The Mathematics of Decline Curves

8/9/2019 The Mathematics of Decline Curves

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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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43/45

.

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.

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The Mathematics of Decline Curves

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Transcript of The Mathematics of Decline Curves