Sensitivity analysis of dimensionless parameters for physical simulation of water-flooding reservoir Sensitivity analysis of dimensionless parameters for physical simulation of water-flooding reservoir

Sensitivity analysis of dimensionless parameters for physical simulation of water-flooding reservoir

  • 期刊名字:中国科学E辑(英文版)
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  • 论文作者:BAI Yuhu,Li Jiachun,Zhou Jifu
  • 作者单位:Division of Engineering Sciences
  • 更新时间:2020-07-08
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Science in China Ser. E Engineering & Materials Science 2005 Vol.48 No.4 441- -453Sensitivity analysis of dimensionless parametersfor physical simulation of water-floodingreservoirBAI Yuhu, LI Jiachun & ZHOU JifuDivision of Engineering Sciences, Institute of Mechanics, Chinese Academy of Sciences, Beiing 100080,ChinaCorrespondence should be addressed to Bai Yuhu (email: byh21 @ sina.com)Received July 15, 2004Abstract A numerical approach to optimize dimensionless parameters of water-floodingporous media flows is proposed based on the analysis of the sensitivity factor defined asthe variation ration of a target function with respect to the variation of dimensionlessparameters. A complete set of scaling criteria for water-flooding reservoir of five-spot wellpattern case is derived from the 3-D governing equations, involving the gravitational force,the capillary force and the compressibility of water, oil and rock. By using this approach,we have estimated the influences of each dimensionless parameter on experimentalresults, and thus sorting out the dominant ones with larger sensitivity factors ranging from104 to 100.Keywords: sensitivity analysis, model scaling, numerical experiment, water-flooding, two phase flow.DOI: 10.1360/04ye0290Petroleum industry in China is confronting challenge in oil supply. On the one hand,the total storage is hardly to meet emergent demands. On the other hand, the oil fieldsdeveloped in early years are costly operated due to the high average water ratio even upto 80%. The lack in energy resource considerably promotes the enhancement of oil re-covery. Experts' estimated that with the oil recovery enhanced by 1 %, the oil produc-tion would be augmented as much as the annual total output with 150 billion Yuan profit.Therefore, a better understanding in the behavior of water, oil and gas underground isindispensable.Physical simulation is an important approach to reveal mechanisms of porous mediaflows for design and optimization of development programs. Compared with field tests,this approach seems to be cheaper, time-saving and easier to implement. The principle ofsimilarity or scaling law is crucial for physical siml中国煤化工a1.3]derivedthe scaling criteria for flows in cold water, heatedTYHC N M H Gding reservoirCopyright by Science in China Press 2005442Science in China Ser. E Engineering & Materials Science 2005 Vol.48 No.4 441- -453by using both inspectional and dimensional analysis methods, showing that the effect ofinertia force could be neglected in most circumstances. In the heated water-flooding res-ervoir, the effect of the gravitational force can be scaled with porous medium parametersrelaxed. However, the pressure drop, the capillary force, the residual oil saturation andrelative permeability cannot be well modeled. Kimber et al.!4I obtained a set of scalingcriteria for steam and steam additive recovery reservoir. By relaxing the requirement ofgeometric similarity, we may employ the same fluids and porous medium in the modelsas in the prototypes. Islam et al.P) got the scaling criteria for polymer, emulsion andfoam flooding experiments by the inspectional analysis. They focused on the compli-cated flooding processes involving mass transfer among phases, the interfacial