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JON BENEDICTECONOMIC ANALYSTXORTEX GAS & OIL COWANY
HOUSTON, TEXAS
T RE MATHEMATICS
DECLINE CURVES
PREPAAEDY
Usws
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T h e p a p e r T h e Ma t h e ma t i c s o f De e ?i n e Cu r v e s h a s b e e n p r e p a r e d a s a n i nd e p t h a n a l y s i s o f t h e d e v e l o p me n t o f d e c l f n e c u r v e s t h a t q u a n t i f yp e t r o l e u m p r o d u c t i o n t h e v i t a l c o n s t i t u e n t t o t h e e c mmf c e v a l u a t f o no f p e t r o l e u m v e n t u r e s . T h e p a p e r i l 1 u s t r a t e s ma t h e ma t i c a l1y h o w t h r e et y p e s o f e q u a t i o n s o r i g i n a t e f r o m t h e r e l a t i o n s h i p o f p r o d u c t i o n ( Q) a n dt i me ( t ) s h o wn i n e q u a t f o n 1 o n page 4 . T h e e x p o n e n t i a l e q u a t i o n fsd e v e l o p e d f r o m t i t s b a s i c f dea a n d then e x p a n d e d f n t o t h e h y p e r b o l I c .F i n a l l y , t h e h a n a o n f c wh f c h f s a u n f q u e c a s a o f the h y p e r b o l f c i sp r o d u c e d . i. . 1 JT h i s wr f t i n g s h o u l d b e o f v a l u e t o t h o s e f n t e r e s t e d p e r s o n s t h a t h a v e anesd t o h o w t h e b a s i c f d e a o r o r i g f n f r m wh i c h t h e s e f n d u s t ~ -s t a n d a r d e q u a t f o n s h a v e e v o l v e d . A l t h o u g h i n f t f a l I y t h e y ma y a p p e a r t ob e s o i a e wh a t COI Z P 1x , t h e r e a d e r wf l 1 f f n d t h a t t h e e q u a t ~ o n s a r e v e r yb a s f $ . H are p mi f c a t e d o n s o u n d r e a s o n f n g and pmvfde the moste f ~ f c i e n t p r a a f s e f r o m wh i c h t a e v a l u a t e f u t u r e productf o n .P r o g r a mme r s , pmgmm u s e r s , engineers, g e o l o g i s t s , e c o n o mi s t s , c o ~ o r a t aP I a n n e r s a n d o t h e r s wf 1 1 f f n d t h a t t h f s d o c me n t a t i o n provides e x h a u s t i v ed e t a f l t h a t i s s e l d o m a v a f l a b l e for ffttfng, e x t r a p o l a t i n g a n di n t e r p r e t i n g o i l a n d g a s p r o d u c t i o n . A l t h o u g h a c t u a l p r o d u c t i o n p a t t e r n sa r e e x p e c t e d to be trregular, the n e t h e ma t i c s f o r t h e i r d e s c m* p t i o n f ss h o wn t o b e v e r y p r e c f s e . T h i s p r e c i s i o n e n a b l e s t h e u s e r t o e x p l o f tt i n e - s a t i n g b e n e f fts such as d e t e mf n f n g t h e 1 ff e o f a project given ttsd e c l i n e p e r c e n t , f n i t i a l p r o d u c t i o n , r e c o v e r a b l e r e s e r v e s a n d c u r v e & / p e .I t s h o u l d b e n o t e d t h a t t h e final e q u a t i o n s wi t h f n t h f s p a p e r a r ec o mp u t a t i o n a l a n d a r e t h e r e f o r e r e a df l y a d a p t a b l e to both c o mp u t e rp r o g r a m a n d p r o g mx n a a b le c a l c u l a t o r s .
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., .
T & c o n t e n t o f t h i s p a p e r c o u l d b e s e p a r a t e d i n t o a t h r e e - p a r t s e r i e s .p a r t s o n e , t w o a n d t h r e e Wu l d b e e x p o n e n t i a l , h n e r b o l i C a n d h a mo n i c ~r e s p e c t i v e l y .
Jon BenedictEconomic A n a l y s t
... i .8
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No v e mb e r 1 , 1 9 8 0. .
