analytical model of waterflood sweep efficiency in vertical

Research ArticleAnalytical Model of Waterflood Sweep Efficiency in VerticalHeterogeneous Reservoirs under Constant Pressure

Lisha Zhao12 Li Li2 Zhongbao Wu2 and Chenshuo Zhang2

1School of Earth and Space Sciences Peking University Beijing 100871 China2Research Institute of Petroleum Exploration amp Development PetroChina Beijing 100083 China

Correspondence should be addressed to Lisha Zhao zhaolisa1121163com

Received 20 June 2016 Revised 27 September 2016 Accepted 25 October 2016

Academic Editor Jian Guo Zhou

Copyright copy 2016 Lisha Zhao et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

An analytical model has been developed for quantitative evaluation of vertical sweep efficiency based on heterogeneous multilayerreservoirs By applying the Buckley-Leverett displacement mechanism a theoretical relationship is deduced to describe dynamicchanges of the front of water injection water saturation of producing well and swept volume during waterflooding under thecondition of constant pressure which substitutes for the condition of constant rate in the traditional way Then this method ofcalculating sweep efficiency is applied from single layer tomultilayers which can be used to accurately calculate the sweep efficiencyof heterogeneous reservoirs and evaluate the degree of waterflooding in multilayer reservoirs In the case study the water frontalposition water cut volumetric sweep efficiency and oil recovery are compared between commingled injection and zonal injectionby applying the derived equations The results are verified by numerical simulators respectively It is shown that zonal injectionworks better than commingled injection in respect of sweep efficiency and oil recovery and has a longer period of water freeproduction

1 Introduction

Heterogeneity is a common problem encountered in oil-bearing formationsThemost significant property that affectswaterflooding performance is the matrix permeability andits variation in the vertical direction causes displacing fluidto advance faster in zones of higher permeability and resultsin earlier breakthrough in such layers This phenomenonnegatively affects volumetric sweep efficiency and leads tolow ultimate oil recovery [1ndash5] Therefore it is considerablyimportant to establish a quantitative characterizationmethodfor calculating sweep efficiency by thismethod we can deter-mine the criteria of dividing and reorganizing layer seriesand relieve interlayer interference in stratified waterfloodingreservoir development

Vertical sweep efficiency is the fraction of vertical cross-sectional area of the reservoir between injection and produc-tion wells that is swept by water at a given time It can be usedto estimate and predict the unswept fraction of the reservoirby water injection and the additional oil recovery potential

Buckley and Leverett [6] presented the fractional flowequation based on mass conservation equation Welge [7]proposed that waterflood sweep efficiency could be obtainedby water cut and water saturation curves To multilayerreservoirs Stiles [8] presented the first model for waterfloodcalculations in stratified reservoirs and assumed velocitiesin different layers to be proportional to their absolutepermeabilities Later models for noncommunicating layerswithout crossflow [9ndash11] and models for communicatinglayers with complete crossflow [12 13] were carried out El-Khatib [14 15] investigated the effect of crossflow on theperformance of stratified reservoirs and presented a closedformanalytical solution for communicating stratified systemswith log-normal permeability distributions Zhou et al [1617] developed a model of linear nonpiston waterflood andcalculated the sweep efficiency by production compartmentEl-Khatib [18 19] considered the gravity effect of the dip angleto sweep efficiency and Snyder and Ramey [20] improvedon previous work while changing the property of layers ina logical and consistent manner Prince [21] investigated the

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 6273492 9 pageshttpdxdoiorg10115520166273492

2 Mathematical Problems in Engineering

wh

Le

Figure 1 Schematic representation of stratified system

effect of mobility ratio and the Dykstra-Parsons permeabilityvariation coefficient (VDP) on the performance

Firstly many of the models mentioned above assumedpiston-like displacement when predicting waterflooding per-formance in stratified reservoirs which was not consistentwith the frontal advance theory Secondly the abovemen-tioned study of flooding process was usually based on thecondition of constant rate which provided a constant volumeof water injected into each layer by production compartmentaccording to each layerrsquos property However for multilayeredreservoir development the production pressure betweeninjection and production well is usually stable while the rateof liquid produced in each layer is changeable So in thismodel the reservoir is divided into a number of layers Andeach layer is considered as non-piston-like displacement onthe condition of constant pressure A mathematical modelis developed for calculating waterflood sweep efficiency byextending the Buckley-Leverett displacement mechanism

2 Mathematical Model Buildingand Assumptions

Figure 1 is a schematic representation of the stratified systemThe production pressure between injection and productionwell is equal to a constant value Δ119875 = 119875119908119891 minus119875inj where 119875inj isthe bottom pressure of injection well MPa 119875119908119891 is the bottompressure of producingwellMPaΔ119875 is the pressure drop fromthe injection well to producing well MPa

The following assumptions are made

(1) The system is divided into a number of homogeneouslayers each has a uniform thickness and constantpermeability

(2) The system is linear and horizontal and of constantthickness

(3) The flow is isothermal and incompressible and obeysDarcyrsquos law

(4) The displacement is non-piston-like with a two-phaseregion where both oil and water exist

(5) Capillary and gravity forces are negligible(6) The system is noncommunicating with no crossflow

allowed between adjacent layers(7) The relative permeability characteristics are the same

for all layers(8) The initial fluid saturation is uniform at the irre-

ducible water saturation(9) The porosity is assumed to be constant in all layers

3 Calculation for Single Layer

Theperformance before displacement front reaches the outletface is quite identical in each layer After water breakthroughinjection continues assuming that the front of displacementremains going forward and the outlet saturation rises from119878119908119891 to 1 minus 119878119900119903 where 119878119908119891 is the water saturation of dis-placement front f and 119878119900119903 is the residual oil saturation fFigure 2 illustrates the three stages duringwaterfloodingThisprocess is split into three subperiods and piecewise functionsof sweep efficiency are deduced with the breakthrough timeas the endpoint

Based on the material balance equation and Buckley-Leverettrsquos frontal advance theory the total volume of waterinjected in a given time equals

int1199050119876119889119905 = int119909119891

1199090

120601119908ℎ [119878119908 (119909 119905) minus 119878119908119888] 119889119909