tension,diffusion, dispersion, absorption and mechanical entrapment. They argued that it is nec-essary to relax the geometric similarity in order to properly scale dispersion and pres-sure-induced effects in some circumstances. Islam et al.!O gained the scaling criteria ofsurfactant-enhanced alkaline/polymer multiple flooding flows, which involve transientinterfacial tension and non-equilibrium mass transfer phenomena. They have observedthat in most cases the mass transfer rate in the model is different from that in the proto-type. However, both the rates have been assumed the same in previous studies. Shen etal.! assumed that the ratios of the gravitational and the driving forces to the capillaryforce were essential similarity parameters for water-flooding reservoirs. Generallyspeaking, there are so many parameters involved in the process of water driving oilflows due to the complex mechanisms. Therefore, it is very difficult or sometimes evenimpossible to keep all the similarity parameters identical in the laboratory experiment.Pozzi and Blackwell6l pointed out that precise scaling of transverse dispersion coupledwith the requirement of geometric similarity would impractically require a large modeland a very long time interval for experiments. In particular, some dimensionless pa-rameters are contradictory. To tackle this kind of problems, the efficient and practicalway out is to single out the dominant parameters and to relax the secondary ones inlaboratory experiments. It is not easy or even impossible to theoretically arrange all thedimensionless parameters in the orders of importance. Previous literatures at most haveprovided the importance of part of the dimensionless parameters qualitatively91. There-fore, how to quantify the dominant degree among them is still a problem open to us.The authors of the present paper have suggested a numerical approach of sensitivityanalysis of dimensionless parameters. To begin with, we have defined the sensitivityfactor in section 1. Then, we have derived a set of scaling criteria of water-flooding res-ervoir flows, accounting for the gravitational and capillary forces, and the compressibil-ity of fluids and rock, and then validated the code in sections 2 and 3, respectively. Fi-nally, we have quantified the sensitivity of all the dimensionless Darameters by using theproposed numerical approach. According to the|YH中国煤化工ctors we may se-lect the dominant dimensionless parameters thaC N M H Gin laboratory ex-periments.Copyright by Science in China Press 2005Sensitivity analysis of dimensionless parameters for physical simulation of water-looding reservoir 4431 Sensitivity analysis of dimensionless parametersActually, it is not realistic for us to estimate the role of each dimensionless parameteron experimental results by physical simulation. In contrast, we present a numerical ap-proach to evaluate the effect of each dimensionless parameter on a target function. Thesensitivity factor of a given dimensionless parameter兀is firstly defined as follows:S;=af[f(,2.,.tN)/f](i=1,2,.., N),a(n;/Tp)where f (兀,Tz,..,ItN) denotes a target function concerned in the experiment. S; iscalled sensitivity factor meaning the relative variation ration of the target function withrespect to that of a dimensionless parameter. In a water-flooding experiment, the targetfunction can be expressed as .f(>,2,..N)=。川(T)72..,IN ,tp)dtp,(2)where 7(m,r2...,TN,tp) represents the oil recovery curve as shown in Fig.1, and Tpthe dimensionless time span of development. In our numerical scheme, the sensitivityfactor is written in the following difference formAanCo,(3)[win which w; =Tim - nipstands for the relative distortion of the ith dimensionlessπipparameter, ao=. [° n[(T/p.,2p.