The calculation method of small-scale water injection multiple in water drive reservoirs

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1 Available nline.jcpr.cm Jurnal f Chemical and Pharmaceutical Research, 04, 6(5):04-09 Research Article ISSN : CODEN(USA) : JCPRC5 The calculatin methd f small-scale ater injectin multiple in ater drive reservirs Cui Chuanzhi, Li Kaiai, Gu Jincheng, Ca Gang and Zhang Chuanba Cllege f Petrleum Engineering, China University f Petrleum, ingda, China Research Institute f Gelgy and Science f Shengli Oilfield, SINOPEC, China ABSTRACT The ater injectin pre vlume multiple in cnventinal cre displacement experiment is incnsistent ith that in the entire reservir, hich results in the difficulty in the applicatin f the results f cre displacement experiment t the calculatin f reservir develpment. Therefre, it is required t n the distributin f ater injectin pre vlume multiple in the frmatin hich is cnsistent ith that in the cre displacement experiment. In this study the case f the Five-spt Well Pattern as used t determine the streamline distributin in the frmatin. Fluids fl ne-dimensinally alng the streamline and each streamline can be seen as the cnnectin f a number f cres. The fling resistance n each streamline as measured t determine the ater injectin rate f each streamline. Accrding t the thery f ne-dimensinal il-ater t phase fl, the ater injectin pre vlume multiple f each cre in the streamline culd be calculated. Thus, the cnsistence uld be achieved beteen the multiple f ater injectin in cre displacement experiment and that in the frmatin. The research results as f high theretical and practical significance t the applicatin f the results f cre displacement experiment t the research f reservir develpment. Key rds: Pre vlume multiple; Streamline; Displacement experiment; Water drive reservir; Rc Cre. INTRODUCTION Generally, ater injectin pre vlume multiple refers t the rati beteen cumulative ater injectin and the ttal pre vlume f reservir. When the ilfield develpment reaches an ultra-high ater cut stage, the cumulative ater injectin pre vlume multiple f the entire reservir is generally less than 5 [-]. Cre displacement experiment is a cnventinal and cmmnly used experimental methd. A large amunt f useful infrmatin can be btained thrugh this ind f experiments, such as the displacement efficiency and the variatins f reservir permeability crrespnding t different ater injectin pre vlume multiples. Hever, enrmus difficulty arises in the applicatin f such infrmatin t actual reservirs, because the ater injectin pre vlume multiple btained in cnventinal cre displacement experiment usually reach several tens r hundreds [3-5], far greater than that f actual reservirs, i.e., the ater injectin pre vlume multiple in cre displacement experiment is nt the ater injectin pre vlume multiple f the entire reservir in a cnventinal sense. In this study the ater injectin pre vlume multiple btained in cre displacement experiment as defined as small-scale ater injectin pre vlume multiple in rder t distinguish it frm that f the entire reservir. In rder t mae it pssible fr the applicatin f the results f cre displacement experiment t the studies f actual reservir develpment, it is necessary t btain the distributin f small-scale ater injectin pre vlume multiple in the frmatin hich is cnsistent ith that f cre displacement experiment. Currently, the small-scale ater injectin pre vlume multiple in the frmatin is usually calculated n the basis f reservir numerical simulatin. The small-scale ater injectin pre vlume multiple in the frmatin as btained thrugh a certain grid f by 04

