SPEE Recommended Evaluation Practice #6 Definition of Decline Curve Parameters Background:

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1 SPEE Recommended Evaluaton Practce #6 efnton of eclne Curve Parameters Background: The producton hstores of ol and gas wells can be analyzed to estmate reserves and future ol and gas producton rates and to valdate results of complex reservor studes. Because accurate producton data are commonly avalable on most wells, producton data analyses can be wdely appled. eclne curve analyss relates past performance of ol and gas wells to future performance, but t reures modfcaton to account for changes n performance due to operatng condtons or changes n reservor behavor. eclne curves are smply a plot of producton rate versus tme on sem-log, loglog, or specally scaled paper. The most common plot s sem-log. When the logarthm of producng rate s plotted versus lnear tme, a straght lne often results. Ths phenomenon s referred to as exponental declne or constant percent declne. If the data plot as a concave upwards curve, a harmonc or hyperbolc declne model can be used to model the data. The mathematcal euaton defnng exponental declne has two constants, the ntal producton rate and the declne rate. The declne rate s the rate of change of producton wth respect to tme and, for exponental declne, s constant for all tme. There are two ways to defne the declne rate for exponental declne nomnal and effectve. The mathematcal euaton defnng hyperbolc declne has three constants, the ntal producton rate, the ntal declne rate (defned at the same tme as the ntal producton rate), and the hyperbolc exponent. The declne rate s not a constant, but changes wth tme, snce the data plot as a curve on sem-log paper. The hyperbolc exponent s the rate of change of the declne rate wth respect to tme, or the second dervatve of producton rate wth respect to tme. There are three ways to defne the ntal declne rate for hyperbolc declne nomnal, tangent effectve, and secant effectve. Fgure 1 llustrates the dfference between the tangent effectve and secant effectve defntons of ntal declne rate. Computer programs usually use the nomnal form of the euatons nternally whle nput and output are usually n terms of effectve declne. The declne curve euatons n terms of nomnal declne and the euatons used to convert from one form of declne to another are as shown below. Page 1 of 7

2 eclne Curve Euatons (for consstent unts) Nomnal Effectve: Exponental declne ln = t ( - ) e = for a partcular tme perod, usually 1 year Effectve declne as a functon of nomnal declne s: - e = 1- e Nomnal declne as a functon of effectve declne s: = - ln (1- ) e Nomnal Tangent Effectve Secant Effectve Hyperbolc declne b 1 = bt ( - ) e = where and are read from the tangent lne ( - ) es = where and are read from the secant lne Nomnal declne as a functon of tangent effectve declne s: = - ln (1- e ) Nomnal declne as a functon of secant effectve declne s: b ( 1- es ) 1 =, b 0 b Where: = nomnal exponental declne rate, 1/tme = ntal nomnal declne rate (t=0), 1/tme e = ntal effectve declne rate from tangent lne, 1/tme es = ntal effectve declne rate from secant lne, 1/tme Page 2 of 7

3 = nstantaneous producng rate at tme 0, vol/unt tme = nstantaneous producng rate at tme t, vol/unt tme t = tme e = base of natural logarthms (2.718 ) b = hyperbolc exponent (escrbes how the ntal declne rate,, changes wth tme; vares from 0 to 1 usually. When b = 1, the declne s called harmonc. Ths exponent s sometmes referred to n the lterature as n ) scusson: There are varyng opnons over whether the effectve or nomnal declne euaton should be used. Nomnal declne rate can take on any value from negatve nfnty to postve nfnty where negatve numbers ndcate producton rate s ncreasng rather than decreasng. Effectve declne rate cannot exceed 1.0 snce an endng flow rate of zero results n an effectve declne rate of / or 1. Ths lmtaton on effectve declne rate can lead to problems wth wells that are experencng extremely hgh ntal declnes such as massve hydraulc fracs n tght gas reservors or horzontal wells. It s not unusual to see ntal nomnal declne rates on the order of 30 (3000%) n these cases. The tangent effectve declne rate correspondng to a nomnal declne rate of 30 s (thrteen 9 s followed by 06.) Snce double precson numbers n computers are lmted to approxmately 15 decmal dgts of accuracy, t s mpossble to represent any exponental declne rate sgnfcantly n excess of 30 usng tangent effectve declne. Some computer programs only allow 8 or fewer dgts for declne rate. At 8 dgts the maxmum nomnal declne rate that can be represented s approxmately 20 (2000%.) Normally, extremely hgh ntal declne rates are assocated wth hyperbolc declne curves, often wth ntal values of b at or near 2. When b s 1 or greater t s possble to use the secant effectve declne even at extremely hgh declne rates. If b s 2 and the ntal nomnal declne s 100 (10000%) the value of the ntal secant effectve declne rate s %. Ths number s easly represented even n sngle precson. Table 1 and Fgure 2 show the value of tangent effectve declne (whch s ndependent of b ) and secant effectve declne for varous values of b. Page 3 of 7