--,TNp,tp)dtp is the area under the oil recovery curve of100-. πm equals to πp80----- π m not equal to tp食s 60-后4020 t0.00.51.0202Dimensionless中国煤化工Fig. 1. The sketch of the comparison of oil recovery between a:TYHc N M H Gotype and priallysimilar model, in which the ith dimensionless parameter is given as a small deviation from the prototype.www.scichina.com444Science in China Ser. E Engineering & Materials Science 2005 Vol.48 No.4 441- -453the protoype, Sam = 5.lm M,.,"I. (o) - n(D>,,.5.. )tipmeans the deviation of the oil recovery of the model from the prototype. One curve ap-plies to the situation, in which the ith dimensionless parameter of the model deviatesfrom the prototype, and the other to the fully similar model. Subscripts m and p indicatethe model and the reservoir prototype, respectively. Arranging the sensitivity factor S; inorder, we can conveniently exhibit the importance degree of each dimensionless pa-rameter. In the following sections, we take the similarity of water-flooding reservoir asan example to demonstrate the procedure.2 Governing equations and solution procedureLet us consider two-phase porous media flows of water and oil. Suppose the flowobey Darcy's law. Taking into consideration the fluids of constant viscosity, the capillaryand gravitational forces and the compressibility of fluids and rock, the governing equa-tions are as follows:v.|P。Kev(p+ P。gVz)+Q0 =-(中P。s。)(4)Ho2ta(中Pwsw)μwV(p+Pw gVz)|+Qw =(5)Q。=no(Po (2-2(-)-(-p,),.(6)2。ln(ro/%)ew = P.918(x)8()+ Puwkw(Pwt二Pw28(x-x)8(y-yp),(7)4h2μw In(ro/ro)where p,从, p, K, and s mean pressure, density, viscosity, effective permeability andsaturation, subscript w and 0 indicate water and oil phases, respectively. And pc is thecapillary force, φ the porosity of rock, g the gravitational acceleration, (xp, yp) the coor-dinate of production well, q1 the injection rate per unit thickness and r。the well radius.The constitutive equations areP。= Poo(1+ C。(P。+ Po8Z - Poo)),(8)Pw =Pwo(1+Cw(Pw +Pw8Z- Pwo)),(9)Pw+P.+Po8Z+Pw8Z_ Pwo+ Pooφ=内(10)(1+e(2中国煤化工in which the symbols have the same meaning asMHCN M H G indicates physi-cal quantities at a certain condition, Co, Cw and Co the compressibility of oil, water androck, respectively.Copyright by Science in China Press 2005Sensitivity analysis of dimensionless parameters for physical simulation of water-flooding reservoir 445The capillary force equation can be expressed aspe=p。- Pw = σcosθ↓φ/KJ(sw),(11)where K denotes the absolute permeability, σ the interfacial tension at the interface ofwater and oil phases, θ the contact angle between water and oil phases, and J(sw) thecapillary force function.The saturation relation reads ;s。+sw =1.(12)Initial conditions are given asPo l=o= Poi, .(13)sw l=o= swi,(14)and boundary conditions areap=0,ap.+P18=0 l= o,W.(15)axdyaThe constitutive equations can be further simplified. According toP。= Poo(1+ C。(Po + Po8z - Poo))= Poo(1+ C。(P。+ Poo(1+C。(Po + Po8Z- Poo))gz- Poo)),omitting the term C。2(p。 +P.gz- poo), eq. (8) can be reduced toP。= Poo(1+ C。(P。+ Poo8Z - Po)).(16)Analogously, we havePw =Pwo(1 +Cw(Pw + Pwo8Z- Pwo)),(17)φ=帕|1+C。(Pw +P。t P.08Z +Pw08Z_ Pwo+ Poo 1.(18)2The governing equations are discretized with the finite difference method and solved bythe implicit pressure-explicit saturation methodl. The discrete implicit oil pressureequation turns out:a.j,kPoi-1.j.k +b,j.kPoi+1.j.k +Cij.kPoij 1.k +d.j.k Poi,j+1.k(19)+ei;j,k. Poi.j.k-1 + f,j,kPoi,j.k+1 + 8i.jkH中国煤化工YHCNMHGwherewww.scichina.com446Science in China Ser. E Engineering & Materials Science 2005 Vol.48 No.4 441一453Aa.j.k =aoi,jk +Aawij.k,b.j.k =bo;,jk + Abw,j.k,Ci,j,k =Coi,j,k + Acwi,j,k,dij.k =doi;jk + Adwij,k,el.j.k =ei.j.k +Aewij,k, f.j,k = foij.k +Afwij.k,8ij,k = 8oi,j.k + Agwij.k,h.jk = h.ij.k + Ahwij,k + A(Cw.j.k Pei;j-I.k +w.jkPa-1.j.k + bwi.jkPei+t.j.k.