= 120601119908ℎ 119909119891 minus 11990901198911015840119908 (119878119908119891) (1)

where 119876 is the flow rate of oil m3s 119908 is the width of eachlayer m ℎ is the height of each layer m 120601 is the porosityof matrix f 119909119891 is the displacement front position m 1199090 isthe initial position of displacement front m 119891119908(119878119908) is thefractional flow of water f

The calculation of sweep efficiency in a single layer isgiven by

120578 = int119909119891

0((119878119908 minus 119878119908119888) (1 minus 119878119900119903 minus 119878119908119888)) ℎ 119889119909

ℎ119871119890= 119878119908 minus 1198781199081198881 minus 119878119900119903 minus 119878119908119888 sdot

119909119891119871119890

(2)

where 119871119890 is the length of model m Before water break-through average saturation 119878119908 is determined by fractionalflow curve (119891119908 minus 119878119908) [6] which is a constant value So theflooding swept volume is only determined by the position ofthe water frontal advance after breakthrough (2) deforms asfollows

120578 = 119878119908 minus 1198781199081198881 minus 119878119900119903 minus 119878119908119888 (3)

where 119878119908 is no longer a constant and should be determinedby the saturation of the producing well 119878119908119890 Consequently

Mathematical Problems in Engineering 3

1

0

Sw

Sor

Swf

Swc

xf

(a) Before water breakthrough

Le

1

0

Sw

Sor

Swf

Swc

(b) At the time of breakthrough

Le

Swe

Sor

Swc

1

0

Sw

Swf

(c) After water breakthrough

Figure 2 Different stages during wateroil displacement

it is necessary to calculate the waterflood front locationbefore breakthrough and the saturation of the outlet face119878119908119890 after breakthrough Firstly we analyze the waterfloodingperformance in a single layer by studying the pressurerelationship based on the constant pressure condition

31 Pressure in Oil and Water Region By Applying Darcyrsquoslaw for a single layer the total velocity at a certain position 119909119886between injection well and the front location is given by

V = V119908 + V119900 = minus119870(119870119903119908120583119908 +119870119903119900120583119900 )

119889119901119889119909 (4)

where 119870 is the absolute permeability 10minus3 120583m2 119870119903119908 is thewater relative permeability f 119870119903119900 is the oil relative perme-ability f 120583119908 is the viscosity of water mPasdots 120583119900 is the viscosityof oil mPasdots

Based on the Buckley-Leverett frontal displacement the-ory the water cut is expressed as

119891119908 (119878119908) = 119876119908119876119908 + 119876119900 =1

1 + (119870119903119900120583119908119870119903119908120583119900) (5)

Substituting (5) into (4) and integrating we obtain thepressure drop between the positions of 119909 = 119909119886

Δ119875119886 = 119875119886 minus 119875inj = V120583119908119870 int1199091198860

119891119908 (119878119908)119870119903119908 119889119909 (6)

The location of the injection well is treated as the positionof119909 = 0 and the frontal displacement function can bewrittenas

119909 = 1198911015840119908 (119878119908)120601119908ℎ int1199050119876119889119905 (7)

Equation (7) can be deformed as

119909 = 119909119891 1198911015840119908 (119878119908)1198911015840119908 (119878119908119891) (8)

From the position of 119909 = 0 to 119909 = 119909119886 the water saturationchanges from 1 minus 119878119900119903 to 119878119908119886 Substituting (8) into (6) we get

Δ119875119886 = V120583119908119870 sdot 1199091198911198911015840119908 (119878119908119891) sdot 119865 (119878119908119886) (9)

where 119865(119878119908119886) is given by

119865 (119878119908119886) = int1198911015840119908(119878119908119886)

0

119891119908 (119878119908)119870119903119908 1198891198911015840119908 (119878119908) (10)

32 Time ofWater Breakthrough The pressure drop from theinjection well to the displacement front location is defined asΔ119875119891 Since the average saturation is a constant before waterbreakthrough setting119865(119878119908119891) to value 119886 and substituting value119886 into (9) yield

Δ119875119891 = 119875119891 minus 119875inj = V120583119908119870 sdot 1199091198911198911015840119908 (119878119908119891) sdot 119886 (11)

Substituting (7) into (11) we get

Δ119875119891 = V120583119908119886119882 (119905)120601119908ℎ119870 (12)

where 119882(119905) represents the accumulated volume of waterinjected from the time of beginning and it is equal to the totalliquid volume produced from the oil well

119882(119905) = int1199050119876119889119905 = int119905

0119908ℎV 119889119905 (13)


In the pure oil region there is only one phase and thepressure drop of this region is

Δ119875119900 = Δ119875 minus Δ119875119891 = 120583119900V119870 (119871119890 minus 119909119891) (14)

Substituting (11) and (12) into (14) we get

Δ119875119891 = 119886Δ119875119882(119905)120601119908ℎ120583119903119871119890 minus 120583119903119882(119905) 1198911015840119908 (119878119908119891) + 119886119882 (119905) (15)

where 120583119903 is the ratio of oil viscosity and water viscosity fTaking the derivative of (13) and substituting it into (12) weobtain

Δ119875119891 = 120583119908119886119882 (119905)120601 (119908ℎ)2119870119889119882(119905)119889119905 (16)

Combining (15) with (16) and integrating and rearranging theequation the accumulated volume of water from 119905 = 0 to 119905 =119905119899 is obtained

119882(119905119899) = radic(120583119900120601119908ℎ119871119890)2 + 2 [120583119908119886 minus 1205831199001198911015840119908 (119878119908119891)] [120601 (119908ℎ)2119870Δ119875119905119899] minus 120583119900120601119908ℎ119871119890

120583119908119886 minus 1205831199001198911015840119908 (119878119908119891) (17)

Substituting (7) into (17) the position of displacement front119909119891 is given by

119909119891 = 1198911015840119908 (119878119908119891)120601

sdot radic(120583119900120601119871119890)2 + 2 [120583119908119886 minus 1205831199001198911015840119908 (119878119908119891)] [120601119870Δ119875119905] minus 120583119900120601119871119890

120583119908119886 minus 1205831199001198911015840119908 (119878119908119891) (18)

When the front reaches the outlet face 119909119891 equals 119871119890substituting it into (18) and rearranging the equation the timeof water breakthrough can be obtained