4 The secant effectve declne rate has the addtonal advantage of beng calculated from two rates read from the smooth lne through the data one rate at tme 0 (whch can be arbtrarly defned) and one rate exactly one year later. Another factor whch must be recognzed s that the producton rate referred to n the above euatons s the nstantaneous rate at a partcular pont n tme. It s not the average rate for a month or the average rate for a year. In cases where the ntal declne rate s hgh the rate at the begnnng of a month may be consderably larger than the average rate durng the month. Any of these methods of defnng the future producton curve wll lead to the correct answer f they are properly appled. The most mportant thng s clear and consstent communcaton between the user and the computer programmer. SPEE Recommended Evaluaton Practce: SPEE recommends that the termnology shown above be used. Further, SPEE recommends that all nput and output be clearly labeled usng the followng names and symbols. Item Nomnal eclne Rate, exponental Intal Nomnal eclne Rate, hyperbolc Effectve eclne Rate, exponental Intal Tangent Effectve eclne Rate, hyperbolc Intal Secant Effectve eclne Rate, hyperbolc Symbo l e e es References: Arps, J.J., (1956). Estmaton of Prmary Ol Reserves. allas: Socety of Petroleum Engneers. Fetkovch, M.J. (1980): eclne Curve Analyss usng Type Curves, J. Pet. Tech., (June 1980) Page 4 of 7

5 Fgure1 eclne Rate efntons for Hyperbolc eclne 100 efnton of eclne Rate Secant Producton Rate, BOP "Actual" ata Tangent Tme, years Page 5 of 7

6 Fgure 2 Effectve eclne Rate as a functon of Nomnal eclne Rate 100% 90% 80% Effectve eclne Rate, 1/tme 70% 60% 50% 40% 30% Tangent Secant, b=0 Secant, b=.5 Secant, b=1 Secant, b=1.5 Secant, b=2 20% 10% 0% 1% 10% 100% 1000% 10000% Nomnal eclne Rate, 1/tme Page 6 of 7

7 Table 1 - Effectve eclne Rate as a functon of Nomnal eclne Rate Nomnal eclne Rate, 1/year Effectve Tangent eclne Rate, 1/year Effectve Secant eclne Rate, 1/year Value of "b" % % % % % % % 2% % % % % % % 3% % % % % % % 4% % % % % % % 5% % % % % % % 6% % % % % % % 7% % % % % % % 8% % % % % % % 9% % % % % % % 10% % % % % % % 20% % % % % % % 30% % % % % % % 40% % % % % % % 50% % % % % % % 60% % % % % % % 70% % % % % % % 80% % % % % % % 90% % % % % % % 100% % % % % % % 200% % % % % % % 300% % % % % % % 400% % % % % % % 500% % % % % % % 600% % % % % % % 700% % % % % % % 800% % % % % % % 900% % % % % % % 1000% % % % % % % 2000% % % % % % % 3000% % % % % % % 4000% % % % % % % 5000% % % % % % % 6000% % % % % % % 7000% % % % % % % 8000% % % % % % % 9000% % % % % % % 10000% % % % % % % Page 7 of 7

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