+ dijpPe.j+I.k + ew.j.k Pci.j./-. + fi.jPei.j.+t + 8wijkPe.,jk),2元2入1bxi--bxi+-byj-aln.j,k =(Ox-1 + Ox,)Qx_' 0u.J.~ (Ox; + Ox-)Qx;’Uu.J.~ (Oy;-1 + Ay;)Qyj='2h22j+一lzk--lzk+5dr,j.k =lij,k =-fi.j.k=-(Oy; + Ay;-)Oy;(Ozk-1 + Ozk )Qzk-l(Ozk- 4 + Szk )Ozk8it.jk=ai.j.k +Bbr.j.k +Ci,.k +dn.j.k +eijk+ fijh + Sij.k (.jkProCi + Pi.jk%oCo),2(1 1P.... 18-h 1ρ18)lzk-li,j,k-;hu.j.k =-Pu.,jkq!-- -Ozk-1 + Szk- Su.,k (.J.kProCi + Pli,kAoC,)Pi.Jk.At"in which n = P1 KK,1/M denotes mobility coefficient, l=o,w, A=p。/Pw.Solving eq. (19) leads to the solution of oil phase pressure. Then we can explicitlyobtain oil saturation by substituting the oil pressure into eq. (4). In order to improve theaccuracy and reduce the mass conservation error, we divide the time step for solving oilpressure into several small steps. In each small step, the saturation equation is solvedexplicitly. The mobility coefficient takes the harmonic mean between the neighboringgrids for solving oil pressure equation. For example, the mobility coefficient in x direc-tion is△x;+1 +△x;i+z,,kAx:+1/A+l,j.k + Ax/4,j,k .so are those in y and z directions. When solving oil saturation, the mobility coefficienttakes the value at the upstream node, accounting for the lag variation of the oil saturationin the flow direction.中国煤化工The validation of the code is made by comparin.MYHCN M H G with the theoryof Buckley-Leverettl2 . The assumptions for Buckley-leverett equation are: one- dimen-sional flow occurring in a homogenous constant thickness rock layer, the negligible cap-CopyihhyScience in China Press 2005Sensitivity analysis of dimensionless parameters for physical simulation of water-flooding reservoir 447illary and gravitational forces, the compressibility of fluids and porous medium beingnot considered, and the constant density and viscosity of fluids. So the position of agiven water saturation x(Sw, t) at time t is determined by the following expression:x(sw ,t)= x(sw ,0)+.r(8o0 [eod,Aφin which x(Sw, 0) represents the initial distribution of water saturation, Q(t) the injectionrate, φ the porosity, A the cross section area of the reservoir, and f'(sw) the derivativeof the water ratio to the water saturation. Fig. 2 shows the comparison of the displace-ments of water saturation contours between the present numerical results and the solu-tion of Buchley-Leverett equation at t= 3000 s and t= 7000 s, respectively. In Fig. 3plotted is the evolution of water saturation at the two positions ofx= 10 m andx= 15 mcompared with the theory of Buchley-I everett. It is seen that both the spatial distributionand time evolution of the water saturation agree well with the theoretical results, whichcompletes the validation of the numerical code employed.1.0. Initial water saturation contourNumerical contour at 3000 s0.9上Numerical contour at 7000 sTheoretical contour of BL at 3000 sTheoretical contour ofBL at 7000 s0.70.60.50.40.30.2_1」151:22:Distance/mFig. 2. Comparison of the displacement of iso water saturation contours between theoretical and numerical resultsatthetimeoft=3000sand7000s.3 Similarity criteriaConsider a quarter of three-dimensional five-spo中国煤化工-flooding res-ervoir with its length of l, width of w, and thicknesYHCNMHGandinjectionwells locate at the end of a diagonal, their horizontal coordinates being (0, 0) andwww.scichina.com448Science in China Ser. E Engineering & Materials Science 2005 Vol.48 No.4 441- -4531.0Theoretical results at 10 mTheoretical results at 15 m0.9Numerical results at 10 m0.8Numerical results at 15 m0.7g 0.60.50.40.3 F0.20L_J20004000 6000800010000 12000 14000Time/sFig. 3. Comparison of water saturation vs time between theoretical and numerical results at the positions ofx= 10sand 15 m.