production decline analysis
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Introduction
Production decline analysis is a traditionalmeans of identifying well production problemsand predicting well performance and life based
on real production data.
It uses empirical decline models that have littlefundamental justications. These modelsinclude the following:
Exponential decline constant fractionaldecline!
"armonic decline
"yperbolic decline
Production Decline Analysis
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#lthough the hyperbolic decline model is moregeneral$ the other two models are degenerations of
the hyperbolic decline model. These three modelsare related through the following relative declinerate e%uation #rps$ &'()!:
where b and d are empirical constants to be
determined based on production data. *hen d + ,$E%. -.&! degenerates to an exponential declinemodel$ and when d + &$ E%. -.&! yields a harmonicdecline model.
*hen , d &$ E%. -.&! derives a hyperbolic
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Exponential Decline
The relative decline rate and productionrate decline e%uations for the exponential
decline model can be derived fromvolumetric reservoir model.
/umulative production expression isobtained by integrating the production
rate decline e%uation.
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Relative Decline Rate
/onsider an oil well drilled in a volumetric oilreservoir.
0uppose the well1s production rate starts to declinewhen a critical lowest permissible! bottom2hole
pressure is reached. 3nder the pseudo4steady2state 5ow condition$ the production rate at a givendecline time t can be expressed as
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The cumulative oil production of the well after theproduction decline time t can be expressed as
The cumulative oil production after the productiondecline upon decline time t can also be evaluated basedon the total reservoir compressibility:
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Ta6ing derivative on both sides of this e%uationwith respect to time t gives the di7erentiale%uation for reservoir pressure:
8ecause the left2hand side of this e%uation is %and E%. -.9! gives
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Production rate decline
E%uation -.! can be expressed as
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0ubstituting E%. -.&;! into E%. -.9! gives the wellproduction rate decline e%uation:
which is the exponential decline model commonlyused for production decline analysis of solution2gas2drive reservoirs. In practice$ the following form of E%.-.&)! is used:
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Cumulative production
Integration of E%. -.&! over time gives an
expression for the cumulative oil production sincedecline of
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Determination of decline rate
The constant b is called the continuous decline rate.
Its value can be determined from production historydata.
If production rate and time data are available$ the b
value can be obtained based on the slope of thestraight line on a semi2log plot. In fact$ ta6inglogarithm of E%. -.&! gives
which implies that the data should form a straightline with a slope of b on the log%! versus t plot$ ifexponential decline is the right model. Pic6ing up
any two points$ t&$ %&! and t9$ %9!$ on the straight
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If production rate and cumulative production dataare available$ the b value can be obtained based onthe slope of the straight line on an
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=epending on the unit of time t$ the b can havedi7erent units such as month4& and year4&.
The following relation can be derived:
where ba$ bm$ and bd are annual$ monthly$ and
daily decline rates$ respectively.
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Eective decline rate
8ecause the exponential function is not easy touse in hand calculations$ traditionally thee7ective decline rate has been used.
0ince e4x > &4x for small x2values based onTaylor1s expansion$ e4b &4b holds true for smallvalues of b.
The b is substituted by b1$ the e7ective declinerate$ in eld applications. Thus$ E%. -.&!
becomes
# ain$ it can be shown that
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=epending on the unit of time t$ the b1 can havedi7erent units such as month4& and year4&. Thefollowing relation can be derived:
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Example Problem 8.1 ?iven that a well has
declined from &,, stb@day to ' stb@dayduring a &2month period$ use the exponentialdecline model to perform the following tas6s:
&. Predict the production rate after &&more months
9. /alculate the amount of oil producedduring the rst year
;. Project the yearly production for thewell for the next ) years
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olution
&. Production rate after && moremonths:
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In summary$
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!armonic Decline
*hen d + &$ E%. -.&! yields di7erentiale%uation for a harmonic decline model:
which can be integrated as
where %0is the production rate at t + ,.
Expression for the cumulative production isobtained by integration:
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!yperbolic Decline
*hen , d &$ integration of E%. -.&! gives
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Expression for the cumulative production isobtained by integration:
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"odel Identi#cation
Production data can be plotted in di7erentways to identify a representative declinemodel.
If the plot of log%! versus t shows a straight
line Aig. -.&!$ according to E%. -.9,!$ thedecline data follow an exponential declinemodel.
If the plot of % versus
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If the plot of
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$i%ure 8.1 A semilo% plot of & versus t indicatin% an
exponential decline.
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$i%ure 8.' A plot of (pversus & indicatin% an
exponential decline.
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$i%ure 8.) A plot of lo%*&+ versus lo%*t+
indicatin% a
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$i%ure 8.- A plot of (p
versus lo%*&+ indicatin%
a ,armonic
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$i%ure 8. A plot of relative decline rate versus
production rate.
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$i%ure 8./ Procedure for determinin% a0 and
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Determination of "odel Parameters
Bnce a decline model is identied$ the modelparameters a and b can be determined by ttingthe data to the selected model.
Aor the exponential decline model$ the b value canbe estimated on the basis of the slope of the
straight line in the plot of log%! versus t E%.C-.9;D!.
The b value can also be determined based on the
slope of the straight line in the plot of % versus
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The b value can also be estimated basedon the slope of the straight line in the plotof
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Illustrative Examples
Example Problem 8.' Aor the data given inTable -.&$ identify a suitable decline model$determine model parameters$ and projectproduction rate until a marginal rate of 9)stb@day is reached.
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olution# plot of log%! versus t is presentedin Aig. -.$ which shows a straight line.
#ccording to E%. -.9,!$ the exponentialdecline model is applicable. This is furtherevidenced by the relative decline rate shownin Aig. -.-.
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$i%ure 8. A plot of lo%*&+ versus t s,o2in% an
exponential
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$i%ure 8.8 Relative decline rate plot s,o2in%
exponential
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0elect points on the trend line:
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$i%ure 8.3 Pro4ected production rate by aexponential
decline model.
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Example Problem 8.) Aor the data given inTable -.9$ identify a suitable decline model$
determine model parameters$ and projectproduction rate until the end of the fth year.
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olution # plot of relative decline rate isshown in Aig. -.&,$ which clearly indicates a
harmonic decline model.
Bn the trend line$ select
%,+ &,$,,, stb+day at t + ,
%&+ )$-, stb+day at t + 9 years:
Therefore$ E%. -.(,! gives
Projected production rate prole is shown in
Aig. -.&&.
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$i%ure 8.15 Relative decline rate plot s,o2in%,armonic
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$i%ure 8.11 Pro4ected production rate by a
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Example Problem 8.- Aor the data given inTable -.;$ identify a suitable decline model$
determine model parameters$ and projectproduction rate until the end of the fth year.
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olution # plot of relative decline rate is shown
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olution # plot of relative decline rate is shownin Aig. -.&9$ which clearly indicates a hyperbolicdecline model.
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$i%ure 8.1' Relative decline rate plot s,o2in%
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$i%ure 8.1) Relative decline rate s,ot s,o2in%
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Projected production rate prole is shown in Aig.-.&(.
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