119879 = [(120583119908120601119871119890119886) 1198911015840119908 (119878119908119891)]2 minus (120583119900120601119871119890)2

2 [120583119908119886 minus 1205831199001198911015840119908 (119878119908119891)] 120601119870Δ119875 (19)

33 Saturation after Water Breakthrough After the time ofwater breakthrough the producing well starts to producewater according to (9) the pressure drop between the twowells is

Δ119875 = V120583119908119870 sdot 1199091198911198911015840119908 (119878119908119891) sdot 119865 (119878119908119890) (20)

where 119878119908119890 is the saturation of the producing well f Afterbreakthrough water injection continues assuming that thefront of displacement remains going forward and the outletsaturation rises from 119878119908119891 to 119878119908119890

Substituting (7) into (13) and taking the derivative we get

119889119882 (119905) = 119908ℎV119889119905 = 120601119908ℎ119871119890119889 11198911015840119908 (119878119908119890) (21)

Solving (20) and (21) simultaneously we get

119889119905 = 1205831199081198711198902120601119870Δ119875 sdot 119865 (119878119908119890)1198911015840119908 (119878119908119891)1198891

1198911015840119908 (119878119908119890) (22)

Making the integral of (22)

119905 minus 119879 = 1205831199081198711198902120601119870Δ119875 119866( 11198911015840119908 (119878119908119890)) (23)

where 119866(119878119908119890) is given by

119866( 11198911015840119908 (119878119908119890)) = int

11198911015840119908(119878119908)

11198911015840119908(119878119908119891)

119865 (119878119908119890)1198911015840119908 (119878119908119891)119889

11198911015840119908 (119878119908119890) (24)

Getting (8) deformed and substituting it into (24)

119905 minus 119879 = 1205831199081198711198902120601119870Δ119875 119866( 119909119871119890 sdot1

1198911015840119908 (119878119908)) (25)

Equation (25) shows the water saturation of any positionwithin the two wells versus injecting time after water break-through Equation (23) is a critical condition of (25) since itrefers to the position of the oil well

4 Calculation for Multiple Layers

41 Water Breakthrough in the First (Most Permeable) Layer 119894(1) The layer 119894

The time of water breakthrough 119879119894 in layer 119894 can becalculated according to (19) and the sweep volume oflayer 119894 is calculated by

119860 119894 = 119878119908 minus 1198781199081198881 minus 119878119900119903 minus 119878119908119888 sdot 119908ℎ119894119871119890 (26)

where 119878119908 is as in Welgersquos equation [7]

119878119908 = 119878119908119888 + 11198911015840119908 (119878119908119891) (27)

(2) The other layers for example layer 119895Substituting 119879119894 into (18) we can get the front position119909119891119895 of layer 119895The sweep volume of layer 119895 is

119860119895 = 119878119908 minus 1198781199081198881 minus 119878119900119903 minus 119878119908119888 sdot 119908ℎ119895119909119891119895 (28)


(3) Total sweep efficiency

120578 = 119860 119894 + sum119899minus1119895=1 119860119895

ℎ119871119890 (29)

42 Water Breakthrough in the Medium (including the Least)Permeable Layer 119894(1) The layer 119894

It is the same as in Section 41(1)(2) Layers with breakthrough before layer 119894 for example

layer 119895The time of breakthrough119879119895 in layer 119895 is calculated by(19)The saturation and water cut of layers that got break-through before 119879119894 can be calculated by

119879119894 minus 119879119895 = 1205831199081198711198902120601

119870119895Δ119875 119866(1

1198911015840119908 (119878119908119890)) (30)

After water breakthrough Welgersquos equation reformsas follows

119878119908 = 119878119908119890 + 1 minus 119891119908 (119878119908119890)1198911015840119908 (119878119908119890) (31)

Substituting (31) into (26) the sweep volume of layer119895 can be obtained(3) Layers without breakthrough for example layer 119896

This is the same as in Section 41(2)(4) Total sweep efficiency

120578 = 119860 119894 + sum119898119895=1 119860119895 + sum119899minus119898minus1119896=1 119860119896

ℎ119871119890 (32)

5 Case Study and Verification

Take one injection-production unit of Yaerxia oilfield inYumen for the case study the parameters of reservoir andfluid properties are listed in Table 1 and the relative perme-ability curves are plotted in Figure 3 By analyzing the data oflogging five layers are classified of the main layer k1g22-2

There are two kinds of injection programs to be com-pared with commingled injection and zonal injection Incommingled injection all layers are injected and producedunder the same pressure condition while in zonal injectionthe pressure drop and the rate of water flow are different andcan be adjusted according to the waterflooding performanceIn this case considering their property differences five layersare divided into two groups in zonal injection Put a packerbetween layer 2 and layer 3 andmake layer 1 and layer 2 groupI and layers 3sim5 group II In order to achieve a balancedfrontal advance the pressure drop of group I is reduced byhalf and that of group II is tripled Use the method aboveto calculate water frontal position and sweep efficiency ofthese two injection programs and the results are verified bya numerical simulator

Table 1 The parameters of reservoir and fluid properties

Variables ValueLength of the model 119871 119890 (m) 200Width of the model 119908 (m) 100Thickness of each layer ℎ (m) 1Pressure drop of the unit Δ119875 (MPa) 10Oil viscosity in the reservoir 120583119900 (mPasdots) 56Water viscosity in the reservoir 120583119908 (mPasdots) 06Average saturation before breakthrough 119878119908 (f) 047Saturation of front before breakthrough 119878119908119891 (f) 033119865(119878119908119891) before breakthrough 119886 (f) 615Residual oil saturation 119878119900119903 (f) 030Irreducible water saturation 119878119908119888 (f) 0321198911015840(119878119908119891) before breakthrough (f) 665Porosity of the model 120601 (f) 01Permeability of layer 1 1198961 (10minus3 120583m2) 166Permeability of layer 2 1198962 (10minus3 120583m2) 13Permeability of layer 3 1198963 (10minus3 120583m2) 08Permeability of layer 4 1198964 (10minus3 120583m2) 24Permeability of layer 5 1198965 (10minus3 120583m2) 12