2 Cui Chuanzhi et al J. Chem. Pharm. Res., 04, 6(5):04-09 calculating the rati f the ttal vlume f infl ater t the pre vlume f the grid. Hever, the small-scale ater injectin pre vlume multiple is greatly affected by the size f pre vlume f the grid, and is still incnsistent ith that in cre displacement experiments. The case f the Five-spt Well Pattern as studied t investigate the calculatin methd f small-scale ater injectin pre vlume multiple in the frmatin hich as cnsistent ith that in cre displacement experiment.. STREAMLINE DISTRIBUTION OF FIVE-SPOT WELL PATTERN In the Five-spt Well Pattern, the ptential functin in the frmatin culd be btained accrding t the principle f superpsitin f ptential. The stream functin culd be acquired accrding t Cauchy-Riemann cnditins, and the stream functin culd be used in the determinatin f the streamline distributin in the frmatin... Ptential functin f five-spt ell pattern Fr a pint surce r pint sin, the ptential expressin at any pint in the frmatin is shn as bel: j q Φ = ( ) ln r + C () π In the abve frmula, j = fr prductin ells, j = fr injectin ells. When there as nly ne Five-spt ell pattern in the frmatin (as shn in Fig.), accrding t the principle f superpsitin f ptential, the expressin f ptential functin in the frmatin is as flls: Φ = q q ln x y (ln x a y a ln x a y a π + + ( 8π + ) + ( ) + ( ) + ( ) ln ( x a) ( y a) ln ( x a) ( y a) ) c y () x a Figure. Schematic diagram f Five-spt Well Pattern.. Stream functin and streamline f five-spt ell pattern Cauchy - Riemann cnditins is satisfied beteen stream functin and ptential functin, i.e.: Φ ϕ = x y Φ ϕ = y x (3) Accrding t Cauchy-Riemann cnditins, the stream functin f Five-spt Well Pattern can be btained thrugh the derivatin frm ptential functin expressin (): q y q y a y a y + a y + a ϕ = arctg + (arctg + arctg + arctg + arctg ) + c π x 8π x + a x a x + a x a Fig. shs the streamline distributin f Five-spt Well Pattern btained accrding t the abve stream functin. (4) 05

3 Cui Chuanzhi et al J. Chem. Pharm. Res., 04, 6(5):04-09 Figure. The streamline distributin schematic diagram f Five-spt Well Pattern. CALCULATION OF SMALL-SCALE WATER INJECTION PORE VOLUME MULTIPLE Fluids fl ne-dimensinally alng the streamline. Under the cnditin f il-ater t-phase fl, the ater saturatin distributin alng the streamline culd be calculated accrding t the thery f ne-dimensinal il-ater t phase fl. Each streamline in the frmatin culd be seen as the cnnectin f a number f cres, and the determinatin f the distributin f small-scale ater injectin pre vlume multiple in the entire frmatin as achieved thrugh the calculatin f the ater injectin pre vlume multiple f each cre in the streamline... Calculatin f the derivative f ater cut Accrding t Darcy's la, the ater cut equatin culd be btained ithut taing capillary pressure and gravity int cnsideratin: f = r µ + r µ (5) Due t the satisfactin f the flling relatinship beteen the rati f relative permeability f il-ater t phase and ater saturatin: r = b S W e b (6) r Thus, the ater cut equatin culd be expressed as bel: f = = r µ b b S µ + + W e r µ µ The derivative f ater cut culd be btained thrugh the derivatin f rate f ater cut: f S µ = b be µ bs µ + be µ bs (7) S in the abve equatin, i.e., the increasing (8) With given il-ater viscsity and relative permeability curve, the ater cut and increasing rate f ater cut crrespnding t different ater saturatin culd be btained accrding t equatins (5)-(8)... Calculatin f ater saturatin alng the streamline In the ne-dimensinal il-ater t-phase fl, Bucley-Leverett Equatin is shn as bel[6]: 06

4 Cui Chuanzhi et al J. Chem. Pharm. Res., 04, 6(5):04-09 ( ) df dx t = (9) dt φ A ds The result after integratin is shn as flls: f t ( s ) x = ( t) dt φ A (0) Accrding t Equatin (0), the derivative f ater cut culd be btained at any lcatin (x) in the streamline at a specific time (t ). The ater saturatin ( S ) and ater cut ( f ) culd be btained ith nledge f the relatinship beteen the derivative f ater cut and ater saturatin..3. Calculatin f small-scale ater injectin pre vlume multiple A segment as selected alng a certain streamline and its length shuld be equivalent t that f the cre in the displacement experiment. Thus, the small-scale ater injectin pre vlume multiple f this sectin culd be btained at the time T : v = t = T ( t) f t = 0 ALφ dt () The calculatin f ater injectin pre vlume multiple f each cre alng the streamline as cnducted in rder t determine the distributin f small-scale ater injectin pre vlume multiple f the entire frmatin..4. Calculatin f the allcatin f ater injectin rate n each streamline Fr each ell grup, the ater injectin rate f each streamline culd be determined ith given ttal ater injectin rate, accrding t the streamline number fled frm the ater ell and the fling resistance f each streamline. The numbers f cres f each streamline varies ith its length. It as assumed that N streamlines fled frm the ater ell, f hich -th streamline as frmed by cnnectin f m cres. If m r r R = + h () j j= µ µ j Then, the ater injectin rate f -th streamline as as flls: t = R N inj R j j= (3) Where inj represents the ttal ater injectin rate f ater ell. Accrding t the mbility f il phase and ater phase at the exit f -th streamline, the il prductin rate and ater prductin rate f this streamline culd be calculated respectively, i.e.: λ = λ + λ t, = t (4) The fling resistance f each streamline varied ith ater saturatin, hich meant the allcated ater injectin rate f each streamline as different at different time. 3. APPLICATION AND EXAMPLE Based n the abve calculatin methds, the cmputing prgram as develped t calculate the distributin f small-scale ater injectin pre vlume multiple in frmatins at varius develpment time. It as assumed that Five-spt Well Pattern as 400m in ell spacing, 8 m in reservir thicness, 0.6 in prsity, um in 07