THE MATHEMATICS
DECLINE CURVES
Production decline curve equations as presented inexploration literature very often appear in some finalform without the benefit of theis derivations. Therelevancy of the authors as~ertions may be restricted,.1 .~to a single application; however, to realize the maximumpotential of these equations, the mathematicti reasoningbehind them should be apparent to the use%. This oppor-tunity to follow the sequentialmathematical constructionof a particular idea gives the reader greaterof the model and aUows him te manipulateacc~odateThis paperproduction
and, in so
his own ends.the
expands the mathematical reasoningfunctions:
I. EXPONENTIALII. HYPERBOLICIII. HARMONIC
understandingequations to
behind tkee
doing, derives five variations common to each(decline function, cumulative function, initial pmductiontlife of project and periodic function). In addition, AeEXPONENTIAL includes a method of determining ~, the declinerate.
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3ecline Curvest ? OV e . I I L b C 1 , 1 9 8 0q Page 2
.
J.J. Arps ~eviewed a chronology of publications from authorsthat dealt with decline c-e analysis.x Among them wereR.B. Johnson and A.L. Bollens who in 1927 introduced theloss-ratiomethod of extrapolating oil-well decline curves.Their loss-ratio concept isper unit of time divided by
defined as thethe difference
production ratein t h a t production
rate from,.thatof the preceding Sime period. This is iUu-st=ated in Table 1.
1J.J. -S, Analysis of Decline Curves, PETROLEUMTECHNOLOGY, T.P. 1 7 5 8 (September,1944)G
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Decline CurvesNovember 1, 1980Fage 4.
I* EXPONENTIALA. The Exponential Decline Function
Suppose a quantity Q ch-g@S at a rate which atany instant of time t is proportional to theamount of Q present at that instant- This isexpressed by the
where l/a is theamount of itq u a n t i t y a t
prevail for
differential equation
4LAQ dt u (EQ. 1)proportionalityconstant. If the
present a t t = O (time zero) is Q* (thetime zero), the initial conditionsQ=QO atevaluating the
C, which enters when EQ. 1with declining productionof the immediate probl~equation
constant of integration,is integrated.the mathematical statement
consists of the differential
(EQ. 2 )
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De c l i n e Cme sNo v e mb e g 1 , 1980Page 5q
with the i n i t i a l condition
Q-Q o att=ooThe problem is to determine a statement fo% Q.Hence, beginning with EQ. 2, rearrange andintegrate
,..!
(EQ. 3)nQ+t+c
During indefinite integration the a s b i t mr y constantC, which always enters when a differential equationis integrated, pxoduces a family of parallel cuzvesas C is assigned v a x i o u s values. This causesshifting of the cume which is illustrated in
vezticlefigure 1.
.
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...7
Decline Cu~esNovember 1, 1980?age 6
Q
+t
\\
i
.
FIGURE 1
Therefore, imposing an initial condition ordesignating any point (te, Qa) est~lishes thatthere is one and only one cuzve of the familythat passes through tiaatparticular point.It folLows that specifying Q = QO at t = Osingularly defines the value of C. Introducingthis idea and solving for C in EQ. 3 produces~
lnQlnQOlnQ~
1t +a-+(0)
c
+Cc.
*
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Decline C-esNovember 1, 1 9 8 0P a g e 7.
.
Substituting this value of C back into EQ. 3 yieldshQ - .& + lnQClnQ - lnQO = ~
,...,
(EQ. 4)
This isdecline
the expression that describes an exponentialcurve 0= negative expmential funotion, where
QQ.
atin
A quantity present at the tThat(Q -
quantity present atintercept).
Proportionality constantdecline pexcent.Loss - ratio.TimeLog to Base e
timet=O
or exponential
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De c l i n e Cu r v e sJ { o v e mb e s 1 , 1980?age 0.q
B. The Exponential Cumulative Function: QcThe exponential cumulative fwction~ Qct is fo~d byintegrating EQ. 4 subject to the initial cendition
t=to=()
This in effect specifies C a s t h e Q-Intercept of,.thecurve at a ttie whqm cumulative production equalsze%o~ or Qc = 0. It follows that Srom Ea. 4,
~Qc = fQ,dt= fQ@e dt.
Relying on the following standa%dfedu = e + C
where
u=+du = -+dt = -adu
fem for expediency,
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.Decline Curves:lovember1, 1980Page 9
t h e n ,~
Q= = fQdt = fQOe dt = -aQ6feudut
Q= = Qt = aQoeY +C(0)
o = Q(o) = -@oe = + C
(EQ. 5)
Substituting tlxisvalue of C back&
Q= = -sQoe Q + UQo
into EQ. 5 yields
(EQ. 6)
which is the exp~ession for the exponential cumu-lative function. In effect, this function measuresthe area bounded by the t-axis and the exponentialcurve~ between tO = O and t = tl. Here, area maytake the meaning of r e c o v e r a b l e reserves as illustratedin Figure 2. By stipulating that Q= = R, where Rrepresents recoverable reserves and L sepxesents theLife of a project, EQ. 6 abovetakes the formof EQ. 7.