0

02

04

06

08

1

0 02 04 06 08 1

WaterOil

Sw

Figure 3 Relative permeability curves of the case

51 Water Frontal Position Use the equations for single layerto calculate the time of water breakthrough of the five layersby (19) for both commingled injection and zonal injection1198791is the breakthrough time of themost permeable layer (layer 1)1198791 is equal to 2141 days in the commingled injection programwhile it is 4282 days in the zonal injection program becauseof the pressure variation The water frontal position of alllayers on 1198791 is calculated by (18) The corresponding sweepvolume of each layer is calculated by (26) and (28)The resultsof equation calculation are listed in Table 2 and results ofnumerical simulation are listed in Table 3 We plot the waterfrontal position on1198791 for the two injection programs (Figures4 and 5)The calculation results are comparedwith numericalsimulation results where the former are plotted on the top


Table 2 The results of analytical model in commingled injection and zonal injection

LayersCommingled injection Zonal injection

Breakthroughtime 119879 (day)

Frontal position on1198791 119909119891 (m)Sweep volume on1198791 119860 (m3)



Layer 1 2141 2000 79857 4281 2000 79857Layer 2 2733 1573 62806 5467 1573 62806Layer 3 44418 98 3923 14806 587 23421Layer 4 14806 294 11745 4935 1739 69454Layer 5 29612 147 5881 9871 877 35028

Table 3 The results of numerical model in commingled injection and zonal injection







1

2

5

4

3

(a) Results of equation calculation

1

2

5

4

3

Oil sat

029500

038963

048426

057889

067351

(b) Results of numerical simulation

Figure 4 Results comparison of water frontal position of commin-gled injection on 1198791

and the latter are plotted right below The comparisons showthat the simulation results match the calculation results wellwhich verify the equations of single layer

52 Water Cut and Sweep Efficiency Use the equations formultiple layers to calculate water cut of the model in both of

1

2

5

4

3


1

2

5

4

3

Oil sat

029500

038963

048426

057889

067351


Figure 5 Results comparison of water frontal position of zonalinjection on 1198791

the injection programs by (5) and calculate sweep efficiencyby (29) and (32) and oil recovery can be obtained Thecurves versus time are shown in Figures 6ndash8 respectivelyThecalculation results are compared with numerical simulationresults In the pictures the solid lines refer to equationcalculations and dashed lines refer to numerical simula-tions Generally in oil field the ultimate sweep efficiency is


0

20

40

60

80

100

0 200 400 600 800 1000 1200 1400 1600

Wat

er cu

t (

)

Time (day)

Commingled injection (equation)Commingled injection (simulation)Zonal injection (equation)Zonal injection (simulation)

Figure 6 Results comparison of water cut versus time in commin-gled injection and zonal injection

0

10

20

30

40

50

60

70

Swee

p effi

cien

cy (

)

0 200 400 600 800 1000 1200 1400 1600Time (day)


Figure 7 Results comparison of sweep efficiency versus time incommingled injection and zonal injection

obtainedwhen thewater cut of the reservoir reaches 98Thecomparisons show that whenwater cut of themodel comes to98 the sweep efficiency of commingled injection is 462while it is 581 for zonal injection and the oil recovery ofcommingled injection is 278 while it is 349 for zonalinjection Figures 9 and 10 illustrate the comparison of watersaturation profile between commingled injection and zonalinjection after 12 months and 60 months respectively

Since the numerical simulator has considered the effectof gravity and capillary forces the values of water cutcalculated by numerical simulation are a bit higher than the

0

5

10

15

20

25

30

35

40

Oil

reco

very

()

0 200 400 600 800 1000 1200 1400 1600Time (day)


Figure 8 Results comparison of oil recovery versus time incommingled injection and zonal injection

03

04

05

06

07

08

0 20 40 60 80 100 120 140 160 180 200x (m)

Commingled injectionZonal injection

S w

Figure 9 Comparison of water saturation between commingledinjection and zonal injection after 12 months

results of equation calculations and the breakthrough timeof the simulation is earlier than that of equation calculationHowever the general trends of these curves are in completeagreement which verify the equations of multilayers

53 Results Analysis Thecomparison results of the two kindsof water injection show the following in zonal injectionthe producing degree of the less permeable layers increasesapparently and achieves a much higher sweep efficiencyand oil recovery compared with commingled injection Byusing zonal injection program it can restrain the monolayerbreakthrough of higher permeable layers and solve interlayer


03

04

05

06

07

08

0 20 40 60 80 100 120 140 160 180 200x (m)


S w


conflicts Therefore a balanced frontal advance is achievedIn oil field development engineers shall subdivide the layerseries with zonal injection based on the reservoirrsquos propertydifferences which could significantly reduce heterogeneityand postpone the time of water breakthrough As a conse-quence better oil production is achieved

6 Conclusion

(1) An analytical model is developed to explain thewaterflooding process in the condition of constantpressure which substitutes for the condition of con-stant rate in traditional methods We deduce thetheoretical relationship between the frontal positionof waterflooding and water saturation of producingwell versus time to describe the dynamic changes ofthese variables

(2) By applying this model from one single layer to mul-tilayers this method gives a quantitative expressionof sweep efficiency of the three stages during water-flooding which can be used to accurately calculatethe sweep efficiency of heterogeneous reservoirs andevaluate the degree of waterflooding in multilayerreservoirs

(3) By comparing the performances of commingledinjection and zonal injection in the case study zonalinjection works better than commingled injection inrespect of sweep efficiency and oil recovery and has alonger period of water free production So it is essen-tial to subdivide the layer serieswith zonal injection inorder to reduce heterogeneity and postpone the timeof water breakthrough

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

This work was supported by China National Petroleum CoLtd Major Science and Technology Projects (2012E-3304)

References

[1] N A Berruin and R A Morse ldquoWaterflood performance ofheterogeneous systemsrdquo Journal of Petroleum Technology vol31 no 7 pp 829ndash836 1979

[2] M R Carlson ldquoThe Effect of reservoir heterogeneity on pre-dictedwaterfloodperformance in theDodslandfieldrdquo Journal ofCanadian Petroleum Technology vol 34 no 10 pp 31ndash38 1995

[3] A K Permadi I P Yuwono andA J S Simanjuntak ldquoEffects ofvertical heterogeneity on waterflood performance in stratifiedreservoirs a case study in Bangko field Indonesiardquo in Proceed-ings of the SPE Asia Pacific Conference on Integrated Modellingfor Asset Management pp 157ndash168 Kuala Lumpur MalaysiaMarch 2004

[4] S Ghedan Y A Boloushi K Khan and M B F Saleh ldquoDevel-opment of early water breakthrough and effectiveness of watershut off treatments in layered and heterogeneous reservoirsrdquoin Proceedings of the SPEEAGE Reservoir Characterizationand Simulation Conference SPE-125580-MS pp 872ndash882 AbuDhabi UAE October 2009

[5] B Rashid A H Muggeridge A-L Bal and G J J WilliamsldquoQuantifying the impact of permeability heterogeneity onsecondary-recovery performancerdquo SPE Journal vol 17 no 2 pp455ndash468 2012

[6] S E Buckley andM C Leverett ldquoMechanism of fluid displace-ment in sandsrdquoTransactions of the AIME vol 146 no 1 pp 107ndash116 2013

[7] H JWelge ldquoA simplifiedmethod for computing oil recovery bygas or water driverdquo Journal of Petroleum Technology vol 4 no4 pp 91ndash98 1952

[8] W E Stiles ldquoUse of permeability distribution in water floodcalculationsrdquo Journal of Petroleum Technology vol 1 no 1 pp9ndash13 2013

[9] H Dykstra and R L Parsons ldquoThe prediction of oil recovery bywaterfloodingrdquo in Secondary Recovery of Oil pp 160ndash174 APIDallas Tex USA 2nd edition 1950

[10] A A Reznik R M Enick and S B Panvelker ldquoAnalyticalextension of the Dykstra-Parsons vertical stratification discretesolution to a continuous real-time basisrdquo Society of PetroleumEngineers journal vol 24 no 6 pp 643ndash655 1984

[11] R Tompang and B G Kelkar ldquoPrediction of waterflood per-formance in stratified reservoirsrdquo in Proceedings of the PermianBasin Oil and Gas Recovery Conference pp 213ndash224 MidlandTex USA March 1988

[12] W N Hiatt ldquoInjected-fluid coverage of multi-well reservoirswith permeability stratificationrdquo Drilling and Production Prac-tice API-58-165 American Petroleum Institute New York NYUSA 1958

[13] J E Warren ldquoPrediction of waterflood behavior in a stratifiedsystemrdquo Society of Petroleum Engineers Journal vol 4 no 2 pp149ndash157 1964

[14] N A F El-Khatib ldquoWaterflooding performance of commu-nicating stratified reservoirs with log-normal permeabilitydistributionrdquo in Proceedings of the Middle East Oil Show andConference Bahrain March 1997


[15] N A F El-Khatib ldquoThe application of Buckley-Leverett dis-placement to waterflooding in non-communicating stratifiedreservoirsrdquo in Proceedings of the SPE Middle East Oil Show pp91ndash102 Manama Bahrain March 2001

[16] Y F Zhou Z J Liu C J Mao X D Wang and B ZhangldquoA new calculation of water displacement efficiency for layeredreservoirrdquo Special Oil and Gas Reservoirs vol 15 no 3 pp 72ndash75 2008

[17] Y F Zhou Y J Fang X D Wang and C C Zhou ldquoA newmethod for calculating non-piston displacement efficiency inmultilayered reservoir with waterfloodingrdquo Petroleum Geologyamp Recovery Efficiency vol 16 no 1 pp 86ndash93 2009

[18] N A F El-Khatib ldquoWaterflooding performance of communi-cating stratified reservoirs with log-normal permeability distri-butionrdquo SPE Reservoir Evaluation amp Engineering vol 2 no 6pp 542ndash549 2013

[19] N A F El-Khatib ldquoWaterflooding performance in inclinedcommunicating stratified reservoirsrdquo SPE Journal vol 17 no 1pp 31ndash42 2012

[20] R W Snyder and H J Ramey ldquoApplication of Buckley-Leverettdisplacement theory to noncommunicating layered systemsrdquoJournal of Petroleum Technology vol 19 no 11 pp 1500ndash15062013

[21] A O Prince ldquoBuckley-Leverett displacement theory for water-flooding performance in stratified reservoirrdquo PetroleumampCoalvol 56 no 3 pp 267ndash281 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


wh

Le

Figure 1 Schematic representation of stratified system

effect of mobility ratio and the Dykstra-Parsons permeabilityvariation coefficient (VDP) on the performance

Firstly many of the models mentioned above assumedpiston-like displacement when predicting waterflooding per-formance in stratified reservoirs which was not consistentwith the frontal advance theory Secondly the abovemen-tioned study of flooding process was usually based on thecondition of constant rate which provided a constant volumeof water injected into each layer by production compartmentaccording to each layerrsquos property However for multilayeredreservoir development the production pressure betweeninjection and production well is usually stable while the rateof liquid produced in each layer is changeable So in thismodel the reservoir is divided into a number of layers Andeach layer is considered as non-piston-like displacement onthe condition of constant pressure A mathematical modelis developed for calculating waterflood sweep efficiency byextending the Buckley-Leverett displacement mechanism

2 Mathematical Model Buildingand Assumptions

Figure 1 is a schematic representation of the stratified systemThe production pressure between injection and productionwell is equal to a constant value Δ119875 = 119875119908119891 minus119875inj where 119875inj isthe bottom pressure of injection well MPa 119875119908119891 is the bottompressure of producingwellMPaΔ119875 is the pressure drop fromthe injection well to producing well MPa

The following assumptions are made

(1) The system is divided into a number of homogeneouslayers each has a uniform thickness and constantpermeability

(2) The system is linear and horizontal and of constantthickness

(3) The flow is isothermal and incompressible and obeysDarcyrsquos law

(4) The displacement is non-piston-like with a two-phaseregion where both oil and water exist

