1.(' rSfILe)P=Po+ 4'1tkh lEl.-"4!Ct,

I r.lflLe)!-y \41Ct.1 (XIII)

where for simplicity we define the function y by

y(u) =Ei(-u) + Ee·uu

It is' of importance to be able to consider thisfunction y as a known function. Accordingly a plotof y for a large range in values of u is appended(fig. 10).

Now this equation XIII is not even a mathematicalsolution ,of the basic flow equation II. However,the closeness of the approximation of this equationto equation XII may be easily shown to be usuallysatisfactory, the ,errQr is certainly never more thana few percent. As an example fig. 4 shows theaccuracy of equation XIII by comparison withequation XII in the particular case where r./r. =IOCX>.

Just as was done in paragraph I, 3a, we super­impose two solutions of the ft)rm of equation XIlIand write r. for r to give our approximation to thebuild-up equation in a finite reservoir as

Now until & is very large the second y-function willbe almQst zero and _the first will be nearly constant,and the equation XIV reduces to

qlL I to+& lp.=Po- 4 nkh In -&-+y(uJ f.. · .. (XV)

h f . . 6 r.'flL ewere or convenIence we wnte U1 .or 4

kt••

Thus we still have that, over the range of valuesof & which will normally be measured, a plot of p.

against In t.t & wijl be linear and, its slope will

still be 4:::h' However, linear extrapolation to

In t..t&= 0 will indicate a false value for the final

build up (which false value we may indicate by p*)defined by

SECTION II, l'REPRINT 7

p. = p.- 4~:h y(uJ ," ,(XVI)

Thus, knowing P. and having determined 4~khand p* from the build-up curve, we may solve equa-

tion XVI for Y(Ul) and thus we may derive a valuer.1fp.c

of ul -= ---'4 k t"

Now if we return to the former equation XIVand let & become infinite, the equation becomes

o'~

~~

~

A ~ 'II

/1/

I !;

• II !

i

7/-

II7fII I

jI !I

• I ..~{ I I,

i I·-i----··_- -1

--~-I

------.._-_.-i.._---. +-

I -.

. ICurwI - ~~iaICI1SI1 or''lullll()n X/Ieurrr:o:· AplJf'f1ximlllion""/llis rrporl, 'lJUlIlion XIII80111r"plDJIN /la-o III, "p,ei.1 us, 'b .10011

I r"I!

••

•5

'0

•$

10

to10 ,s .~

(Po -.Pw ' I~hFig. 4. Comparison between precise theory and apJl.I'Oximation of this report for the

case of a well in a finite circular reservoir.

31

PROCEEDINGS THIRD WORLD PETROtEU}{ CONGRESS

qt. ( qIL )p. = p. 1tr.2 fhc = p.- 41tkh

. (r.2f IL C)f-:- 4k t. . . . .. . . .. (XVII)

where p is the final static (i.e. fully built up)

closed in pressure, and then, substituting the known

qIL dr.lflLc.values of Po. 41tkh an 4 kt• In XVII we may

obtain a value of p.. thereby correctly extrapolatingthe build-up curve.

PART II-APPLICATIONS

32

I-Introduction

Part II of this report is intended to be largelycomplete in itself. The object of this part is, bythe use of several examples, to illustrate the methodsof applying the. more important equations developedin Part I. These equations are collected for easyreference in the Summary of Equations Chart, Ap­pendix A; each equation is repeated in two formsnamelv exactlv as derived in Part I and also asnlodified for ~se with practical oil-field units. Alltne symbols used in the report are collected in atable '(Appendix B) to which reference should bemade for the units which are to be used for thevarious quantities involved, and for the values ofthe numerical constants ·of conversion (A, B. D, Fand G) to be used with the particular units em­ployed:

The examples chosen to illustrate the use of themethods here presented are all from wells in theCasabe field, Colombia, and have been selected froma batch of 66 pressure surveys in this field madebetween August, 1948, and April, 1950. Those whichhave been selected for inclusion in this report havebeen so chosen because they clearly emphasize certainpoints of particular interest; it must be emphasizedthat the accuracy and compatibility of the exper­imental data obtained from the five wells here con­sidered is in no way superior to that of the other 61which are not included.

It is perhaps of interest to state that the oilbear­ing formations of the Casabe field consist of threeseries of lenticular multiple sands 'A', 'B' and 'e'of which the 'A' and 'B' sands are of Oligocene andthe 'e' sand is of Eocene age. They are fine j?;rainedto silty, rather argillaceous, and fairly well con­solidated. The oil is heavy (200 -21 0 API) andviscous (40. cp ±). Some doubt exists as to thebubble point. It was first thought that. the oil inall sands was undersaturated, but it now appearsthat pressures in the 'A' sands at least are below

t It is of interest to observe that this limiting form ofequation XIV may be expressed thus(Present Reservoir Pressure) = (Initial Pressure) -

(Total Volume Withdrawn) X I iJ'ch •(Total ReservOir Pore Volume) Compressibility" I ISprecisely what we should expect from elementary consider­ations.

the bubble point. This, howe\'er, does not appear toaffect the build-up curves.

2-Examples of the Pressure Build-up in aNew Well

The earlier theory developed in Part 1" (I, 3)is designed to apply to a single well in an infinitefield. Such a condition never obtains, of course, butit is further pointed out (I, 3d) that the caseof a lIevJ well in a finite field is similar so long asthe total withdrawal from the well has been keptsmal!. It is difficult to lay down a criterion for theorder of this smallness-practical applications seemto indicate that a normal well may be allowed toproduce for a period of weeks or even of severalmonths and it will still obey the "infinite reservoir"theory. Thus we may apply the infinite reservoirtheorY to the following case.

CB-16I was completed to the 'A' sands on 7thFebruary 19."0, and it was closed in from 16thFebruary to 8th March while continual bottom holepressure meas~rements were bein~ taken. As willbe app:trent from the analysis of the results, it wasnot really necessary to leave the well closed in forsuch a long- period in order to obtain adequate. data,althou~h it is interesting- to observe how closely thepressure in the well followed the predicted courseover such a long time.

This beinJ1: a new well, we apply the theory for asingle well in an infinite reservoir, as has been al­ready explained. Reference to the Summary of Equa­tions Chart, Appendix A, shows that the eouationfor the build-up in this case is number V, which inpractical units is

GqIL t.+& (V )P,,=P.-C.khlogl0-&- ..... a

Reference to the table of Appendix B gives a def­inition of these symbnls and of the units in whichthey are to be measured. The. U.S. system of unitsbeing in use. in Casabe, we must use the value of Ggiven in the tabulation as pertaining to this system,namely 162.6. Thus equation Va is modified to

I62.6q/L. 1.+& (Vbp,,=p. C.kh logle-&- ... )

which is thus the equation which we must attempt

SECTION II, PREPRINT 7

to fit to the experimental data. As the method of

I · • fl' I t.+8o . tana YSIS consIsts 0 p ottmg oglO -8o- agams p.,

we firstly require a value for to. As has been pre­viously explained (I. 3b) the· eql.;lation V (orits modified form, Vb) assumes that q. the rate ofproduction. has remained constant for the whole lifeof the well (to). This not being in fact true. themethod of correction is to take the last availableproduction rate-in this particular case. 641 bbl[day­as the value for q. and to use a corrected time tl'instead of the theoretical to. where t. is obtainedby dividing the total cumulative production of thewell (in this case. 5847 bbl) by this last productionrate.

Hence for t. in Vb abo\'e we substitute

t Total cum. prod. 5847 d hI = =-- ays = 219 oursLast prod. rate 641Thus the build-up equation Vb may be modified

by the insertion of this value of t l for to' This gives

q fL 219 +&p. = po - 162.6 Co k h loglO & •• (Vc)

The value of q = 641 bblJday could now be in­~rted. but it is perhaps more convenient to leavethis until a later stage.