(xp, yp), respectively. To derive the similarity criteria, we introduce the dimensionlessindependent variables:qrtqjtxp=~, yp=一, Zp=~, tp=-YkRφxpyxh(1-Sew -s) φxk Yk hSsdependent variables:pcKrowhPw Krowh .p。Krowh.PcD, PwD=上,PoD9[Hw9{rHw9rMwsome other dimensionless parameters:HoD =,PooDCoD =Coquw, c_ C. qrwUwPwo .K rowhKrowhCq1HwPwoKrowhPooKrowhCoD =-PwoD =PoD =914wPo;KrowhPwf Krowh。.Poin=9iHwPwD='9yHwTCoD =roD =-and the normalized saturation and permeability:中国煤化工CNMHG,s。一SroMHs。=KoDAwD=vAsKcwoKrowCopyright by Science in China Press 2005Sensitivity analysis of dimensionless parameters for physical simulation of water-flooding reservoir 449in which XR, yR and ZR are reference lengths in three directions, respectively, Sro and Scwdenote the residual oil saturation and the irreducible water saturation, respectively, Keworefers to the effective permeability of oil phase under the condition of the irreduciblewater saturation, Krow the effective permeability of water phase under the condition ofthe residual oil saturation, and the subscript D means dimensionless parameter. Substi-tuting all these variables into the governing equations, constitutive equations, and theinitial and boundary conditions, we get a set of twenty-four scaling criteria of water-flooding system as follows:Kw. Yk,趣、上、上、,互、Sew. Sro,KrowKcwoKrow xRzR xR YkR’xR’ Os’ Ospo。Swi-Scw ovVK: cosθKrowhCoq.Ww Cw qy从w9jHwHwPwo9IMwPwo8ZR,KrowhrowhCp91Mw PwoKrowh PooKrowh PwfK row hPo;Krowh,J(5.).9il4w9nlHw9q1HwThey are defined as不,E.,T24 hereafter. From the physical point of view, π|denotes the ratio of the oil permeability under the condition of the irreducible watersaturation to the water permeability under the condition of the residual oil saturation, T2and兀3 the dimensionless permeability of water and oil phase, respectively, 兀s and πsthe similarity of geometry, π6,兀7, T8 and πg the similarity of well position and wellradius, π 10 the ratio of the irreducible water saturation to the mobile oil saturation, π11the ratio of the residual oil saturation to the mobile oil saturation, π 12 the reduced initialwater saturation, π13 the ratio of the capillary force to the reservoir pressure differenceinduced by the injection rate q1, π14 and π1s the ratios of the viscosity and density ofwater to oil respectively, π16 the ratio of the gravitational force to the driving force, π17,π18 and π19 the relative volume variation rations of oil, water and rock caused by thereservoir pressure difference induced by the injection rate qI, respectively, 兀20,兀21,兀22and T23 the respective ratios of the reference pressure of oil and water, the pressure ofwell bottom, and the initial pressure to the reservoir pressure difference and π24 the cap-illary force function.Consider a prototype with its parameters listed in Table 1. According to theabove-derived scaling criteria, we may design a completely similar model with its pa-rameters listed in Table 1, too. Performing numerical calculations for the model and theprototype, we obtain the oil recovery curves and water ratio at production well in bothcases, as plotted in Figs. 4 and 5. The results of the"中国煤化rearefoundingood agreement with each other, implying that the:ling criteria iscomplete. However, we can also see from Table 1 uidl ausiipii osiuidrity:0HCNMHGmeans anwww. scichina.com450Science in China Ser. E Engineering & Materials Science 2005 Vol.48 No.4 441- -453Table 1 Parameters of the prototype and the corresponding scaled modelPara-LnhkeworoC。CWmeters /m_/m/m's-1_X 10-12/m2 X 10-'/m2/m_X 10~1%/Pa-1 X 10-1/Pa-1Prototype 1401401(8X 1030.370.780.15.0Model0.70.054X 1050.4070.8585X 10+8.8xp8PwTmeters /m/m_ /ms-2__ X 10-1%/Pa-1__ X10/Pa X 10/Pa X 10/PaX 10/Pa X 10-/Nm~9.86.012.010.02.5017816.610.99.092.271.0 r0.8-已0.6t宫0.4Prototype9-0.2十o0203040Dimensionless timeFig. 