Verification of Decline Curve Analysis Models for Production Prediction Kewen Li and Roland N. Horne, SPE, Stanford University

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1 SPE Verfcaton of Declne Curve Analyss Models for Predcton Kewen L and Roland N. Horne, SPE, Stanford Unversty Copyrght 2005, Socety of Petroleum Engneers Inc. Ths paper was prepared for presentaton at the 2005 SPE Western Regonal Meetng held n Irvne, CA, U.S.A., 30 March 1 Aprl Ths paper was selected for presentaton by an SPE Program Commttee followng revew of nformaton contaned n a proposal submtted by the author(s). Contents of the paper, as presented, have not been revewed by the Socety of Petroleum Engneers and are subject to correcton by the author(s). The materal, as presented, does not necessarly reflect any poston of the Socety of Petroleum Engneers, ts offcers, or members. Papers presented at SPE meetngs are subject to publcaton revew by Edtoral Commttees of the Socety of Petroleum Engneers. Electronc reproducton, dstrbuton, or storage of any part of ths paper for commercal purposes wthout the wrtten consent of the Socety of Petroleum Engneers s prohbted. Permsson to reproduce n prnt s restrcted to a proposal of not more than 300 words; llustratons may not be coped. The proposal must contan conspcuous acknowledgment of where and by whom the paper was presented. Wrte Lbraran, SPE, P.O. Box , Rchardson, TX , U.S.A., fax Abstract Reserves and producton n petroleum reservors can be estmated usng emprcal and analytcal declne curve analyss models. There have been many such models. In ths study, three representatve models (exponental, harmonc, and the mechanstc L-Horne models) were chosen to predct and match ol and gas producton from both core samples and reservors wth dfferent permeablty. The recovery and reserves were estmated usng the three models and the results were compared. The comparson demonstrated that the recovery at an assumed economc lmt predcted usng the L- Horne model was greater than the exponental model but was less than the harmonc model. Note that the exponental model tends to underestmate reserves and producton rates whle the harmonc model has a tendency to overpredct the reservor performance. It was also found that the reserves predcted usng the harmonc model were greater than one pore volume n some cases, whch s physcally mpossble. The model predctons usng the expermental data of recovery n the core samples were also compared to the true values. The results demonstrated that the L-Horne model had the best estmaton of the recoverable recovery compared to the exponental and the harmonc models n the cases studed. Introducton Several approaches can be used to estmate reserves and predct producton n reservors. Numercal smulaton technques may facltate the predcton of ol producton but may also fal because of the great uncertanty and the complexty of reservors. Another frequently used technque s the declne curve analyss method. However many of the exstng declne curve analyss models are heurstc and are based on the emprcal Arps equatons 1 : exponental, hyperbolc, and harmonc models. The exponental declne curve tends to underestmate reserves and producton rates; the harmonc declne curve has a tendency to overpredct the reservor performance (Agb and Ng 2 ). In some cases, as ponted out by Camacho and Raghavan 3, producton declne data follow nether the exponental nor the harmonc model but cross over the entre set of curves. Fetkovch 4 provded a theoretcal bass for the Arps equaton n the case of exponental declne. The assumptons made were as follows: (1) sngle-phase flow; (2) no water njecton and no water producton; (3) producton rate s proportonal to p p (p s the average pressure n the wf reservor and p wf the bottom pressure n the producton well). However two-phase flow and water producton often occur n reservors developed by water floodng, especally at the later perod of producton. In ths case, relatve permeablty and capllary pressure should be honored n a producton predcton model wth a theoretcal bass. Note that the exponental and harmonc models often work better at the later perod of producton than durng the early perod. L and Horne 5 proposed a mechanstc declne model based on prevous theoretcal and expermental studes (L and Horne 6-7 ). The model reveals a lnear relatonshp between the ol producton rate and the recprocal of the ol recovery or the cumulatve ol producton. Two-phase flow propertes such as relatve permeablty and capllary pressure are ncluded n ths model. L and Horne 5 showed that the mechanstc model worked satsfactorly n dfferent reservors. Later Reyes et al. 8 appled ths model to the producton