Section II, Preprint 7

CopYright 1931 by E. ]. Brill, Leiden, Netherlands. All rights reserved, including the right to translate or to reproduce this paper or parts thereof in any fo~

PRESSURE BUILD-UP IN WELLS BY

D. R. HORNER'"

Synopsis The report presents a method of analysis of the pressure build-up curve obtained from a closed-in well 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 of this curve. Mehods are also given for extrapolating the recorded pressures to infinite dosed-in time for the cases of I-a new well far· from. al.lY reservoir boundary II-a new well close to a fault, but far from any other boundary III-a well in a fiti'ite reservoir. Several illustrath'e examples are discussed. Resume Le rapport presente une methode d'analyse de la courbe de remontee de pression dans un puits ferme obtenue en representant la pression de fond en fonction du logarithme de to

t

.&, a etant Ie temps

ecouIe apres la fermeture et to Ia duree de la vie productive passee dupuits. II en ressort qu'il est possible dans tous les cas de determiner Ia permeabilite de Ia formation a par.tir de l'inclinaison de .

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

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

p.=Po- 4 qlL nk h

I

t o+& In -&-+y(uJ 6

l · .. (XV) f.. r.'flL e

.or 4 t•• k Thus we still have that, over the range of values of & which will normally be measured, a plot of p. against In t.

..-9.L~Ei(_r.lflLe)+~e- 4kT ( 4'1tkh

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

r. 1 fILe)

.

rSfILe)

lEl.-"4!Ct,

. . h f or convenIence we wnte were

hydrocarbon-filled reservoir pore volume compresslblhty culated to be

1.('

still be In t..

t

t

4:::h'

&=

0

&

U1

wijl be linear and, its slope will

However, linear extrapolation to

will indicate a false value for the final

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

SECTION II, l'REPRINT

p. = p.- 4~:h y(uJ

7

tion XVI for Y(Ul) and thus we may derive a value

," ,(XVI)

of

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

ul

r.1fp.c 4 t"

-= ---' k

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

o'

~

• • A

~ ~ 'II

~

~

/

I

•

'0

! ;

II

!

i

-

7/

• 5

1/

II

7f

I

I

I

j

!

I

I

I

• {

10

.. ~I

,I

i

I

·-i----··_- -1

--~- -----.. _-_.-

•

$

I

i . _---. +-

.

I

-.

I

r"

CurwI - ~~iaICI1SI1 or''lullll()n X/I eurrr:o:· AplJf'f1ximlllion""/llis rrporl, 'lJUlIlion XIII 80111 plDJIN /la-o III, "p,ei.1 us, 'b .10011

,s .~

10

(Po -.Pw '

I

I !

r"

to

I~h

Fig. 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

p.

where p

=

p.

qt.

1tr.2 fhc

= p.-

( qIL )

41tkh

. (r.2f C)f . . . .. . . .. (XVII) -:- 4k ILt. is the final static (i.e. fully built up)

closed in pressure, and then, substituting the known

qIL

values of Po. 41tkh an

dr.lflLc. 4 kt • In XVII we may

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

PART II-APPLICATIONS I-Introduction Part II of this report is intended to be largely complete in itself. The object of this part is, by the use of several examples, to illustrate the methods of applying the. more important equations developed in Part I. These equations are collected for easy reference in the Summary of Equations Chart, Appendix A; each equation is repeated in two forms namelv exactlv as derived in Part I and also as nlodified for ~se with practical oil-field units. All tne symbols used in the report are collected in a table '(Appendix B) to which reference should be made for the units which are to be used for the various quantities involved, and for the values of the numerical constants ·of conversion (A, B. D, F and G) to be used with the particular units employed: The examples chosen to illustrate the use of the methods here presented are all from wells in the Casabe field, Colombia, and have been selected from a batch of 66 pressure surveys in this field made between August, 1948, and April, 1950. Those which have been selected for inclusion in this report have been so chosen because they clearly emphasize certain points of particular interest; it must be emphasized that 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 61 which are not included. It is perhaps of interest to state that the oilbearing formations of the Casabe field consist of three series of lenticular multiple sands 'A', 'B' and 'e' of which the 'A' and 'B' sands are of Oligocene and the 'e' sand is of Eocene age. They are fine j?;rained to silty, rather argillaceous, and fairly well consolidated. The oil is heavy (200 -21 0 API) and viscous (40. cp ±). Some doubt exists as to the bubble point. It was first thought that. the oil in all sands was undersaturated, but it now appears that pressures in the 'A' sands at least are below t It is of interest to observe that this limiting form of equation XIV may be expressed thus (Present Reservoir Pressure) (Initial Pressure) (Total Volume Withdrawn) X I iJ'ch • (Total ReservOir Pore Volume) Compressibility" I IS precisely what we should expect from elementary considerations.