(5) Capillary and gravity forces are negligible(6) The system is noncommunicating with no crossflow

allowed between adjacent layers(7) The relative permeability characteristics are the same

for all layers(8) The initial fluid saturation is uniform at the irre-

ducible water saturation(9) The porosity is assumed to be constant in all layers

3 Calculation for Single Layer

Theperformance before displacement front reaches the outletface is quite identical in each layer After water breakthroughinjection continues assuming that the front of displacementremains going forward and the outlet saturation rises from119878119908119891 to 1 minus 119878119900119903 where 119878119908119891 is the water saturation of dis-placement front f and 119878119900119903 is the residual oil saturation fFigure 2 illustrates the three stages duringwaterfloodingThisprocess is split into three subperiods and piecewise functionsof sweep efficiency are deduced with the breakthrough timeas the endpoint

Based on the material balance equation and Buckley-Leverettrsquos frontal advance theory the total volume of waterinjected in a given time equals

int1199050119876119889119905 = int119909119891

1199090

120601119908ℎ [119878119908 (119909 119905) minus 119878119908119888] 119889119909

= 120601119908ℎ 119909119891 minus 11990901198911015840119908 (119878119908119891) (1)

where 119876 is the flow rate of oil m3s 119908 is the width of eachlayer m ℎ is the height of each layer m 120601 is the porosityof matrix f 119909119891 is the displacement front position m 1199090 isthe initial position of displacement front m 119891119908(119878119908) is thefractional flow of water f

The calculation of sweep efficiency in a single layer isgiven by

120578 = int119909119891

0((119878119908 minus 119878119908119888) (1 minus 119878119900119903 minus 119878119908119888)) ℎ 119889119909

ℎ119871119890= 119878119908 minus 1198781199081198881 minus 119878119900119903 minus 119878119908119888 sdot

119909119891119871119890

(2)

where 119871119890 is the length of model m Before water break-through average saturation 119878119908 is determined by fractionalflow curve (119891119908 minus 119878119908) [6] which is a constant value So theflooding swept volume is only determined by the position ofthe water frontal advance after breakthrough (2) deforms asfollows

120578 = 119878119908 minus 1198781199081198881 minus 119878119900119903 minus 119878119908119888 (3)

where 119878119908 is no longer a constant and should be determinedby the saturation of the producing well 119878119908119890 Consequently


1

0

Sw

Sor

Swf

Swc

xf


Le

1

0

Sw

Sor

Swf

Swc


Le

Swe

Sor

Swc

1

0

Sw

Swf





V = V119908 + V119900 = minus119870(119870119903119908120583119908 +119870119903119900120583119900 )

119889119901119889119909 (4)



119891119908 (119878119908) = 119876119908119876119908 + 119876119900 =1

1 + (119870119903119900120583119908119870119903119908120583119900) (5)


Δ119875119886 = 119875119886 minus 119875inj = V120583119908119870 int1199091198860

119891119908 (119878119908)119870119903119908 119889119909 (6)


119909 = 1198911015840119908 (119878119908)120601119908ℎ int1199050119876119889119905 (7)


119909 = 119909119891 1198911015840119908 (119878119908)1198911015840119908 (119878119908119891) (8)


Δ119875119886 = V120583119908119870 sdot 1199091198911198911015840119908 (119878119908119891) sdot 119865 (119878119908119886) (9)

where 119865(119878119908119886) is given by

119865 (119878119908119886) = int1198911015840119908(119878119908119886)

0

119891119908 (119878119908)119870119903119908 1198891198911015840119908 (119878119908) (10)




Δ119875119891 = V120583119908119886119882 (119905)120601119908ℎ119870 (12)


119882(119905) = int1199050119876119889119905 = int119905

0119908ℎV 119889119905 (13)



Δ119875119900 = Δ119875 minus Δ119875119891 = 120583119900V119870 (119871119890 minus 119909119891) (14)


Δ119875119891 = 119886Δ119875119882(119905)120601119908ℎ120583119903119871119890 minus 120583119903119882(119905) 1198911015840119908 (119878119908119891) + 119886119882 (119905) (15)


Δ119875119891 = 120583119908119886119882 (119905)120601 (119908ℎ)2119870119889119882(119905)119889119905 (16)



120583119908119886 minus 1205831199001198911015840119908 (119878119908119891) (17)


119909119891 = 1198911015840119908 (119878119908119891)120601


120583119908119886 minus 1205831199001198911015840119908 (119878119908119891) (18)


119879 = [(120583119908120601119871119890119886) 1198911015840119908 (119878119908119891)]2 minus (120583119900120601119871119890)2

2 [120583119908119886 minus 1205831199001198911015840119908 (119878119908119891)] 120601119870Δ119875 (19)


Δ119875 = V120583119908119870 sdot 1199091198911198911015840119908 (119878119908119891) sdot 119865 (119878119908119890) (20)



119889119882 (119905) = 119908ℎV119889119905 = 120601119908ℎ119871119890119889 11198911015840119908 (119878119908119890) (21)


119889119905 = 1205831199081198711198902120601119870Δ119875 sdot 119865 (119878119908119890)1198911015840119908 (119878119908119891)1198891

1198911015840119908 (119878119908119890) (22)


119905 minus 119879 = 1205831199081198711198902120601119870Δ119875 119866( 11198911015840119908 (119878119908119890)) (23)

where 119866(119878119908119890) is given by

119866( 11198911015840119908 (119878119908119890)) = int

11198911015840119908(119878119908)

11198911015840119908(119878119908119891)

119865 (119878119908119890)1198911015840119908 (119878119908119891)119889

11198911015840119908 (119878119908119890) (24)


119905 minus 119879 = 1205831199081198711198902120601119870Δ119875 119866( 119909119871119890 sdot1

1198911015840119908 (119878119908)) (25)







119878119908 = 119878119908119888 + 11198911015840119908 (119878119908119891) (27)





120578 = 119860 119894 + sum119899minus1119895=1 119860119895

ℎ119871119890 (29)




119879119894 minus 119879119895 = 1205831199081198711198902120601

119870119895Δ119875 119866(1

1198911015840119908 (119878119908119890)) (30)


119878119908 = 119878119908119890 + 1 minus 119891119908 (119878119908119890)1198911015840119908 (119878119908119890) (31)




ℎ119871119890 (32)