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,. .
.
Decline Cues~ovember 1, 1980Page 10 .
starting with EC).4gration by imposing
which is
and performing definite inte-the limits t =o*QAe u
R
R
R
u
&&Qdt = $~QOe dt
-k ~aQO{-e a + C - (-e ai qJ~
aQO{l - e 3
o
+
as
to t~ = L,
(see Figuse
(EQ.
2)
7)
exactly the same equation EQ. 6 with theexception that L has zeplaced t. Notice that sincelimits wese imposed to find EQ. 7, the constant afintegration cancels. This isintegration.
c. Initial Production: Q.
as definite
By rearranging EQ. 7, the initial productionbeginning of decline is
QO=RAa(l-e a, (SQ. 8)
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Decline CurvesNovember 1, 1980Page 11
D. Life of the Project: LThetheEQ.
life ofdecline7
L-ea
L
the project from the beginning ofto the economic limit, L, is, from
aQo{l -L
L-ea
1-+ o
ln{l -
-ain{l
L--e a}
J
Rq}
.7 $
(EQ. 9)
.. .
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Decline CurvesXovember 1, 1980Page 12.
.
Q
f~Qdt =
4 ;
,Area under curvebetween zero and. Lt.-
{Curve
IL t
FIGURE 2RECOVE~LE RESERVES
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Decline Cu2vesNovember 1, 1980Page 13..
E. Exponential Periodic Function: Qi
The periodic (monthly,annual~may be genesated by taking the
etc.) production, Qi,difference in cumulative
p~oduction between successive time periods, ti-l andtip hy applying E Q. 6 whe%e Qi represents the quantityproduced in the ith time pe~iod. Instead of e m p l o y t i gt h i s method, however, EQ. 10, below, would provide amore direct calculation*OS Qi.Proceeding from the expression for
+:
exponential decline,EQ. 4,Again,during
i n t e g %a t e with r e s p e c t to t from t+.-,to t,.imposing Mmits implies definite i;&ati&which the constant of integration cancels.Therefore, progressing through the integration
brings aboutprocess
Qi =
Qi =
.
*b=Qe To
ti ~= r* Q-e adt
)
(&Q. 10)
Q iti-~ Qdt
k. z. t. 1-1aQo{-e = - (-e = )
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. 12eclineCurvesNovember 1, 1980, Page 14
Q
Q.
(
\ / Qi
.
J .
ti ~ .AreabetweenQ e dt ti-l and i= ti-~ O
FXCXIRE 3PERIODIC PRODUCTION
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Decline CurvesNovember 1, 1980Page 15 q
which is the expression for the periodic functionfor exponential decline between the limits ti-land ti. Qi is depictsd in Figure 3.
F. 1The Exponential Decline Rate: ~
As an addendum to the foregoing material, thefollowing problem will emphasize the essenceof understanding the intrinsic value,;un~erlyingJ*the exponential equation.
1Suppose it becomes necessary to solve foz ~, theproportic~ality constant, or the exponential decliner a t e . E Q. 7 wo u l d by necessity be the base equationfsom which to work. The reasen is that * is forcedto be qique to the recoverable reserves R. EQ. 7,however, poses a cliffculty. Since a appears in theexponent and is a multiplier at the same time in thefunctional relation of EQ. 7, it would be difficultif not impossible to solve for explicity. EQ. 7 isstated as
-...&R = GQQ{l - e a}
q
Instead of using a time-consumingitexative process1to converge on the value of ~, the following method
i s proposed.
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~ecline CumresNovember 1, 1 9 8 0Page 1 6..
Assume that reserves in place, T, and recoverablereserves, R, have been predetermined. Define Kas the percent of reserves in place that arerecoverable. The relationship is described as
R = KT.
Furthermore, as shown in figure 4, define reservesin place, T, as J. ,; :
&T = \~QOe dtO
Although this integralmeasuzes an infinite areain theory, fos the immediate psobLem it has beendefined to equal T when extended to its Mmit.This importantcauses ~ to bean exponential
concept is interesting because itunique in describing the decline offunction that
reserves in place. The areaafter integration is
T = aQO.
when integrated equalsdescsibed by T above
This may be seen in the fo120wing calculations:
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Decline CuevesNovember 1, 1980. Page 1 7
wheseo--
ea=l
and
1= infinitelylarge number.=0
.