=

32

the bubble point. This, howe\'er, does not appear to affect the build-up curves. 2-Examples of the Pressure Build-up in a New Well The earlier theory developed in Part 1" (I, 3) is designed to apply to a single well in an infinite field. Such a condition never obtains, of course, but it is further pointed out (I, 3d) that the case of a lIevJ well in a finite field is similar so long as the total withdrawal from the well has been kept smal!. It is difficult to lay down a criterion for the order of this smallness-practical applications seem to indicate that a normal well may be allowed to produce for a period of weeks or even of several months and it will still obey the "infinite reservoir" theory. Thus we may apply the infinite reservoir theorY to the following case. CB-16I was completed to the 'A' sands on 7th February 19."0, and it was closed in from 16th February to 8th March while continual bottom hole pressure meas~rements were bein~ taken. As will be app:trent from the analysis of the results, it was not really necessary to leave the well closed in for such a long- period in order to obtain adequate. data, althou~h it is interesting- to observe how closely the pressure in the well followed the predicted course over such a long time. This beinJ1: a new well, we apply the theory for a single well in an infinite reservoir, as has been already explained. Reference to the Summary of Equations Chart, Appendix A, shows that the eouation for the build-up in this case is number V, which in practical 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 which they are to be measured. The. U.S. system of units being in use. in Casabe, we must use the value of G given in the tabulation as pertaining to this system, namely 162.6. Thus equation Va is modified to I62.6q/L. 1.+& (Vb

p,,=p.

C.kh logle-&- ...

)

which is thus the equation which we must attempt

SECTION II, PREPRINT

to fit to the experimental data. As the method of ' IoglO -8ot.+8o agams . t p., · consIsts • 0fplottmg anaIYSIS we firstly require a value for to. As has been previously explained (I. 3b) the· eql.;lation V (or its modified form, Vb) assumes that q. the rate of production. has remained constant for the whole life of the well (to). This not being in fact true. the method of correction is to take the last available production 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 obtained by dividing the total cumulative production of the well (in this case. 5847 bbl) by this last production rate. Hence for t. in Vb abo\'e we substitute Total cum. prod. 5847 d tI = = -- ays = 219 h ours

Last prod. rate

641

Thus the build-up equation Vb may be modified by the insertion of this value of t l for to' This gives

p.

21 9 + & q fL 162.6 Co k h loglO & •• (Vc)

= po -

The value of q = 641 bblJday could now be inbut it is perhaps more convenient to leave this until a later stage. The method of treating the experimentally obtained pressure data may be most easily explained by reproducing the measured values exactly as they were 'reported from well CB-16I. ~rted.

Pressure

p.

c.1.

time

&

(psig)

(hours)

1192

19 25 31 37 43 49 55 61 67 73 79 85 91 97 103 109 U5 121 127 133 139 145 151 157 477

1200

1206 1212 1216

1220 1223 1227 1230 1232 1235 1236 1237 1239 1241 1242 1241 1243 1244 1245 1247 1249 1249 1250 1267

21 9

+ & &

12·53 9.760 8.065 6·\)[9 6.093 5469 4.982 4·590 .4-269 4-000 3.772 3·576 3407 3·258 3.126 3·00\) 2·904 2.810 2·724 2.647 2.576 2·510 2450 2·395 1459

IOglO

21 9+&

&

1.0g80

.9894

.go66

.8400 ·7848 .7379 .6974 .6618 .6303 .6021

·5766 .5534 ·5324 .5 130 4950 4784 4630 4487 4352 4228 410\) 03997 03892 03793 .1641

7

Columns I and 2 are derived directly from the bottom hole pressure readings. columns 3 and 4 are calculated from column 2. and finally the values of P. from column 1 are plotted against the corres. pondlng values of 10giO

21 9

+&

&

from column 4

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

+

. I manner. but to pot I IoglO 2"19& & horJzon. ventlona tally from right to left. i.e. with the zero on the right hand side as has been done in fig. 5. as this gives a' more vivid impression of rising pressure. If agreement is to be obtained with the theory. and thus with the derived equation Vc. the points when so plotted should fall on a straight line, excepting only possibly the points corresponding to small & when the effects of the after-production (see footnote on page 3) may be still felt. As can be seen from fig. 5. the accuracy with which these experimental points do in fact plot on a straight line in this case-and particularly the very last point which represents nearly 20 days closed inis really quite remarkable. The interpretation of this figure 5 is simple. Firstly we may deduce a value for Po. which is done by extending the straight line plot to the point corresponding to 10giO 219: - &

=0

and reading the

correspondillg pressure which gives in this case a value for Po of 1280 psig. Although we have defined Po to be the initial reservoir pressure. this must be interpreted somewh~t widely. In this case, for example. the well considered was drilled as an· infilling well into an already heavily drilled field, and so the pressure Po = 1280 psig must be interpreted as the reservoir' pressure at the well at the moment of its cOIn/letion. In addition to this pressure determination we may determine the permeability k from the slope of the 'straight line. This can perhaps best be done by selecting two arbitrary and fairly widely separated points A and B on the line (fig: 5). The pressure difference between A and B is 1272-n82 = 90 pSi, and the corresponding difference in 10giO

21

9&+ &

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

33

PROCEEDINGS THIRD WORLD PETROLEUM CONGRESS

o

-x /250

I I

./

VI

I V

Pw _psig

I I

1200

I

I

IISO

vr

1

I

-r-!I , I

Vi

I I

I

II

I 1/. ,

1.5

%

I

I

Fig, (VI

I

to 219.,.7/

'" r:r

/09-

,

I

IC8."2.5 o

0

0.5

I I OD~:I~~s

/300

I

./

o.

I

/'

ft.

1/00

/' .Y

V

t.5

V

I Y

~/

V

/

V

fig. (vii

1.0 "

0.5 7U .. z)

.O$.;,~ Fig. 5 and 6. Observed presslJre bilild-up curves in new wells.

34

~

V

/" 4'

V

o

SECTION II, PREPRINT

Difference in pressure

Slope =

Difference in 10gi0 21;9~t .&

;:=:

and

thi~

10gi0

219

9° =82 1.1

slope we now equate to the coefficient of

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

have 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 1 From PVT data Shrmkage factor Co = .93 ~ Pay thickness h = 349 ft From Electric log and if w~ "Substitute these values of p., Co and h in the above equation ;XVIII we have 8 ·93 X k X 349 _ - 1·9 641 X4 0 that is

.0127 k =

l.gB

I.gB = I56mD . 012 7 Thus we have derived a value of 1280 psig for the reservoir pressure and 156 rnd fQr the permeability, both of which are in good agreement with other data for the Casabe field. Figure 6 shows an example of the build-up of another new well. This requires no further comment, except to note that it is included only because it is an exceptionally good example of a very long period closed in. The data relevant to this example are: Well No CB-125, A sands q = 280 bbl[day t. = 764 hours (31.8 days) Max. C. I. time 2847.7 hours (n8.7 days) k (from the slope of the line) = 88 md or

7

3-Examples of the Pressure Build-up in a Well near a Linear Barrier Fault The theory which has been developed for a linear barrier fault is strictly only applicable to a well in an otherwise infinrte reservoir. However, we may approximate to this condition by a new well close to a fault and considera~ly farther from any other barrier. Such a well is CB-123. completed to the Csands at the beginning of September 1949; it was closed in for test from 4th November 1949 to 5th January 1950. Reference to Appendix A shows that the relevant equations are numbers IX, X and XI, which, subsituting the values of A, D, F and G appropriate to U.S. units from Appendix B, become p.. =p.- C::h {I62.610gI0 t.;.&

. ( -7o.6oEI

+70.60Ei(_3793ka;fILC)

I .. (IXa)

which is· valid for all ranges of .&, and which may be approximated to by qIL \ t. +.& p.. = p. - C. kh l 162.6 loglo -.&-

-70.60Ei(

k =

The points corresponding to very large cl.osed-in times do not, of course, plot with the same accuracy on the straight line. This is because the earlier group of 17 points were obtained from one (or perhaps two) runs of the pressure gauge, while the; last 8 points are eight spot readings taken at widely spaced values of .&.

3793 allfp.c) k(t.+.&)

I ... (Xa)

3793ka~fILC)

for all save very large values of &, and by p..

= P. -

q P. t. + .& (XI) 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, the two approximations Xa and XIa are used. Firstly we require a value of to, which is derived as for the previous example. We use q ='275 bbl[day and to = 1353 hours. The value of q. as before, we do not yet substitute. However, we insert the value' of to in the two approximate equations which become 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)

35

til CI'I

) 23001

CB-/23 ('

2200) - _

en -./ V ~~

Oos.~I"""SSuNS

'0

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

IX IJ

.1/" ~pe

2100)

-

-

-

-

-

- -- - ' - - -- . - -- _'- . - --I~ "

2000

.~.~ ~

IPw-p& 1900

1800\

1700)

1000

~

---

~

~

--

2.5

-----

~

q::;- .....:::::::

--

::;;:::::0

1:0'"

.~ ~

~

./

&

~ ::-~ -

i/ ./

.