The method of treating the experimentally ob­tained pressure data may be most easily explainedby reproducing the measured values exactly as theywere 'reported from well CB-16I.

Pressure c.1. time 219 + & 219+&p. & & IOglO &(psig) (hours)

1192 19 12·53 1.0g801200 25 9.760 .98941206 31 8.065 .go661212 37 6·\)[9 .84001216 43 6.093 ·78481220 49 5469 .73791223 55 4.982 .69741227 61 4·590 .66181230 67 .4-269 .63031232 73 4-000 .60211235 79 3.772 ·57661236 85 3·576 .55341237 91 3407 ·53241239 97 3·258 .51301241 103 3.126 49501242 109 3·00\) 47841241 U5 2·904 46301243 121 2.810 44871244 127 2·724 43521245 133 2.647 42281247 139 2.576 410\)1249 145 2·510 039971249 151 2450 038921250 157 2·395 037931267 477 1459 .1641

Columns I and 2 are derived directly from thebottom hole pressure readings. columns 3 and 4 arecalculated from column 2. and finally the values ofP. from column 1 are plotted against the corres-

. 219 + &pondlng values of 10giO & from column 4

as in figure 5.It is convenient to plot. P. vertically and in a con-

. I I I 2"19 + & h .ventlona manner. but to pot oglO & orJzon-

tally from right to left. i.e. with the zero on theright hand side as has been done in fig. 5. as thisgives a' more vivid impression of rising pressure.

If agreement is to be obtained with the theory.and thus with the derived equation Vc. the pointswhen so plotted should fall on a straight line, ex­cepting only possibly the points corresponding tosmall & when the effects of the after-production (seefootnote on page 3) may be still felt.

As can be seen from fig. 5. the accuracy withwhich these experimental points do in fact plot on astraight line in this case-and particularly the verylast point which represents nearly 20 days closed in­is really quite remarkable.

The interpretation of this figure 5 is simple. Firstlywe may deduce a value for Po. which is done byextending the straight line plot to the point cor-

responding to 10giO 219:- & = 0 and reading the

correspondillg pressure which gives in this case avalue for Po of 1280 psig. Although we have definedPo to be the initial reservoir pressure. this must beinterpreted somewh~t widely. In this case, for exam­ple. the well considered was drilled as an· infillingwell into an already heavily drilled field, and so thepressure Po = 1280 psig must be interpreted as thereservoir' pressure at the well at the moment of itscOIn/letion.

In addition to this pressure determination wemay determine the permeability k from the slope ofthe 'straight line. This can perhaps best be done byselecting two arbitrary and fairly widely separatedpoints A and B on the line (fig: 5). The pressuredifference between A and B is 1272-n82 = 90 pSi,

and the corresponding difference in 10giO21

9&+ &

is 1.2 - .1 = I.I. The slope of the line is then ob­tained by division thus

33

o-x

PROCEEDINGS THIRD WORLD PETROLEUM CONGRESS

/250I I ./I

VIPw _psig I V I

I

I %I I I 1/. I I I II

1200 vr 1 I Fig, (VI

Vi -r-!II, I

IISO ,01.5 to 0.5

/09-219.,.7/

, '" r:r

IC8."2.5 II I Io OD~:I~~s ./

o.~

I /'V

VI ~/

V

Y/"

4'/

ft.V

V Vfig. (vii

.Y/'

34

/300

1/00t.5 1.0 0.5

" 7U .. z).O$.;,~

Fig. 5 and 6. Observed presslJre bilild-up curves in new wells.

o

SECTION II, PREPRINT 7

or

that is

which is· valid for all ranges of .&, and which may beapproximated to by

3-Examples of the Pressure Build-up in a Wellnear a Linear Barrier Fault

The theory which has been developed for a linearbarrier fault is strictly only applicable to a well inan otherwise infinrte reservoir. However, we mayapproximate to this condition by a new well closeto a fault and considera~ly farther from any otherbarrier. Such a well is CB-123. completed to the C­sands at the beginning of September 1949; it wasclosed in for test from 4th November 1949 to 5thJanuary 1950. Reference to Appendix A shows thatthe relevant equations are numbers IX, X and XI,which, subsituting the values of A, D, F and Gappropriate to U.S. units from Appendix B, become

p.. =p.- C::h {I62.610gI0 t.;.&

. ( 3793 allfp.c)-7o.6oEI k(t.+.&)