5 Cui Chuanzhi et al J. Chem. Pharm. Res., 04, 6(5):04-09 permeability, and0 mpa.s and 0.5 mpa.s in il and ater viscsity respectively. In the cre displacement experiment, the cre as 8 cm in length, hile cm in radius and 0.08 cm 3 /s in ater injecting rate. Fig.3 shs the relative permeability curve adpted in the calculatin. Kr Figure 3. S The relative permeability curve Kr Kr Fig.4 is the cntur map f the calculated small-scale ater injectin pre vlume multiple in the frmatin ith the recvery percent f reserves f 0%. This map is ne quarter f ne Five-spt ell pattern. It can be seen that the distributin f small-scale ater injectin pre vlume multiple in the frmatin alng the mainstream beteen the ater ell and il ell as greater. The small-scale ater injectin pre vlume multiple adjacent t the ater ell is greater than that in the else regin f the frmatin and gradually decreased frm the mainstream t the flans ith even nn-existence in certain regins amng the il ells. Oil ell Water ell Figure 4. The cntur map f the small-scale ater injectin pre vlume multiple CONCLUSION Based n the distributin f streamlines and the results f cre displacement experiment, the calculatin methd f small-scale ater injectin pre vlume multiple in the frmatin hich as cnsistent ith that in cre displacement experiment as established in this study, hich as f great practical significance t the applicatin f the results f cre displacement experiment t the research f reservir develpment. NOMENCLATURE a half the distance beteen t il ells, cm A the sectin area f streamline, cm b, b the cefficient c the cefficient 08

6 Cui Chuanzhi et al J. Chem. Pharm. Res., 04, 6(5):04-09 f H r ater cut reservir thicness, cm abslute permeability f reservir, il phase relative permeability µ m r ater phase relative permeability L cre length, cm q the prductin rate ith ne unit thicness, cm 3 /s ater injectin rate f ne ater ell, cm 3 /s inj il prductin rate, cm 3 /s ater prductin rate, cm 3 /s ( t ) ater injectin rate f ne streamline, cm 3 /s r the distance frm any pint t pint sin, cm fling resistance, µ m mpa s R v x, y Φ ϕ φ cm small-scale ater injectin pre vlume multiple the crdinate ptential functin stream functin prsity µ il viscsity, mpa s µ ater viscsity, mpa s Acnledgement This r as Supprted by Prgram fr Changjiang Schlars and Innvative Research Team in University (IRT94) and China Natinal Science and Technlgy Majr Prjects (Tpic name: The technlgy t imprve il recvery at high ater cut perid in Shengli ilfield, Tpic number: 0ZX050). REFERENCES [] SUN Huan-quan., OGRT, 000,7(): [] HUANG Tian-un; CHEN Jun-bin; WANG Zhi-hua, West-China Explratin Engineering, 008,0(): [3] DENG Rui-jian, Jurnal f Jianghan Petrleum Institute,003,5(3),-3. [4] DU Yu-hng, ZHANG Ji-chun; HOU Cui-fen, Special Oil and Gas Reservirs, 004,(5),5-55. [5] ZHAO Zeng-yi; LIAO Jian-de; LENG Yan, et al., Jurnal f Yangtze University, 009,(6): [6] Bucley,S.E. and Leverett,M.C., Trans., AIME, 94,46: