Estimating the Stabilized Deliverability of a Gas Well Using the Rawlins and Schellhardt Method: An Analytical Approach

duration of the shut-in tiis often is not long enough to reach the . , This paper introduces a direct method to use the results of true average reservoir pressure in the well% drainage area. Houpeuztdeliverability analysis to derive the constants “C” and Although isochronaland modifkd isochronaltests weredevelo P “n” in the Rawlins and Schellhardt gas well deiiversbllity to circumvent the long flow times required in low permesbl ity equation. The motivation for this effort is the need to report the reservoirs, these tests may still require a single, stabili=d fiOW resultsof Rawlins and ScheUhardtanalysisto regulatory agencies, period at the end of the test in order to estimate the srrzfdfized and the widespread use of their deliverabi lity equation by producingcapacityof the well. engineers. We present a detailed procedure which shows how these results can be applied to deliverability forecasting. This The conventional deliverability test analysis technique was paper includes an ?lustmtive example in which the new nwthod is proposed by Rawlins and Schellhardt.} They observedthat a logapplied to field data from the literature. This example presents log plot of the difference between the squares of the average comparisons between Houpeurt and Rawlins and Schellhsrdt reservoir pressure and the bottomhole flowing pressure against analysesand showsthe correlation betweenthe two methods. gas flow rate can be representedby a straightline defined by qg = c G* “P;f)n . . .. . . .. . . . . . . . . , ..,.,,,,....,,, . . . . . . . . . . . .(1) INTRODUCTION The purpose of deliverability testing is to determine a gas well’s where C is defined as the stabdizai performancecoefficient,and n is the reciprocal of the slope of the straight line, Extrapolation of production capabilities under’specific reservoir conditions. A this line to the difference between the squares of ths average common prmiuctivity indicator obtained from these tests is the reservoir pressure and the bottomhole flowing pressure equal to absolute open flow (AOF) potential, which is defined as the atmosphericpressuredefines the AOF. maximum rate at which a well could flow against a theoretical atmospheric backpressure at the sandface. Although in practice Eq, 1 was developed empirically from the observation of a the well cannot produce at this rate, the AOF is often used by number of gas well tests. Extrapolation of Eq. 1 over large regulatory agencies for establis%tg field proration schedulesand variationsin pmssum can result in incorrectestimatesof the AOF, settingmaximumallowableproductionrates for individualwells. Subsequent theoretical developments by Houpeurd have shown A number of testing techniques have been developed to assess a that a more accurateanalysisfor gas flow is possiblewith gas well’sdeliverabilitycharacteristics, Flow+fter-flowl tests am P2-p~f=aq8+bqi ,,, ., .. . . . . . . . . . . . .. . .. . . . . .. . . . ,, .,4,.. (2) conducted by producin$ the well at a seriesof different flow rates and measuring the stabdized bottomholeflowin$ pressures, Each where the flowcoefficients,a and b, are defined by flowrate is establishedin successionwithoutan intermediateshutin period. The primary limitation of these tests is the long time a= L422x106~ Z T required to reach stabilization in low permeability reservoirs. Consequently, the isochrorta12and modified isochronal tests k,ll g ‘ [’151’~g(%)-i+’] .b.,(3, were developed to shotten test times. ~ = L422x106 jit Et TD kt h An isochronal testis conductedby alternativelyproducing the .,, ,!,.,.,.,,, ,,, ,,, ,,, ,,, ,, ., .,... (4) well, then shutting it in md allowing it to build up to the average 7 2 is a solution to the diffusivity equation for radial flow, reservoir ressure prior to the beginning of the next flow period. R A ;hough t$e Houpeurt equation has a theoretical basis and is The rnodi ed isochronal test is conducted similsriy, except the rigorously correct, the more famiiiw but em ifically b~ed Rawlins and Schellhardt equation continues to L used, inkd Referencesm~Uustradons at end of paper favored, b I the natural gas industry. Consequently, we have i combined e two analysistechn4uea and havedevelopeda mm 2 Estimatingthe Stabtized Deliverabilityof a Gss WellUsing the SPE 23440 Rawtinsand SchellhardtMethod: An AnalyticalAppioach . accurate version of the Rswlins-Schellhardtmethod which can be used in deliverability forecasting. Our technique can be used to estimate the stabihzed performance coefficient, C, without requiring stabilizedflowingconditions and is especiallyuseful for analyzing isochronal and modified isochronal tests without a stabilized flow period. This is a simple method which rquires only data from a modified isochronal test to develop a performance prediction for the well without a priori estimates of reservoir properties. I?ZFINITIONS AND THEORETICAL DEVELOPMENT Our deliverability test analysis technique is derived by equating derivatives of log (qg) with respect to log (AP ) from both the Rawlins-Schellhardt and Houpeurt equations, 1 similar method was used by Brigham,s Duong,s and Poettmann and KazemiTto develop quations in terms of pressures-squared for estimating reservoir properties from gas deliverability tests. Because of the pressure-dependentgas properties, the pressure-squaredforms of the deliverability equations (Eqs, 1 and 2) often are inaccurate at high pressures. Therefore, in the subsquent derivation, we use the pseudopressure transformation introduced by A1-Hussainy,et al,a J P pp=2 P& pb /404 ‘go) . . . . . . . . . . . . ! ..,...,.,,, . . . . . . . . . . . . . . .(5) Our method is a!su applicable, however, for deliverability quations written with pnssure-squared as the dependentvariable. In terms of pseudopressure,the Rawlins and Schellhardtequation becomes qg = C ~P@) PP@wf)l” %’ ,,, ., .,,.,,... . . . . . . . . . . . . . . . . . . . . . . .(6) Similarly, the Houpeurtquation is Pp@) PP(IW) = a qg + b q~ . . . . . .. . . . . .. . .. . .. . . . . . .. . .. ...(7) where the flowcoefficients,u and b, are defined by 6 b ~x10 = D kg h , .,, ., ...,..,, . . . . . . . . . . . . . . ,.! ...,,,0,,,, (9) Taking the logarithmof both sidesof Eq, 6 yields /og(q8) =/og(C)+ fl/og~~@ -PP@w/)] .. ...0.. . .. .. ..(lO) Reatmnging Eq. 10 and solving for n shows that n is the slope of a log-log plot of qg vs. *P Alternatively, n can be expressed as the derivativeof log (q8) with respect to log (A@: ‘=m&3i$m=hm#kz-T”( ll) Similarly, taking the logaxithmof both sides of E@7 yields ‘OtiP@-pp@~j)] =10~aq8+ bql]... !...!..! . .. ..!. (12) Differentiatinglog (@P) with respect to q, gives . a+2bq r ,,..,.,.,,,..,.,,,,,., (13) aqg+bq~ Substituting F@ 13 into Eq 11 yields an quation for n in terms of the Houpeurtflow eoefftciem [1 ~=1 ~g+%l -a+bg qg a+2bqg (14) a+2bqg To develop an expression for the performance coefficient, C, in terms of the Houpeurt flow coefficients,we combine Eqs, 10 and 12to obtain In c‘(a~,~bqf~ = (a f~qg)” . . . . . . . . . . . . . . , ...,,...,,, (15) Eq. 15 is similar to a result derived by Poettmann and Kazemi.T We show the applications and importance of this development in the procedure in the next section of this paper, The flow rate required in Eq, 15 is defined by solving Eq. 14 for the gas flowrate, qg, a(l -n) ‘g= b(2n-1) .. ... .. . . . . . .. . .. . . . . . . . .. . . .. . .. . . . . .. . ... ... (16) Implicit in our derivation is the assumption of radial flow of a single-phase gas in a homogeneous, isotropic reservoir, For naturally fractured reservoirs, our method, like conventional deliverability analysis techniques, is valid only after the matrixfracture system has begun to behave like a single, homogeneous unit. Similarly, our method is valid only after pseudoradial flow is exhibited in hydraulically fractured wells, We also assume wellborestorageeffects are negligible. I“ DELIVERABILITY TEST ANALYSIS Application of our method assumes that the slo~, l/n, of the empirical deliverability plot remains constant wnh time. This assumption implies that, if we can calculate values of a and b (I@. 8 and 9, respectively) for given reservoir properties, we also can calculatea flow rate with@. 16, We then substitute this flow rate into Eq. 15 and calculate a stabilized C value, and assuminga constantvalue for n, calculatethe AOF AOF = C ~p(ji) Pp@b)r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (17) To apply our new deliverability analysis technique to field data, we present an analysis procedure below. We will then apply this procedureto a fieldexample, General Analysis Procedure. We recommend the following proceduro to analyze isochronal and modified isochronal tests using our technique, Although presented in terms of pseudopressures, this procedure also is applicable with the pressure-squaredvariables, 1. Plot 4P = Pp@) Pp@wf) vs. qg on log-log graph paper for the measuredflow data. 2, For each flow time, construct the best tit line through the data points, Typically, some of the earl data points will not J’ agree with the general trend of the ata, so these points should be ignored in all subsequentanalyses. 3. Determine the deliverability exponent, n, for each best-fit line by least-squaresregressionanalysis using the following equation: I !i1OW.5(10,4P)J N$(fOgq810g&p~-, , ‘=-$=’’”(’8) 4, For M isochronal tests, compute the arithmeti~ average deliverabilityexportcn4ii,