+70.60Ei(_3793ka;fILC) I .. (IXa)

q p. { 1353+'&p.. = p. - C. k h 162.6 loglo .&

-70.6oEi (_37;;~;f:C)} ... (Xb)

for all but very large .&, and

p.. = p. - 325.1 C;:h IOglO I35~+.& (XIb)

qIL \ t. +.&p.. = p. - C. kh l 162.6 loglo -.&-

-70.60Ei( 3793ka~fILC)I ... (Xa)

for all save very large values of &, and by

q P. t. + .& (XI)p.. = P. - 325·1 C. k h loglo -.&- . . a

for very large.&.The exact equation IXa is not of great interest as

far as the applications are concerned. Instead, thetwo approximations Xa and XIa are used.

Firstly we require a value of to, which is derivedas for the previous example. We use q ='275 bbl[dayand to = 1353 hours. The value of q. as before, wedo not yet substitute. However, we insert the value'of to in the two approximate equations which become

and thi~ slope we now equate to the coefficient of

10gi0219&+ &in the build-up equation Vc. Thus we

have

Difference in pressureSlope =

Difference in 10gi0 21;9~t .&

;:=: 9° =821.1

82 = 162.6 C;:h

from which we may deduce

C. k h = I682.~ = 1.98 ..... (XVIII)qIL 2

In the particular case of the well CB-161 the fol­lowing additional data are available:Rate of production q = 641 bbl/day (Ref. page 9)Vis~osit)" p. = 40 cp 1From PVT dataShrmkage factor Co = .93 ~Pay thickness h = 349 ft From Electric logand if w~ "Substitute these values of p., Co and h inthe above equation ;XVIII we have

·93 X k X 349 _ 8641 X40 - 1·9

.0127 k = l.gB

k = I.gB = I56mD.0127

Thus we have derived a value of 1280 psig for thereservoir pressure and 156 rnd fQr the permeability,both of which are in good agreement with other datafor the Casabe field.

Figure 6 shows an example of the build-upof another new well. This requires no further com­ment, except to note that it is included only becauseit is an exceptionally good example of a very longperiod closed in.

The data relevant to this example are:Well No CB-125, A sands

q = 280 bbl[dayt. = 764 hours (31.8 days)

Max. C. I. time 2847.7 hours (n8.7 days)k (from the slope of the line) = 88 md

The points corresponding to very large cl.osed-intimes do not, of course, plot with the same accuracyon the straight line. This is because the earlier groupof 17 points were obtained from one (or perhapstwo) runs of the pressure gauge, while the; last 8points are eight spot readings taken at widely spacedvalues of .&.

35

tilCI'I

o0.51.02.0 1.5

109",·15siJv

----'----Fig. ,. Observed pressure bui,d.up ccrve in well affected by a linear barrier fault.

2.5

)CB-/23

(' ~

'0 Oos.~I"""SSuNS en-./V

)-_ -...,-_. Th~/;t:lIl ks/eu~WJ~;'.9 Ao ~'1ultl;M IXIJ

~~)

.1/"~pe:/" ®- - - - - - -- - '- - --.--- _'- .---I~

I~' l,-)~" ope ,;Z5 ---

.~.~~ ---~I-"--~::-~-&~ -

.~~ ./::;;:::::01:0<>'"

i/.....:::::::--\~

q::;- /"'- - - ._,

----- ./

~~

) --- - - - -- - - - --- - -- -

---'-- - .

2100

2200

23001

1700

1800

1000

2000

IPw-p&1900

SECTION II, PREPRINT 7

whence.530 X 10-6 a2 = .0468

a2 = 8.83 X 104

or a = 297 feetHowever, this figure is not in·agreement with the

present (rather obscure) subsurface picture whichplaces the fault at about .1100 feet from the well.