0

02

04

06

08

1

0 02 04 06 08 1

WaterOil

Sw


















1

2

5

4

3


1

2

5

4

3

Oil sat

029500

038963

048426

057889

067351





1

2

5

4

3


1

2

5

4

3

Oil sat

029500

038963

048426

057889

067351





0

20

40

60

80

100

0 200 400 600 800 1000 1200 1400 1600

Wat

er cu

t (

)

Time (day)



0

10

20

30

40

50

60

70

Swee

p effi

cien

cy (

)

0 200 400 600 800 1000 1200 1400 1600Time (day)





0

5

10

15

20

25

30

35

40

Oil

reco

very

()

0 200 400 600 800 1000 1200 1400 1600Time (day)



03

04

05

06

07

08

0 20 40 60 80 100 120 140 160 180 200x (m)


S w





03

04

05

06

07

08

0 20 40 60 80 100 120 140 160 180 200x (m)


S w



6 Conclusion




Competing Interests


Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

0

Sw

Sor

Swf

Swc

xf


Le

1

0

Sw

Sor

Swf

Swc


Le

Swe

Sor

Swc

1

0

Sw

Swf





V = V119908 + V119900 = minus119870(119870119903119908120583119908 +119870119903119900120583119900 )

119889119901119889119909 (4)



119891119908 (119878119908) = 119876119908119876119908 + 119876119900 =1

1 + (119870119903119900120583119908119870119903119908120583119900) (5)


Δ119875119886 = 119875119886 minus 119875inj = V120583119908119870 int1199091198860

119891119908 (119878119908)119870119903119908 119889119909 (6)


119909 = 1198911015840119908 (119878119908)120601119908ℎ int1199050119876119889119905 (7)


119909 = 119909119891 1198911015840119908 (119878119908)1198911015840119908 (119878119908119891) (8)


Δ119875119886 = V120583119908119870 sdot 1199091198911198911015840119908 (119878119908119891) sdot 119865 (119878119908119886) (9)

where 119865(119878119908119886) is given by

119865 (119878119908119886) = int1198911015840119908(119878119908119886)

0

119891119908 (119878119908)119870119903119908 1198891198911015840119908 (119878119908) (10)




Δ119875119891 = V120583119908119886119882 (119905)120601119908ℎ119870 (12)


119882(119905) = int1199050119876119889119905 = int119905

0119908ℎV 119889119905 (13)



Δ119875119900 = Δ119875 minus Δ119875119891 = 120583119900V119870 (119871119890 minus 119909119891) (14)


Δ119875119891 = 119886Δ119875119882(119905)120601119908ℎ120583119903119871119890 minus 120583119903119882(119905) 1198911015840119908 (119878119908119891) + 119886119882 (119905) (15)


Δ119875119891 = 120583119908119886119882 (119905)120601 (119908ℎ)2119870119889119882(119905)119889119905 (16)



120583119908119886 minus 1205831199001198911015840119908 (119878119908119891) (17)


119909119891 = 1198911015840119908 (119878119908119891)120601


120583119908119886 minus 1205831199001198911015840119908 (119878119908119891) (18)


119879 = [(120583119908120601119871119890119886) 1198911015840119908 (119878119908119891)]2 minus (120583119900120601119871119890)2

2 [120583119908119886 minus 1205831199001198911015840119908 (119878119908119891)] 120601119870Δ119875 (19)


Δ119875 = V120583119908119870 sdot 1199091198911198911015840119908 (119878119908119891) sdot 119865 (119878119908119890) (20)



119889119882 (119905) = 119908ℎV119889119905 = 120601119908ℎ119871119890119889 11198911015840119908 (119878119908119890) (21)


119889119905 = 1205831199081198711198902120601119870Δ119875 sdot 119865 (119878119908119890)1198911015840119908 (119878119908119891)1198891

1198911015840119908 (119878119908119890) (22)


119905 minus 119879 = 1205831199081198711198902120601119870Δ119875 119866( 11198911015840119908 (119878119908119890)) (23)

where 119866(119878119908119890) is given by

119866( 11198911015840119908 (119878119908119890)) = int

11198911015840119908(119878119908)

11198911015840119908(119878119908119891)

119865 (119878119908119890)1198911015840119908 (119878119908119891)119889

11198911015840119908 (119878119908119890) (24)


119905 minus 119879 = 1205831199081198711198902120601119870Δ119875 119866( 119909119871119890 sdot1

1198911015840119908 (119878119908)) (25)







119878119908 = 119878119908119888 + 11198911015840119908 (119878119908119891) (27)





120578 = 119860 119894 + sum119899minus1119895=1 119860119895

ℎ119871119890 (29)




119879119894 minus 119879119895 = 1205831199081198711198902120601

119870119895Δ119875 119866(1

1198911015840119908 (119878119908119890)) (30)


119878119908 = 119878119908119890 + 1 minus 119891119908 (119878119908119890)1198911015840119908 (119878119908119890) (31)




ℎ119871119890 (32)






0

02

04

06

08

1

0 02 04 06 08 1

WaterOil

Sw


















1

2

5

4

3


1

2

5

4

3

Oil sat

029500

038963

048426

057889

067351





1

2

5

4

3


1

2

5

4

3

Oil sat

029500

038963

048426

057889

067351





0

20

40

60

80

100

0 200 400 600 800 1000 1200 1400 1600

Wat

er cu

t (

)

Time (day)



0

10

20

30

40

50

60

70

Swee

p effi

cien

cy (

)

0 200 400 600 800 1000 1200 1400 1600Time (day)





0

5

10

15

20

25

30

35

40

Oil

reco

very

()

0 200 400 600 800 1000 1200 1400 1600Time (day)



03

04

05

06

07

08

0 20 40 60 80 100 120 140 160 180 200x (m)


S w





03

04

05

06

07

08

0 20 40 60 80 100 120 140 160 180 200x (m)


S w



6 Conclusion




Competing Interests


Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Δ119875119900 = Δ119875 minus Δ119875119891 = 120583119900V119870 (119871119890 minus 119909119891) (14)


Δ119875119891 = 119886Δ119875119882(119905)120601119908ℎ120583119903119871119890 minus 120583119903119882(119905) 1198911015840119908 (119878119908119891) + 119886119882 (119905) (15)


Δ119875119891 = 120583119908119886119882 (119905)120601 (119908ℎ)2119870119889119882(119905)119889119905 (16)