90- 1ea= 9
ze \
In other words,
aQoieaQo{laQ*
total
- o}
reserves in place, T, hasbeen explained in terms of t he initial production
1rate, QO, and the decline rate, ; or loss ratio a.R has alxeady been defined in EQ. 7 where Rarea of re~overable reserves. From this itthat
is thefollows
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Qecline Curves qNovember 1, 1980Page 18
Q
o
FIGURE 4RECOVERABLE RESERVES
RESEhVES IN PLACE
I
Where:Q
t
k Recoverable
u: Reserves in
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Decline CurvesNovember 1, 1980Page 19.
RK
= KTR==&
and by substitution, -&
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~Decline CurvesNovember 1, 1900Page 20..
II. HYPERBOLICA. The Hyperbolic Decline Function
From Table 1, the loss-ratio, a, is defined asthe production rate per unit of time dividedby the difference in that production r a t e fromthat of the preceding time period. Translatedmathematically, J. ;
If u is a positive constan~, then theis exponen~ial. The reasoa becauseis identical to EQ. 2 which provesIf, however, the first differences6, are constant, the expression ishyperbolic type shown below
where 8This is
Qd@j7@= -sdt ..is the first derivative ofshown in Table 2.
toof
(EQ. 15)expressionEQ. 15be exponential.the loss-ratio,
that of the
(EQ. 16)the loss-ratio.
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Decline Curvesq November 1, 1980Page 21
q
TABLE 2MONTH
JANmJLYJANJULYJANmYJ.AN.mYJANnLYJANmYJAN~LYJANJULYJANJULYJANJULYJANJULYJANJULY
I
193719371938193819391939194029401941194119421942194319431944194419451945194619461947194719481948
MONTHLYPRODUCTIONRATEQ
28,20015,6809;7006 , 6 3 54 , 7 7 53 , 6 2 82 , 8 5 02 , 3 0 01 , 9 0 51,6101,3651,1771,027904802717644582529483442406375347
;0SSIN>RODUCTIONWTE DURING; MONTHS
- 1 2 , 5 2 0- . 5 , V 9 8 0~ 3 , 0 6 5- 1 , 8 6 0- 1 , 1 4 7778550395295245188150123102857362534641363128
... I
q
LOSS RATIO3N MONT=YBASISL=6AQ
- 7 . 5 2T 9 + 7 2- 1 2 . 9 7- 1 5 . 3 9- 1 8 . 9 6- 2 1 . 9 6- 2 5 . 0 8- 2 8 . 9 5- 3 2 . 7 6- 3 4 . 4 3- 3 6 . 9 7- 4 1 . 1 5-44.20-47.2S-50.30-53.35-S6.40-59.45-62.50-65.55-68.60-71.65-74.70
FIRSTDERIVATIVEOF LOSSRATIO
-0.37-oq 54-0.40-0.59-0.50-0.52-0.64-0.63-0.28-0.42-0.70-0.508-0.508-0.508-0.508-0.508-0.508-0.508-0.508-0.508-0.508-0.508
Table taken from Analysis of Decline Cues by J.J. Arps,T.P. 1758, Sept., 1944
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..,Decline CurvesNovember 1, 1980 .P a g e 2 2.
AfteZ equating B to the change in the loss-ratio,the problem once more becomes one of finding arepresentation of Q. IntegrationofEQ. 16 leadsto
QamEt3t+~ q (EQ.17)J. d
InterpretingC, the cunstant of integration,find
att=O,Z&Z.with EQ. 17 n
=Cbut from EQ. 2 and EQ. 15
& = -a.Therefore, ....,
whese aO is translated a s positive constantrepre==nting the loss-ratio a t t = O or?graphically, the Q - intercept fos EQ. 17.
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Decline CuriesNovember 1, 1980Page 23q
substituting this value back
.
& = -Ot -a. qrearranging,
%% = -St -aoQ -$t aOm=~ J .
~=- a tQ .FinaLly, a second integrationdifferential will spec i fy Q:
q
q
into EQ. 17 yields
the
Substituting the above e q u a t i o n i n a
above
standardform in terms of u and du, and realizing thatsince the following formulation
.fq~lnu+cu
where
holds,
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Dec l ine CurvesNovember 1, 1980Page 24
then:
lnQ = -#nu + C
=1-#n(l i=$%+ C0
.