4. Comparison of oil recovery between the prototype and the corresponding scaled model.1.0r0.80.6 I0.4 F中0.2 F中国煤化工.MHCNMHGDimensionless tihnFig. 5. Comparison of the water ratio between the prototype and the corresponding scaled model.Copyright by Science in China Press 2005Sensitivity analysis of dimensionless parameters for physical simulation of water-flooding reservoir 451impractically large gravitational acceleration. Therefore, we have to relax some of themin actual modeling. Nevertheless, we should identify which one is negligible at first.4 Sensitivity factors of dimensionless parametersAccording to the definition of the sensitivity factor aforementioned, we can find thatsensitivity factor S: varies with dimensionless parameter πi. For the prototype defined bythe parameters listed in Table 1, the dimensionless parameters are evaluated as follows:π1=2.108, π4=1, πs=14, π6=1, π7=1, πg=7.143x10 , π1o=0.294, π11=0.177, π12=0, π13= 5.781x10-, π14=5, π1s=0.8, π16= 4.533x10-, π17= 1.730x10-, π18=1.081x10-, π19= 1.297x10-3, π20=π21=π23=5.55, π22 =4.625, π2 and π3 are normal-ized permeability of oil and water phases, and π24 the capillary force function. Set thedeviation coefficient Wi of each dimensionless parameter to be 1% and -1% respectivelyexcept for those representing the similarity of well locations, and keep the others identi-cal between the model and the prototype. Then, we can readily derive the sensitivityfactors as listed in Table 2.Table 2 Sensitivity factors of dimensionless parametersS;1wWI= 1%1.847X 10-11.845X 10-1.864X10-1 3.713X10-2 4.065X10-3 1.148X10-3 7.885X 10-w:=-1% .1.867X 10-1.870X 10-1.861X10-' 3.461X10~2 3.990X10-3 1.167X10-3 7.885X 10-3S11121314151617wi= 1% 9.974X 10-1.160X100 3.952X10-4 1.848X10-1 1.014X10-1 4.742X 10~2.164X 10w=-1%,9.974X 10+1.260X 1003.952X 1041.864X101 1.012X10-1 4.724X10-3 2. 164X10-318920222W:= 1%2.879X10-3 3.049X10-3 1.560X10-3 1.272X10~2 3.260x10-2 2.755X10-2 3.952X 10+wi=-1%2.879X10-3 3.048X 10~1.560<10-3 1.274X10-2 3.244X102 2.751X10-2 3.952X104We can see from Table 2 that the order of sensitivity factors ranges from 10 4 to 10.Apparently, the larger the sensitivity factor is, the more important the corresponding di-mensionless parameter is. Hence, we can easily make a choice of dominant dimen-sionless parameters based on the numerical results. If we just reserve those parametersranging from 10-' to 10', the scaling law looks likeKKw swi-S&w μ。PooArow.Kcwo K rowAs从w Pw0This implies that the ratio of the oil permeability under the condition of the irreduci-ble water saturation to the water permeability under the condition of the residual oilsaturation, the dimensionless permeability of water and oil, the density and viscosityratios of water and oil, and the reduced initial wat中国煤化工lost importantparameters in water-flooding modeling and should*1HC N MH Gse results ex-actly reflect what the most important and natural Tactors are in water-ilooding flows.www.scichina.com452Science in China Ser. E Engineering & Materials Science 2005 Vol.48 No.4 441- -453Among the attributes related to porous medium, the permeability exerts more influenceon flows than others do. Among the properties of fluids, the density and viscosity aremore important. Other factors, such as the gravitational force, the compressibility ofwater, oil and rock and the capillary force, may be relaxed if they contradict to thedominant ones. The reason is that the error induced by relaxing the secondary dimen-sionless parameters is far less than the dominant ones. We can also qualitatively drawthese conclusions from the value of the dimensionless parameter itself. For example, π13means the relative importance between the capillary force and reservoir pressure differ-ence induced by the injection rate qI. Its value implies that the capillary