data from sx Kern County ol felds and found that physcal parameters derved from the declne curve analyss could be used to descrbe regonal and reservor propertes. The man purpose of ths study was to conduct a comparson between the exponental model, the L-Horne model 3, and the harmonc model. To do so, both expermental data from core samples and producton data from reservors wth dfferent values of permeablty were analyzed usng the three models. Values of maxmum recovery and recovery at an assumed economc lmt were nferred usng the three models and the results were compared. The model predctons were also compared to the true values n the cases of core samples. Mathematcal Background The emprcal Arps 1 declne equaton represents the relatonshp between producton rate and tme for ol wells durng pseudosteady-state perod and s shown as follows: q q( t) = (1) 1/ b (1 + bd t)

2 2 SPE where q(t) s the ol producton rate at producton tme t and q s the ntal ol producton rate. b and D are two constants. Eq. 1 can be reduced n two specal cases: b=0 and b=1. b=0 represents an exponental declne and b=1 represents a harmonc declne n ol producton. For 0<b<1, Eq. 1 s defned as the hyperbolc model. The two specal cases of b=0 and b=1 wll be dscussed as follows. The For b=0, an exponental declne n ol producton can be obtaned and expressed as follows: Dt q ( t) = q e (2) The exponental declne model can also be expressed n terms of cumulatve producton: N p 1 = ( q q) (3) D where N p s the cumulatve ol producton. In the case of exponental declne, one should get a lnear trend by plottng the cumulatve ol producton versus the producton rate. Accordng to Eq. 3, the maxmum producton can be obtaned by settng q=0. The producton at an economc lmt of producton (q mn ) can also be estmated from Eq. 3. The In the Arps equaton (Eq. 1), b=1 represents a harmonc declne n ol producton, whch can be expressed as follows: q q( t) = (4) (1 + D t) In terms of cumulatve producton, the harmonc declne can be expressed as: N p q q = ln (5) D q A lnear trend between the cumulatve ol producton versus the logarthm of producton rate wll be expected accordng to Eq. 5. One can not nfer the maxmum producton by settng q=0 from Eq. 5. However one may obtan the producton at an economc lmt of producton (q mn ). The The L and Horne producton declne curve analyss model 5 s expressed as follows: 1 q( t) = a0 b0 (6) R( t) where R(t) s the recovery at tme t, n the unts of pore volume (R=N p /V p, V p s the pore volume). a o and b o are two constants assocated wth capllary and gravty forces respectvely. The detals on dervng Eq. 6 and calculatng a o and b o were descrbed by L and Horne 5-7. The two constants a o and b o are expressed as follows 7 : a 0 e AM ( Swf Sw ) = Pc (7) L b0 = AM e ρg (8) where A and L are the area and the characterstc length (or heght) of the reservor or the core, S w s the ntal water saturaton and S wf s the water saturaton behnd the water front, ρ s the densty dfference between water (the wettng) and ol (the nonwettng) phases, g s the gravty constant, M e s the capllary pressure at Swf, and s the global moblty n whch relatve permeablty data of ol and water are ncluded. In summary, all three models, the exponental model (Eq. 3), the harmonc model (Eq. 5), and the L-Horne model (Eq. 6), represent the relatonshp between producton rate and the cumulatve producton n dfferent forms. If the declne trend s exponental, one should get a lnear trend by plottng q versus cumulatve producton. If the declne trend s harmonc, we expect a lnear trend by plottng the logarthm of q versus cumulatve producton. If the declne trend follows the L- Horne mechanstc model (Eq. 6), one can obtan a lnear trend by plottng q versus the recprocal of cumulatve producton (or the recovery R). Therefore we can compare the three models by calculatng the producton at an economc producton lmt of q mn. The maxmum recovery or the recoverable reserve s defned as follows: R 1 S S w or max = (9) 1 Sw where S w s the ntal water saturaton and S or s the resdual ol saturaton. Note that R max can be nferred from the exponental model and the L-Horne model but not from the harmonc model. Results Both expermental data from floodng core samples and producton data from reservors were used to verfy and compare the three models. For all the comparsons, the producton rate and recovery data appled were the same, ncludng the number of data ponts. The advantage n usng expermental recovery data from the core floodng experments s that the reserves or the maxmum producton are known exactly. Therefore the accuracy of the producton predcton models can be evaluated and compared. P c