,; (EQ. 18)
The constant of integration,C, is evaluatedsubject te the initial conditions
Q= QOatt=O
t o yield
= -+(l) + C..,.
=O+c
= CSubstituting lnQO = C back into EQ. 181 Q isdetermined:
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:Decline CurvesNovember 1, 1980Page 25.
lnQ - lnQ~ = -lLn(l + ~)oin(+) = -~~n(l + +)o 0
1--0&al ;E**Which is the expression for the herbolic declinefunction.
B. The Hyperbol ic Cumulative FunctionThe h~erbolic cumulative function Q. is obtained
k
by integrating EQ. 19,
1Q = 00(1 + $)-Fo 1
Q= = tQdt = fQO(1 + +).Fdt.oBy setting the above equation in standard fonuin terms of u and du, the f~liowing formula
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Decline Curves Novem ber 1, 1 9 8 0Page 26.
where
.
may be used to evaluate Q= in the followingJ ,: ~.*
UOQOQc=Qt= ~fundu
manner:
, (EQ. 20)
The constant of integration,C, is evaluatedsukject to the initial.condition
t=o...
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Decline CurvesNovember 1, L99UPage 27q
Therefore:
q
UOQC,()= -(1) + c$(1 - ~)UOQOc=-
0(2 - + q
Substituting the value of C.back into.~EQ.20, yields
aoQo +Q= = (1 + +) aoQo3(1 - 13) o - 8(1-+)(EQ. 21)
%*hichis the expression for the hyperbolic cumulativefunction.By the same reasoning that was used for the exponentialcumulative function, designate 50Z the hyperbolic thatQ= = R where R is the symbol fos recoverable reservesand set limits for t = O to t= L which constrains &hearea under the hyperbolic curve to the Life (L) of theproject. By starting with &Q. 19 and perfoming definiteintegration,
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Decline Curves qNovember 1, 1980q Page 28
R=
and reviewing
1--6QO(l + ~)o
the a b o v e equation in terms of u and du,
where
U=(L+ +)odu = *to
dt = ~u
then,
1aQ 1-*~= 00 (1++) 1:!3(l- +) 0 l-~aoQo ,R=m-=Tf(l+~ -l}. (EQ. 22)
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.
Recline CurvesNovember 1, 1980Page 29.
This duplicatesOf EQ. 22, t iSEq. 22 measures
EQ. UL, the
except thatl i f e of the
recoverab le
c . Initial ProductionQ. may beQ. is theu.-
R
Q.
D. Life of
segregatedvalue of Q
i
by the.att=
+
.$
1g,aoR(6 - 1)
.
in the caseproject.
hyperbolically.
rearrangement of EQ.o :
22q
UOQO= m)
a R(3 - 1)~{ (~8 00
1-*8LUo{(l + &o Q 1} . (EQ. 23)Project
. 1}
toin
ThetheEQ.
life of the project from beginning of declineeconomic220 EQ.
limit is detemined by solving for L22 is used here because recoverable
reserves, R,
R
L
expressed in terms of Life, L.
[(1+ o99 -1+ 1) -1) q (EQ. 24)
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Decline Cusves .:{ovember1, 1980.Page 3!I
E. The H~exbolic Periodic FundtienAS in the exponential case, hyperbolic periodicproduction, Qi# may be generated bydifference in cumulative production
t*ing thebetween the
time periods ti-l ~d ti wi* the fid of EQG 21?the cumulative function. Iastead of engaging thismethodr however? EQ. 25 should provide a direct
i Jcalculation foe Q:.*Beginning with the expression for h~erbolicdecline, EQ. 19, integrate with respect to tusing definite integration, thus eliminatingthe constant of integration,C. Ewoeeeding,
1--Q= Qo(l+# 0
1Ati 7Q* = f:i m)Q(l+1 Qdk = ti-L o ai-1 o
in &ems of u and duti ~ ~n+l t.ti-~ ladu=n+lti-~ ,n$l
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Decline Cu~esNovember 1, 1980 ~Page 3 1.
where
then
StU=(l+T)o
d~ a -@o
491UGQO (10(1- ~)
q
~oQo ti~ r undui-l
1001=ti-~l+ l-f
UOQO BtiQi= ~r{(l + ~) $ti-l-(1+ ) }
o a o(EQ. 25)
the expression for the periodic function for
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Decline CurvesNovember 1, 1980Page 32..