---'-

1.5

109",·

iJ v

-

1.0

15s

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

®

"

-- ---

" ' -

/

- - - -- - - - -- 2.0

I~'

:/ ope

,;Z5

~I-"

._,

-

-

-

-- -

0.5

---

l,-) ~

o

SECTION II, PREPRINT

for,very: large &. Just as in the previous example, we plot p .. ~gainst 10g10 1 353; ; &. The resulting plot is

7

c -= 5.1 X IO·e volslvollPsi k=270 md the above equation XIX becomes

shown in fig. 7. Theoretical requirements are that xatx .25X40X5· 1 Xlo·e) = 2. the early part of the curve be a straight line (repre- -Ei ('_3793 1353 X270 53 sented by equation Xb) and that the latter part be also a straight line (according to equation XIb) of i.e. -Ei (-.530 X 10-& all) = 2.53 twice the stope of the earty part. These two straight and reference to a table of Ei-functions (2, 3) shows lines should be Joined by a transition zone. that, for this value of -Ei (-.530 X 10-8 a 2 ) the There are, indeed, two well defined straight lines I quantity and II, but their slopes are respectively 330 and 125, .530 X 10-6 a 2 = .0468 i.e. in a ratio of 2.64: I, instead pf 2: I. This may be whence a2 = 8.83 X 104 explained, however, by the fact that the particular or a = 297 feet well in question is subjected to the influence of two However, this figure is not in·agreement with the intersecting faults, the effect of which may be. present (rather obscure) subsurface picture which theoretically shown to be to increase the ratio or the places the fault at about .1100 feet from the well. two slopes. Instead of proceeding in the manner .just detailed, As Before, a value of Po may be obtained by e.'Ctra- however, we may modify out approach thus. polating the. line I to give Po = 2263 psig. Firstly, we accept the' possibility that the stopes The value of k is obtained from the slope of the of the lines I and II (figure 7) may not be exactly straight line II, and proceeds just ·as before: in the ratio of 2: I. If we suppose this ratio to be b: I a first approximation to the build-up equation 12 162.6 C:th = slope of liile II = 5 may be obtained by modifying equation IX to read (in practical units) and substituting the k-nown values q = 275 bblfday (Ref. page II) q!L t.+.& p.. =P.-C.kh G log10-.&p. = 40 cp 1from PVT data Co =·93 ~ . ( Da'f!LC) -(b-l}AEl -k(t +'&) h = 57 ft (Electric log) o we have (b-l) AEi (_ D ~;P.C)} (rXb) 1:62.6 275 X 4 0 = 125 ·93XkX57 The problem of fitting this equation to the ob00 i.e. 33,7 = 125 served points can then be performed thus. k An approximate value of "b" is derived from the whence k = 270 md. ratio of the two slopes, a value of Po is obtained by We may also estimate the distance of the fault the extrapolation of the last part of the build-up curve and a value of "a" is obtained just as previously from the well by the following method. described Le. by equating the -Ei function to We equate the function -Ei (_3793al~C) . 1353 at the point of intersection of the 2.3 X IOg10

I

+

to the value of 2.3 logto I35i,+.&

t

tot.&

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

at

_Ei(_3793 ;:C) = 2.3XI.I = 2.53 .. (XIX) 1353 Now·if we substitute the known values f = .25 ' P.-=40cP

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

two straight lines. It is convenient that, given this point 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'! (for in practice .& must usually be very large indeed before line I is thoroughly established, and so it is possible for some of the later points in the transition zone, to be mistaken for points on this line I) it may not now be taken as definite that the b~t possible yalues of a, b and Po have now ~een chosen, although it may be ~pected that theywitI not differ too widely. from the final values. Thus the equation IXb is now plotted to se.e how nearly the calculated curve fits the observed pointsi.

37

PROCEEDINqS THIRD WORLD PETROLEUM CONGRESS

! !

I

I ---r--

! ! !T

!

I I I 1300

fAI lilY

1200

I

V 1/ Vi@ I--I--+--+--I--Ir--+-,I--;-+/'--I---1---+---t-f+I/""7"1--+-+-; V I I j--:- i IY I I

I

1I00

II I L\~ I