Instead of proceeding in the manner .just detailed,however, we may modify out approach thus.

Firstly, we accept the' possibility that the stopesof the lines I and II (figure 7) may not be exactlyin the ratio of 2: I. If we suppose this ratio to beb: I a first approximation to the build-up equationmay be obtained by modifying equation IX to read(in practical units)

q!L I t.+.&p.. =P.-C.kh G log10-.&-

. ( Da'f!LC)-(b-l}AEl -k(t

o+'&)

+ (b-l) AEi (_ D ~;P.C)} ., (rXb)

c -= 5.1 X IO·e volslvollPsik=270 md

the above equation XIX becomes

-Ei ('_3793 xatx .25X40X5·1 Xlo·e) = 2.1353 X270 53

i.e. -Ei (-.530 X 10-& all) = 2.53and reference to a table of Ei-functions (2, 3) showsthat, for this value of -Ei (-.530 X 10-8 a2) thequantity

The problem of fitting this equation to the ob­served points can then be performed thus.

An approximate value of "b" is derived from theratio of the two slopes, a value of Po is obtained bythe extrapolation of the last part of the build-upcurve and a value of "a" is obtained just as previouslydescribed Le. by equating the -Ei function to

2.3 X IOg10 tot.& at the point of intersection of the

two straight lines. It is convenient that, given thispoint of intersection, the value of "a" thereby deter­mined is independent of the slopes of the lines.

Due to some uncertainty in the value of "b'! (forin practice .& must usually be very large indeed beforeline I is thoroughly established, and so it is possiblefor some of the later points in the transition zone, tobe mistaken for points on this line I) it may notnow be taken as definite that the b~t possible yaluesof a, b and Po have now ~een chosen, although it maybe ~pected that theywitI not differ too widely. fromthe final values.

Thus the equation IXb is now plotted to se.e hownearly the calculated curve fits the observed pointsi.

1:62.6 275 X40 = 125·93XkX57

i.e. 33,700 = 125k

whence k = 270 md.We may also estimate the distance of the fault

from the well by the following method.

We equate the function -Ei (_3793al~C). 1353

to the value of 2.3 logto I35i,+.& t at the point of

intersection of the two straight lines I and II (fig. 7).

These lines intersect at logJo·I35~+.&= 1.1, and

so we 'have the' equation

_Ei(_3793at

;:C) = 2.3XI.I = 2.53 .. (XIX)1353

Now·if we substitute the known valuesf = .25 'P.-=40cP

for,very: large &. Just as in the previous example, we

plot p ..~gainst 10g10 1353;; &. The resulting plot is

shown in fig. 7. Theoretical requirements are thatthe early part of the curve be a straight line (repre­sented by equation Xb) and that the latter part bealso a straight line (according to equation XIb) oftwice the stope of the earty part. These two straightlines should be Joined by a transition zone.

There are, indeed, two well defined straight lines Iand II, but their slopes are respectively 330 and 125,i.e. in a ratio of 2.64: I, instead pf 2: I. This may beexplained, however, by the fact that the particularwell in question is subjected to the influence of twointersecting faults, the effect of which may be.theoretically shown to be to increase the ratio or thetwo slopes.

As Before, a value of Po may be obtained by e.'Ctra­polating the. line I to give Po = 2263 psig.

The value of k is obtained from the slope of thestraight line II, and proceeds just ·as before:

162.6 C:th = slope of liile II = 125

and substituting the k-nown valuesq = 275 bblfday (Ref. page II)p. = 40 cp 1from PVT data

Co =·93 ~h = 57 ft (Electric log)

we have

t Note that the factor 2.3 is a universally applicable con­stant for this method (actually it is In JO).

37

PROCEEDINqS THIRD WORLD PETROLEUM CONGRESS

II---r--!