3 SPE Hgh Permeablty Sandstone Fg. 1 shows the analyss results usng the three models for expermental data of producton from Berea sandstone (L and Horne 5 ). The Berea sandstone sample had a permeablty of around 1200 md and a porosty of about 24.5%; ts length and dameter were 43.5 cm and 5.06 cm. The Berea sandstone sample was fred at a hgh temperature to stablze the clay. The relatonshp between the producton rate and the recovery was plotted n Fg. 1a to perform the exponental producton declne analyss. There s a lnear trend at the later perod. The maxmum recovery nferred usng the exponental model was OOIP (orgnal ol n place). The recovery at an assumed economc lmt of 1 g/mnute calculated usng the exponental model (Eq. 3) was also about OOIP n ths case. Note that the actual maxmum recovery or the recoverable reserve calculated usng Eq. 9 was OOIP. The dfference between the predcton by the exponental model and the true value s sgnfcant. These data are also lsted n Table 1 n whch R eco represents the recovery at the assumed economc lmt. Fg. 1b shows a good lnear trend of the relatonshp between the producton rate and the recprocal of recovery for most of the data ponts, as predcted by the L-Horne model. The maxmum recovery nferred usng the L-Horne model was OOIP. The recovery at an assumed economc lmt of 1 g/mnute calculated usng Eq. 6 was about OOIP. The predcton by the L-Horne model s close to the true value (0.696 OOIP) of the maxmum recovery. The relatonshp between the logarthm of producton rate and recovery s shown n Fg. 1c, and was used to conduct the harmonc producton declne analyss. The trend s also lnear at the later perod. As stated prevously, the maxmum recovery cannot be nferred from the harmonc model. The recovery at an economc lmt of 1 g/mnute calculated usng Eq. 5 was OOIP, whch s far from the true value (0.696 OOIP) and s also physcally mpossble because the recovery can not be greater than. Note that the entre set of data ponts show a lnear trend when the L-Horne model s used but only the data ponts at the later perod of producton show a lnear trend when the exponental and the harmonc models are appled. Low Permeablty Chalk Fgs. 2a, b, and c show the comparson of the three models n a low permeablty chalk sample usng the expermental recovery data of L and Horne 9. The permeablty of the chalk sample was around 5 md and the porosty was 36.2%; ts length and dameter were 7.5 cm and 2.54 cm, respectvely. As n the Berea cores, the L-Horne model shows a better lnear trend than the exponental and the harmonc models. The maxmum recovery and the recovery at an economc lmt were calculated and the results are lsted n Table 1. The L- Horne model gave the best estmaton of the maxmum recovery and the recovery at an economc lmt, n comparson to the measured value. Note that the maxmum recovery lsted for the exponental model was nferred usng only the data ponts that show a lnear trend (see Fg. 2a). The magntude would be much smaller f the entre data set were used as was done n Fg. 2b n whch the L-Horne model s appled. Fgs. 1 and 2 show the comparson of the three models usng expermental recovery data obtaned from floodng core samples. In order to further verfy and compare the three models, ol producton data from complex reservors developed by water floodng were also used and the results are shown n Fg. 3 through Fg. 5. Naturally-Fractured Low Permeablty Reservors The ol producton data by water floodng from the E.T. O Danel lease n Spraberry (Baker et al. 10 ) were analyzed usng the three models and the results are shown n Fgs. 3a, b, and c. The Spraberry olfeld s a naturally-fractured low permeablty reservor wth a hgh densty of fractures. Water breakthrough occurred at producton wells shortly after water njecton began because of the hgh-densty fractures. The ol recovery by water floodng n the olfeld s beleved to be domnated by countercurrent water mbbton because of the early water breakthrough and the hgh-densty fractures 9. One can see from Fgs. 3a, b, and c that all of the three models show a lnear trend at the later perod of ol producton. However the values of ol recovery predcted by the three models are dfferent (see Table 1). For the ol recovery at an economc lmt of 1 OOIP/year, the magntudes calculated usng the exponental, the L-Horne, and the harmonc models was 0.358, 0.455, and OOIP respectvely. The predcton of recovery by the L-Horne model s n between the exponental and the harmonc models, smlar to the observaton n the core samples. Complex Reservors Another example chosen was the ol producton data n an offshore waterdrve feld reported by Dake 11 (p. 417). Ths ol feld s not a naturally-fractured reservor but has a large