120583119908119886 minus 1205831199001198911015840119908 (119878119908119891) (17)


119909119891 = 1198911015840119908 (119878119908119891)120601


120583119908119886 minus 1205831199001198911015840119908 (119878119908119891) (18)


119879 = [(120583119908120601119871119890119886) 1198911015840119908 (119878119908119891)]2 minus (120583119900120601119871119890)2

2 [120583119908119886 minus 1205831199001198911015840119908 (119878119908119891)] 120601119870Δ119875 (19)


Δ119875 = V120583119908119870 sdot 1199091198911198911015840119908 (119878119908119891) sdot 119865 (119878119908119890) (20)



119889119882 (119905) = 119908ℎV119889119905 = 120601119908ℎ119871119890119889 11198911015840119908 (119878119908119890) (21)


119889119905 = 1205831199081198711198902120601119870Δ119875 sdot 119865 (119878119908119890)1198911015840119908 (119878119908119891)1198891

1198911015840119908 (119878119908119890) (22)


119905 minus 119879 = 1205831199081198711198902120601119870Δ119875 119866( 11198911015840119908 (119878119908119890)) (23)

where 119866(119878119908119890) is given by

119866( 11198911015840119908 (119878119908119890)) = int

11198911015840119908(119878119908)

11198911015840119908(119878119908119891)

119865 (119878119908119890)1198911015840119908 (119878119908119891)119889

11198911015840119908 (119878119908119890) (24)


119905 minus 119879 = 1205831199081198711198902120601119870Δ119875 119866( 119909119871119890 sdot1

1198911015840119908 (119878119908)) (25)







119878119908 = 119878119908119888 + 11198911015840119908 (119878119908119891) (27)





120578 = 119860 119894 + sum119899minus1119895=1 119860119895

ℎ119871119890 (29)




119879119894 minus 119879119895 = 1205831199081198711198902120601

119870119895Δ119875 119866(1

1198911015840119908 (119878119908119890)) (30)


119878119908 = 119878119908119890 + 1 minus 119891119908 (119878119908119890)1198911015840119908 (119878119908119890) (31)




ℎ119871119890 (32)






0

02

04

06

08

1

0 02 04 06 08 1

WaterOil

Sw


















1

2

5

4

3


1

2

5

4

3

Oil sat

029500

038963

048426

057889

067351





1

2

5

4

3


1

2

5

4

3

Oil sat

029500

038963

048426

057889

067351





0

20

40

60

80

100

0 200 400 600 800 1000 1200 1400 1600

Wat

er cu

t (

)

Time (day)



0

10

20

30

40

50

60

70

Swee

p effi

cien

cy (

)

0 200 400 600 800 1000 1200 1400 1600Time (day)





0

5

10

15

20

25

30

35

40

Oil

reco

very

()

0 200 400 600 800 1000 1200 1400 1600Time (day)



03

04

05

06

07

08

0 20 40 60 80 100 120 140 160 180 200x (m)


S w





03

04

05

06

07

08

0 20 40 60 80 100 120 140 160 180 200x (m)


S w



6 Conclusion




Competing Interests


Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120578 = 119860 119894 + sum119899minus1119895=1 119860119895

ℎ119871119890 (29)




119879119894 minus 119879119895 = 1205831199081198711198902120601

119870119895Δ119875 119866(1

1198911015840119908 (119878119908119890)) (30)


119878119908 = 119878119908119890 + 1 minus 119891119908 (119878119908119890)1198911015840119908 (119878119908119890) (31)




ℎ119871119890 (32)






0

02

04

06

08

1

0 02 04 06 08 1

WaterOil

Sw


















1

2

5

4

3


1

2

5

4

3

Oil sat

029500

038963

048426

057889

067351





1

2

5

4

3


1

2

5

4

3

Oil sat

029500

038963

048426

057889

067351





0

20

40

60

80

100

0 200 400 600 800 1000 1200 1400 1600

Wat

er cu

t (

)

Time (day)



0

10

20

30

40

50

60

70

Swee

p effi

cien

cy (

)

0 200 400 600 800 1000 1200 1400 1600Time (day)





0

5

10

15

20

25

30

35

40

Oil

reco

very

()

0 200 400 600 800 1000 1200 1400 1600Time (day)



03

04

05

06

07

08

0 20 40 60 80 100 120 140 160 180 200x (m)


S w





03

04

05

06

07

08

0 20 40 60 80 100 120 140 160 180 200x (m)


S w



6 Conclusion




Competing Interests


Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
























1

2

5

4

3


1

2

5

4

3

Oil sat

029500

038963

048426

057889

067351





1

2

5

4

3


1

2

5

4

3

Oil sat

029500

038963

048426

057889

067351





0

20

40

60

80

100

0 200 400 600 800 1000 1200 1400 1600

Wat

er cu

t (

)

Time (day)



0

10

20

30

40

50

60

70

Swee

p effi

cien

cy (

)

0 200 400 600 800 1000 1200 1400 1600Time (day)





0

5

10

15

20

25

30

35

40

Oil

reco

very

()

0 200 400 600 800 1000 1200 1400 1600Time (day)



03

04

05

06

07

08

0 20 40 60 80 100 120 140 160 180 200x (m)


S w





03

04

05

06

07

08

0 20 40 60 80 100 120 140 160 180 200x (m)


S w



6 Conclusion




Competing Interests


Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

20

40

60

80

100

0 200 400 600 800 1000 1200 1400 1600

Wat

er cu

t (

)

Time (day)



0

10

20

30

40

50

60

70

Swee

p effi

cien

cy (

)

0 200 400 600 800 1000 1200 1400 1600Time (day)





0

5

10

15

20

25

30

35

40

Oil

reco

very

()

0 200 400 600 800 1000 1200 1400 1600Time (day)



03

04

05

06

07

08

0 20 40 60 80 100 120 140 160 180 200x (m)


S w





03

04

05

06

07

08

0 20 40 60 80 100 120 140 160 180 200x (m)


S w



6 Conclusion




Competing Interests


Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










03

04

05

06

07

08

0 20 40 60 80 100 120 140 160 180 200x (m)


S w



6 Conclusion




Competing Interests


Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









analytical model of waterflood sweep efficiency in vertical

Documents