! !! !! T

I

I I1300

f A IlilY

II I L\~I I <,IO?Y

I / VI II ! 1/ Vi@

I--I--+--+--I--Ir--+-,I--;-+/'--I---1---+---tV-f+I/""7"1--+-+-;

I I j--:- i IY1I00 1--I--+--+-+-+--l-,-+-i-+-:-t-+--+V-/;/."""!-4--+-+--+--1

I--+-+---t-+-+--:-I - r -+-f'-+--Il-.'/q.--+'-+-t-+-+-I~-+--+-+--t.__J i I

I' /' I I

1200

Y I IlOOOf--1-+-+-f---1-+-+7"4--4-f---1-+--+'-f-+-+-+-1

/. .

SlOO l---'--'---::!.oI;:--'--'----JL-..l--=20~...L-L--.!L-..L-LOL....L-l.--.!L-..L.--:!o

"';: f,j p,}.. '"Fig. 8. Observed prl:Ssure build-up curve in well possibly affected by a linear barrier Iault

in the event of an unsatisfactory fit, the. three quan­tities a,i>an~ Po ~ay be sO!l1ewhit modiIied, and apr~ess (Sf trial and error will finally lead to the bestvalues. Fortunately this seems to be fairly quick inapplication.

The end result of such a process may be seen in

figure 7, in which the broken line"has been calculatedto fit the observed points using the method outlinedabove.

The final "best" values of the quantities a, b andPo were not widely divergent from the valpes alreadyobtained, as may be seen in the following tabulation

38

~ECTION II, PREPRJNT 7

Best values Previously'from Egn IXb obtained values

tained to calculate k. However, linear extrapolationof the straight line region of this build-up curve doesnot give 'the value of the reservoir press,ure.

As an example of the method we include the resultsof a long survey made on well CB-IIO (p" plotted

against 10giO to;;&) figure 9. The relevant equations

are XV, XVI and XVII. We pr.oce~d thus:The slope of the initial straight part is measured

to be 121, and then from equation XV we equate this

value to the coefficient of 10gl/O~.& to give an

equation 121 = GC::hand taking G = 162.6 (U.S.

units, Appendix B) we get

q!L 121C. kh = 162.6 = ·744

which may be soh'ed for k, knowing q, po, Co andh, as before.

If we extrapolate t~e straight line, howeyer, weget a false pressure reading p* = 1313 psig. Tocorrect this we also need to know the value of po·'vVe take in this case 1343 psig (as measured in a

. I"" ,1,s~.'Ps;I I

I I (hem_sri-/,i '!.',SI3 !,..;;I

0C8 'lfO •...i. V

,.J.i I ...... V

I I ,/-+-- '

A CI,w",«I iT _II V I

'("$psi.1'/';--.

I--;

I p:/ I I . No,

I .- 'Sit

......v :-- .-so'-~v

:/.«

y "'"~

/', ,/

V,

... '.0 u

1100

130

'000

Fig. 9. Observed pressure build·up curve in well in a finite reservoir.

39

PROCEEDINGS THIRD WORLD PETROLEUM CONGRESS

Now we go to equation XVII, which we write

q(L • (Br.!f(LC)p.=p.-A C• kh .... kto ...... (XVIIa)

This y-function is plotted in fig. 10. Reference tothis figure shows that if y(11) = .571, then u = .537.

Thus we have for this case

in this equation XVIIa to give

p. =! 1343-70.00 X .744 -:- .537= 1343-98= 1245 psig

as the calculated static pressure at the well afterinfinite time closed in.

Agreement is here not too good-the apparenterror in P. is about 5S psi as may be seen fromfigure 9. This is, as yet, the only well which we havehad closed in for long enough to be able to makesuch a check. Accordingly this method must still betreated as unproven.