III. HARMONICA. The Harmonic Decline Function
By postulating that 6 = 1 in EQ. 19 the Harmonicfunction materializes
1Q=o(l+& -tCJo... i
+(1) t)=Qo (l+=o
(EQ. 26)
which is the expression for the Harmonic Decline.B. The Harmonic Cumulative Function
The cumulative function may be realized by theindefinite integration method beginning with EQ. 26:
Q= = fQdt Q.1~(1+ t q+)o
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.ecline CurvesNovember 1, 1980Page 33.
Again, Performing the above integration using u and duas surrogates to expedite the procedure via therelationship,
whese
,*
Then it follows that,
duQ= = Qt = aOQOf~
(EQ. 27)
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Decline Curves. N& mn b e r 1 , 1 9 8 0Page 34q
By the initial condition
t = o
the constant of integration becomes
Q= = Q(o) = u Q In(l + +)+C000..,. i .J s
O = aoQo~(l + 0) + c
= aoQoh(l) = C
= aoQo(0) + C
C=()
substituting C = O back into EQ. 27\
Q= = aoQoln(l + ~) +oI Q= = @oQo~n(l + ~)o I
which is the expressionFunction.
the
o
Harmonic
.
(EQ. 28)
Cumulative
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Decline CurvesNovember 18 1980Page 35..
*
As before, in Exponential and Hyperbolic, by declaringthat Q= = R, subjecting EQ. 26 to the limits t = O tot = L, perfo~ng definite integration,
Q.Q= 1++ o
Lao
and USiIlg the standard
~Ldu~~ = lnu[:
form with u and du,
wheretu=l+ a0
du = +o
dt = aodu
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.Decline C-esNovember 1, 1 9 8 0Page 3 6
The following is revealed:
.
.,.1
This isR as Q=
= aoQo\n(l + +/ - aoQoln(l + +).J o 0
= t zoQoln(l + +1-0o(EQ. 2 9 )
a duplication of EQ. 28, except for portrayingand L as t. In othes words, this relationship
states the Recoverablec * Initial Production
Reserves (R) in
The in.itiaLrearranging
productionEQ. 2 9 .
rate, Q.t
.
Q. = RUoln(l + +)o
terms of
derived
Life
by
(L).
9
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P
D. Life of the !?%ojectThe l i fe of t he project
Decline CurvesNovember 1, 1980P a g e 3 7
from the beginningdetermined by solving
,,.
1 +~=u oL=
* L_c)o
q
of the29.in EQ.
13 q
1.
(EQ. 32)E. The Harmonic Pe%iodic Function
theor the same reasoning that was established foederivation of the Exponential and Hyperbolic curves,a periodic f~ction for the Harmonic curve shouldalso be created. Beginning with the expression.the Harmonic Decline EQ. 2 6 , in tegrate withrespect to t from ti-l to t..a
for
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d
Q s ,+,(l+ td0Qi = JtiQdt = f? o ~t i - 1 t.L-1(.I+ +)o
in terms of u and du
q
u= L++o
du = titodt = a,du&
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.
t
to
ti
c.L-1a
SYMBOLS
A quantity (productionrate) present at time t.
Initial production rate. QO is that quantity present att=O or the Q-intercept as shown in Figure 2, Page12.Cumulative production from t=O to t=t.Periodic production frommonth~y? annual~ etc.).Constant: of integration.Life of a project.Recoverable reserves.Resesves i n place.
A
t i-~ t o t i . (periodicmay mean
Time.Time zero. tO oz t=O may be found on the t-axis inFigure 2 , Paqe i2..Time period i.TheTheper
time perind thatloss-ratio which
directly precedesis defined as the
time period i.production rate
unit of time divided by the difference in thatproduction rate from that ofa ,maybe shown as an examplequantatively as:
the preceding time period.on Table 1, Page 3 .ar
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symbo ls (Centd)
w - i
Proportionality constantFirst difference otan exam p le on Table
d(a)dtin Natural
e Base of
or Exponential decline
q
percenta (the loss ratio). 6 may be shown as2, page21 0s quantitatively as:
10~MithXt or log t o ba se e .
naturale = . 2 . 7 1 8 3
logarithm.i .J
,
a.
Denotes a differentiablefunction of some independentv a x i a b l e ( s a y o f t). u is used to simpli.Eythe integrationprocess.
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BIBLIOGRAPXY
.ARPS, JJ. Analysis of Decline Cxmres.?etro~e~ Tec~o~oqv. T.P. 1758Septembe%, 1944.
i
.