force is threeorders less than that of the reservoir pressure difference under the condition of highpermeability and large gradient of the driving force. n 16 represents the relative impor-tance between the gravitational and the driving force. Its value of 4.533X 10-2 impliesthat the gravitational force is two orders less than that of the driving force for the case ofnot very thick reservoir and the small density difference between water and oil. Never-theless, it is hard to qualify the dominance degree of all the dimensionless parameters inthis way. Particularly, this qualitative analysis is of no help for quantifying the impor-tance of the dimensionless parameters. This is exactly where the advantage of the pro-posed numerical approach lies.5 ConclusionsIn the present paper, the authors have proposed a numerical approach to analyze thesensitivity of each dimensionless parameter. A universally physical definition of the sen-sitivity factor is given to quantify the dependence degree of a target function on the di-mensionless parameters. In regard to a water-flooding reservoir, we have derived a com-plete set of scaling criteria, which involves the effects of the gravitational force, the cap-illary force and the compressibility of water, oil and rock. By calculating the sensitivityfactor numerically, we have further quantitatively identified the dominant dimensionlessparameters. The results show that in the case of water-flooding reservoir, the order of thesensitivity factors ranges from 10~ to 10, and the most important scaling parameters arethe ratio of the oil permeability under the condition of the irreducible water saturation tothe water permeability under the condition of the residual oil saturation, the density andviscosity ratios between water and oil and the reduced initial water saturation. Verylikely, the numerical optimization can be applied to other physical problems.Acknowledgements This work was supported by the National 973 Project (Grant No. G1999022511), and theInnovation Project of Chinese Academy of Sciences.References1. Shen Pingping, Fundamental research on largely enhancing oil recovery, China Basic Science (in Chinese),2003, 2: 9-14.2. Sedov, L. I, Similarity and Dimensional Methods in Mecl中国煤化工press, 1959.3. Geertsma, J.. Croes, G. A.. Schwart, N., Theory of dimen:YHC N M H Gleum reservoir, Trans.AIME, 1956, 207: 118一127.4. Kimber, K. D.. Faroug, A. S. M., Puttagunta, V. R., New scaling criteria and their relative merits for steamCopyright by Science in China Press 2005Sensitivity analysis of dimensionless parameters for physical simulation of water-flooding reservoir 453recovery experiments, The Journal of Canadian Petroleum Technology, 1988, 27(4): 86一94.5. Islam, M. R., Faroug, A. s. M., New scaling criteria for polymer, emulsion and foam flooding experiments,The Journal of Canadian Petroleum Technology, 1989, 28(4): 79- - -97.5. Islam, M. R., Farouq, A. S. M.. New scaling criteria for chemical flooding experiments, Jourmal of CanadianPetroleum Technology, 1990, 29(1): 29- 36.7. Shen Pingping, The theory and experiment of water and oil flow in porous media (in Chinese), Beijing: OilIndustry Press, 2000.8. Pozzi, A. L., Blackwell, R. J.. Design of laboratory models for study of miscible displacement, SPE, 1963, 4.28- 40. .9. Kong Xiangyan, Chen Fenglei, Similarity criteria and physical simulation for water flooding (in Chinese), .Petroleum Exploration and Development, 1997, 24(6): 56- -60.10. Khalid, A.. Antonin, S., Petroleum Reservoir Simulation, London: Applied Science Publishers, 1979.11. Kong Xiangyan, Advanced Mechanics of Fluids in Porous Media, Hefei: China Science & Technology Uni-versity Press, 1999.中国煤化工MYHCNMHGwww. scichina.com

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