S-Comments

In the application of these methods there arecertain definite dangers. Firstly it is not always clearwhich part of the build-up curve is to be used todetermine k. It is not uncommon for many of theearly pressure readings to fall on a straight line, when

plotted against IOglO to t& although they have been

taken during the period of the after-production. Thisreport does not attempt to cover this period of after­production, and this early straight line, having aslope often many times greater than the true value of

~~~ will thus give incorrect values of k and po.

Secondly, when the after-production has ceased, all

wells show an initial slope of~~~and ret for eXtra­

polation purposes it is necessary to know which casemust be catered for; infinite, semi-infinite or finitereservoir. Thus a good deal of care must be exer­cised in analysing the results of build-up curves ifreliable data are to be obtained.

After analysing a large number of wells by thismethod we have obtained the impression that accept­able values for the permeability are more frequentlyobtained than for the extrapolated pressure. This isprobably due to the fact that whereas one is contentwith only an approximate value of k the limits oferror withID which the reservoir pressure is requiredare very much smaller.

as obtained above

Po = :1:343 psigA=70 •60

q(L - 744C.kh -.

Br,lffLckt. = ·537

previous pressure survey) and use equation XVI,which is (Ref. Appendix A)

• q(L (Br,lf(Lc)p = p.-A C. khY kto .(XVIa)

Substituting then the values

* . ~p = 1313ps~g as abovePo = 1343' pSlgA == 70.00 (ref. Appendix B)

and Cq:h= ·744 (from slope of linear part• of curve, as calculated

above)in equation XVIa, we have

(Br,!f(LC)

1313 = 1343 - 70.60 X ·744 Y k to

which may be solved for y to give

(Br,lf (LC) _ 1343 - 1313 - 5

Y k to -70.60 X .744 - . 71

and we substitute the values

Bibliography

(I) Musical, Flow of Homogeneous Fluids Through PorousMedia, OIap. X, Sec:. 10.2, Eqn I.

(2) Jahnke und Emde, FUIIktionentafein Section I, pp. 6-8.(3) Tables of Sine, Cosine and Exponential Integra1s, Vols

I and II, Federal Work Agency, Works Project Admin­istration for the City of New York.

(4) Musical, Flow of Homogeneous Fluids Through PorousMedia, Chap. X, Sec. 10.13, Eqn 2.

40

APPENDIX A-SUMMARY OF EQUATIONS CHART

_..- -_.,-- ..REF.

OIJANTIT'I'RANGE OF VALUES THE MORE IMPORTANT EQUATIONSEOUATlON

'AGEBOUNDAR'I' OF TIME.,) FOR

No. No. CONDl'I1ONS EVALUATEDWHIOI EON. IS VALlO * .1 EXPRESSED IN THE UNITS OF 'ART I ~I I10DIFIED FOR USE WITH PRACTICAL OIL.FlELD UNITS

, -$...... ,..,.a :.. a. I p,..... at & . ( r'f ue ) P:9.E.. .( Il~fpe)III 2 ..~ ...... An ...... p. P.... 411 ~l-~ P • P. + C.kh f. - I--(........u'" --

&1 1.+'" ~ l.+~y 2 .-----1 AI ".t.... Pw• P. - 11 n -N- P... P. - k log. ~,.....,.,.. .~ t1 l.+.l .(~) 't~)} Pw· P.-&{6Iog. ~-A.Eif~.~)+A.firD~tel}IX • SIoeIo .... ro • Aa ...... Pw·P·- 4lfk n""T-e'-kl. +e. k

.-.....- ww.., ro.;.., (L.o ......., All ••c.... "'Y ~ {I:-Io+.l ,{~)} &t l.~ ~~)}x • ...... ,........ f............ P.. ·P. - 41f n""T -e. - I. P.., - P. - k Glog. --;;;- A. fi I.odootwi......... clo........11

oooonolol V",,.... i'i%l; I +'" P.. ·P.- ~\og..~XI s P.. - P. - 2lf k In T......• ,ttl..... •t

~(·f~J (~J} P .11 ~{t·.(_Dr.r,uc.J_ (D,~rpCllXII .., ..... All va..... P -P'+ 4t k I:t-4kl -y I • + C.kh' kl Y klSlott- _a .. tho £ {t.+'Il (r.. f,uc ~ (ro' f)lC)} Pw • P. - & {61og.~ +A.y(~)-A.y(~nXIV • .......... ,1,11".1... P.. -P.- 41{ In~+Y 4k(I.+:Iil -y~

~"'., c...,.,. of • IIttil. ,"""'""' ..... An ••c.,. .,.,., ~{ t.+'" (~n ~{ 1+4t (Il"..rpcj)XV I......a. P.. - P. - 41f In T" + Y . • P... P.- c"kh 61og.T + A.y kl..

, clfftl.. ,..... * MJ; (~) p* p _~ (Dr.' fpC)XVI E.1r......w,laUe - P -P.- 1f Y 4kl. •• c"kh Y kl.

• c..................A1 '"""". (n) (': fpC) Ps • P. - ( tifh) + ( 1l't(~C.)XVI P. - P. - 41{ + 4 kl.

:t: Note Ihat nant: of the equttlont: which appl, to prcttvtt bulld~ win be .trlctly applicable while the tiled or the art~productlOft I, 11111 .lrntflcML

-l>o...

PROCEEDINGS THIRD WORLD PETROLEUM CONGRESS

APPENDIX B - LIST OF SYMBOLS

UNITSSYMBOL DEFINITION Part I Part II

Metric System I U.S. System

y

rr.

md

psipsipsi

cppsipsi

70.60ft

948.2

bbl/day

325-ra fraction

r62.6ft

vols/vol/psi3793

md

m ftm ft

hour hour

hour hourhour hour

hour hour

kg/em2

kg/em2

kg/an.:!

cpkg/em2

kg/em2

m'/day

43-83a fraction

21·91m

Ivols/vol/kg/em2 I

28jo II

I

I

I!!

secsecsec

em

a fraction

cpatmatm

emgmfccgmfcc

sec

vols/vol/atm

darcy

sec

cm

atmatmatm

cc of subsurfacevolume

cc of subsurfacevolume per sec.cc of subsurfacevolume per sec.

emem

AaBCo

pp.t

xx.

cDEi

eFfGhJoJ1kIn

IOg10pp

Po

A numerical constantDistance from well to faultA numerical constantShrinkage factor, tank volume

subsurface volumeCompressibilityA numerical constantThe exponential integral function (d. footnote

to page 2)The exponential constantA numerical constantPorosityA numerical constant IPay thicknessBessel function of the first kind of order zeroBessel function of the first kind of order unityPermeabilityLogarithm to base eLogarithm to base 10ViscosityPressure at distance r and time t IInitial reservoir pressure, or reservoir pressure

at moment of completion of wellp. Final closed-in static pressure in the wellp. Pressure in the well during build-upp. False extrapolated pressureQ. Total inflow over circle r = h

q Rate of production of well, assumed constant t Iqo. ~. q•• q. Well production rates associated with h, 12, t3

(see fig. I)Distance from well centre-lineRadius of external (assumed circular) reservoir

boundaryRadius of wellDensity of reservoir fluid at pressure pDensity of reservoir fluid at pressure p.Time (measured from time zero at moment of

completion of well)Value of to .corrected for non-constancy of q tValue of Jat moment of closing in the well tVarious times associated with qo. qll q., q.

(see fig. I)Time elapsed since closing in the wellAn independent variabler.2fpc4ktoAn independent variableme nQ root (in order of increasing magnitude)

of J1 (x) =0 1They-function, defined by y(u) =Ei(-u) + -e"

(See fig. 10) U

t For methods of correction when Q. has not been kept constant, see paragraph I, 3b.

42

Fig. 10. Graph of the function y(u).

BIA!Q to

8 .8

8 .6

5 .5

4 .4

3

:3 ~ .•.

2 .,2-

...u

tto.

.8J .08

0..-

- -- :1:.'- ..- ..

.e-i.oo -

.5-i.05

.4 .04

.:s .05

02 .02 __. - - -

. - -

.,+B A

~

"HUf~ljn, I

-.!y( ul = TI e-

tL+ E;( -1.1.)

200 300 "100 500 800

B