permeablty contrast between layers. Water breakthrough happened at the early njecton perod because of the deltac depostonal envronment and the large permeablty contrast. The hgh permeablty layers n between low permeablty layers may functon as fractures. Therefore the ol producton data from ths reservor may also be analyzed usng the L- Horne model. The results obtaned usng the three models are shown n Fgs. 4a, b, and c. Features smlar to Fg. 3 are observed. Data ponts plotted usng all three models show a lnear trend. However the nferred values of the ol recovery at an economc lmt of 1 OOIP/year were dfferent and were 0.348, 0.535, and OOIP respectvely (see Table 1). The predcted maxmum ol recovery was about OOIP when the exponental mode was used and was OOIP when the L-Horne model s used. Note that the maxmum ol recovery estmated by Dake 11 (p. 417) usng a dfferent technque was about OOIP, whch s very close to the result predcted by the L-Horne model. Fault Reservors The last example chosen s the ol producton data after water breakthrough n another North Sea ol feld developed by water floodng, also reported by Dake 11 (p. 443). Ths s an solated fault block of an extremely complex ol feld. It s of the delta top depostonal envronment wth a large permeablty contrast. Dake 11 (p. 443) reported that numercal smulaton modelng faled to produce a hstory match to the

4 4 SPE feld performance. Fgs. 5a, b, and c demonstrate the results analyzed usng the three models. Both the exponental and the L-Horne models match the producton data satsfactorly and predct smlar maxmum ol recovery, and OOIP respectvely. It s dffcult to fnd a lnear trend usng the harmonc model (see Fg. 5c). We have also conducted comparsons usng other producton data, for example, the ol producton data n East Texas ol feld reported by Dake 11 (p. 450). A smlar phenomenon was observed: all of the three models can match the producton data at the later perod of producton. However the values of ol recovery predcted by the three models are dfferent. The results are lsted n Table 1 although the graphs are not presented here. Comparson of the Three Models Fg. 6a shows the comparson between the exponental and the L-Horne models n terms of maxmum recovery. The values of maxmum recovery predcted by the L-Horne model are greater than those predcted by the exponental model. Note that the exponental declne curve tends to underestmate reserves and producton rates. The number of reservor n Fg. 6a s defned n Table 1. Fg. 6b shows the comparson among the three models n terms of the recovery at an economc lmt. One can see that the magntude predcted usng the L-Horne model s n between the exponental and the harmonc models, greater than the exponental model but less than the harmonc model. As stated prevously, the harmonc model has a tendency to overpredct the producton (Agb and Ng 2 ). The values of the recovery at an assumed economc lmt nferred usng the harmonc model are greater than one pore volume n several cases, whch s physcally mpossble. The comparson presented n Fgs. 6a and b demonstrates that the L-Horne model may have a better accuracy than the exponental and the harmonc models for the cases studed. Conclusons Based on the present study, the followng conclusons may be drawn n the cases studed: 1. The maxmum recovery determned n core samples usng the L-Horne model s close to the true value, whereas the results estmated usng the exponental model are sgnfcantly dfferent from the true. 2. The harmonc model overestmates the recovery n many cases. The recovery (n the unts of OOIP at an economc lmt) nferred usng the harmonc model was greater than n some cases, whch s physcally mpossble. 3. The exponental model underestmates the recovery or the reserve, whch s verfed usng the expermental data n both low and hgh permeablty core samples. 4. The recovery at an assumed economc lmt predcted usng the L-Horne model was greater than the exponental model but was less than the harmonc model n all of the cases studed. 5. The comparson of model predctons usng the expermental recovery data n the core samples demonstrated that the L-Horne model had the best estmaton of the recoverable recovery, compared to the exponental and the harmonc models n the cases studed. Acknowledgements Ths research was conducted wth fnancal support from the US Department of Energy under grant DE-FG07-02ID14418, the contrbuton of whch s gratefully acknowledged. Nomenclature a 0 = coeffcent assocated wth capllary forces, m/t A = cross-secton area of the core or reservor, L 2 b 0 = coeffcent assocated wth gravty, m/t D = constant g = gravty constant, L/t 2 k = absolute permeablty, L 2 k ro k rw M e M o M w = relatve permeablty of ol phase at a specfc water saturaton = relatve permeablty of water phase at a specfc water saturaton L = core length, L = global moblty of the two phases at a specfc water saturaton, ml/t = moblty of ol phase at a specfc water saturaton, ml/t = moblty of water phase at a specfc water saturaton, ml/t N p = cumulatve producton rate, L 3 P c = capllary pressure at a specfc water saturaton, 2 m/lt q = ol producton rate, L 3 /t q = ntal ol producton rate, L 3 /t R = ol recovery n the unts of pore volume R max = maxmum ol recovery n the unts of pore volume S wf = water saturaton behnd mbbton front S w = ntal water saturaton t = producton tme, t µ o = vscosty of ol phase, m/lt µ w = vscosty of water, m/lt ρ o = densty of ol phase, m/l 3 ρ w = densty of water phase, m/l 3 ρ = densty dfference between water and ol phases, m/l 3 References 1. Arps, J.J.: Analyss of Declne Curves, Trans. AIME (1945) 160, Agb, B. and Ng, M.C.: A Numercal Soluton to Two-Parameter Representaton of Declne Curve Analyss, SPE 16505, presented at the SPE Petroleum Industry Applcatons of Mcrocomputers, Montgomery, Texas, June 23 26, Camacho, V. and Raghavan, R.: Boundary- Domnated Flow n Soluton-Gas Drve Reservors, SPERE (November 1989), Fetkovch, M.J.: Declne Curve Analyss Usng Type Curves, JPT (June 1980) L, K. and Horne, R.N.: A Declne Curve Analyss Model Based on Flud Flow Mechansms,

5 SPE SPE83470, presented at the 2003 SPE/AAPG Western Regonal Meetng, Long Beach, Calforna, May L, K. and Horne, R.N.: Characterzaton of Spontaneous Water Imbbton nto Gas-Saturated Rocks, SPEJ (December 2001), 6(4), L, K. and Horne, R.N.: A General Scalng Method for Spontaneous Imbbton, SPE 77544, presented at the 2002 SPE Annual Techncal Conference and Exhbton, San Antono, TX, USA, September 29 to October 02, Reyes, J.L., L, K., and Horne, R.N.: Applcaton of a New Mechanstc Declne Curve Method to Kern County Ol Felds, SPE 90212, presented at the 2004 SPE Annual Techncal Conference and Exhbton, Houston, Texas, U.S.A., September L, K. and Horne, R.N.: An Analytcal Scalng Method for Spontaneous Imbbton n Gas-Water- Rock Systems, SPEJ (September 2004), 9(3). 10. Baker, R.O., Spenceley, N.K., Guo, B., and Schechter, D.S.: Usng an Analytcal Declne Model to Characterze Naturally Fractured Reservors, SPE 39623, presented at the 1998 SPE/DOE Improved Ol Recovery Symposum, Tulsa, Oklahoma, Aprl 19-22, Dake, L.P.: The Practce of Reservor Engneerng, Elsever, Amsterdam (2001). Table 1: Values of recovery predcted by the three models No. Name Exponental L-Horne Harmonc Measured R max R eco R max R eco R eco R true Note 1 Fred Berea hgh permeablty 2 Chalk low permeablty 3 Offshore feld hgh k contrast 4 North Sea fault fault 5 East Texas feld before nfll drllng 6 East Texas feld after nfll drllng 7 Spraberry feld naturally fractured Recovery s n the unts of OOIP. Rate, g/mnute Fg. 1a: Analyss usng the exponental model for expermental data from Berea sandstone. Rate, g/mnute / -1 Fg. 1b: Analyss usng the L-Horne model for expermental data from Berea sandstone.

6 6 SPE Rate, g/mnute Fg. 1c: Analyss usng the harmonc model for expermental data from Berea sandstone. Rate, g/mnute Fg. 2a: Analyss usng the exponental model for expermental data from chalk. Rate, g/mnute / -1 Fg. 2b: Analyss usng the L-Horne model for expermental data from chalk. Rate, g/mnute Fg. 2c: Analyss usng the harmonc model for expermental data from chalk. Rate, OOIP/Year Fg. 3a: Analyss usng the exponental model for producton data from Spraberry ol feld. Rate, OOIP/Year / -1 Fg. 3b: Analyss usng the L-Horne model for producton data from Spraberry ol feld.

7 SPE Rate, OOIP/Year Fg. 3c: Analyss usng the harmonc model for producton data from Spraberry ol feld. Rate, OOIP/year Fg. 4a: Analyss usng the exponental model for an offshore ol feld wth a hgh permeablty contrast. Rate, OOIP/year Fg. 4c: Analyss usng the harmonc model for producton from an offshore ol feld wth a hgh permeablty contrast. Rate, OOIP/year Fg. 5a: Analyss usng the exponental model for producton data from a North Sea fault ol feld. Rate, OOIP/year / -1 Fg. 4b: Analyss usng the L-Horne model for producton from an offshore ol feld wth a hgh permeablty contrast. 10 Rate, OOIP/year / -1 Fg. 5b: Analyss usng the L-Horne model for producton data from a North Sea fault ol feld.

8 8 SPE Rate, OOIP/year Fg. 5c: Analyss usng the harmonc model for producton data from a North Sea fault ol feld. Maxmum Number of Reservor 8 Fg. 6a: Comparson between the exponental model and the L- Horne model. Recovery at Eco. Lmt, OOIP Number of Reservor Fg. 6b: Comparson among the exponental, the harmonc, and the L-Horne models.