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Iranian Journal of Chemical EngineeringVol. 9, No. 4 (Autumn), 2012, IAChE

Rapid Estimation of Water Flooding Performance and Optimizationin EOR by Using Capacitance Resistive Model

A.R. Bastami1, M. Delshad2, P. Pourafshary3

1- Graduate Student, Institute of Petroleum Engineering, University of Tehran2- Graduate Student, Institute of Petroleum Engineering, University of Tehran

3- Assistant Professor, Institute of Petroleum Engineering, University of Tehran

AbstractWater flooding, the oldest and most common EOR method, increases the displacementefficiency in a reservoir and also maintains the reservoir pressure for a long period oftime. In Iran, water injection is widely used as a method to enhance recovery from oilreservoirs. Defining the optimized injection rates and injection patterns, dependent onthe geological structure of the reservoir, is essential in operational and economicaldecisions for reservoir management.In this paper, the Capacitance-Resistive Model is used to find interwell connectivity,and optimized injection rates in a synthetic field. In this approach, the reservoirreceives injector rate variations as an input signal, while the producer responsesdetermine the injector/producer pair connectivity quantitatively. This model is used topredict oil production for a specific reservoir, if the production/injection rate andbottomhole pressure data are available. The results show that the Capacitance-Resistive model has the capability to be used for the production history matching and tooptimize the injection rate in different wells of a reservoir during the immiscibleflooding to maximize the oil production. Moreover, they show that any change in oiland water prices can significantly influence the optimized water injection rates.

Keywords: Waterflooding, Water Injection, Production Forecast, Optimization,Capacitance Resistive Model

Corresponding author: [email protected]

1. IntroductionOil recovery operations have traditionallybeen subdivided into three stages: primary,secondary, and tertiary. Historically, thesestages described the production from areservoir in a chronological sense. Primaryproduction, the initial production stage,resulted from the displacement energynaturally existing in a reservoir. Secondary

recovery, the second stage of operations, wasusually implemented after primaryproduction declined. Traditional secondaryrecovery processes are waterflooding,pressure maintenance, and gas injection,although the term secondary recovery is nowalmost synonymous with waterflooding.Tertiary recovery, the third stage ofproduction, was that obtained after


4 Iranian Journal of Chemical Engineering, Vol. 9, No. 4

waterflooding (or whatever secondaryprocess was used). Tertiary processes usedmiscible gases, chemicals, and/or thermalenergy to displace additional oil after thesecondary recovery process becameuneconomical [1, 2].

2. Capacitance resistive modelThe idea of this model is similar to theelectrical current in the electrical circuits,including a network of capacitors andresistors. Hence, by considering flow stream,storage capacity in porous media, pressuredifference and permeability similar toelectrical current, capacitance, potentialdifference and resistance, respectively, themathematical relations in electrical circuitscan be used in the reservoir. In this model, areservoir is supposed to receive a signal(injection) and deliver a reaction signal(production) to what it receives. Historicalinjection and production rates are the maininput data for this model. Analyzing thesedata could provide a great deal of usefulinformation about the reservoir and reducethe uncertainties in reservoir modeling. Thefinal purpose is the economical andoperational optimization in waterfloodingprojects to maximize the oil recovery [4].Albertoni et al. used a simple model to findthe interwell connectivity. He showed thateven the injection rates of injectors which arefar away from producers can affect theirproduction rates. He estimated the interwellconnectivity by a linear model and estimatedcoefficients by using MLR1 [5]. Yousef et al.added a new parameter and developed themodel to consider both capacitance and

1. Multiple Linear Regression

resistance effects by using compressibilityand transmissibility concepts, respectively [6,7]. Sayarpour et al. defined three differentsimplified models and presented analyticalsolutions for each and validated thesesolutions by applying them for some realfields [8, 9]. Weber et al. reviewed theproblems of using this model in large scalefields with a large number of wells andsuggested some solutions for minimizing theerror caused by these problems [10].Delshad et al. used this model to detect thepresence of fractures in a reservoir andcalculate the fracture permeability [11].

3. Mathematical developmentsThe material balance for a simple reservoirincluding one injector/producer pair asshown in Fig. 1, is as follows:

Figure 1. A control volume including one injector/producer pair

t pdPcV i ( t ) q( t )dt (1)

To find an equation based on only injectionand production rates, linear productivityindex can be used.

)( wfPPJq (2)

By replacing average pressure form equation

(2)

Bastami, Delshad, Pourafshary

Iranian Journal of Chemical Engineering, Vol.9, No. 4 5

(2) in equation (1),

dtdP

Jtitqdtdq wf )()( (3)

where is time constant. The solution of thisfirst order differential equation is as follows:

0

0

0

0

0

0

tt( t t )

t

tt( t t )

wf wf wft

eq( t ) q( t )e e i ( )d J

eP ( t )e P ( t ) e P ( )d

(4)

This is the basic formulation of this model.Now, assume that the reservoir consists ofdifferent control volumes with a producerand 1 injector around it. [12]. A schematicrepresentation of this approach is illustratedin Fig. 2.

Figure 2. A schematic view of a reservoir with 2injectors and 2 producers which describe theparameters of CRM in this modified approach.

In this approach, there is a time constant for

each producer and a weight coefficient foreach pair of injectors and producers. Hence,the equations which describe the relationsbetween injectors and producers are differentin this approach:

dt

dPJtitq

dttdq jwf

ji

I

iij

jj

j

j

)(.1)(1)(

1

(5)

where the time constant is defined as:

j

ptj J

Vc

(6)

The solution of the above equation is asfollows:

0

1

0

0 0

1

( t t )

j j

t tt tIwf , j

ij i jijt t

j

j j j j

q ( t ) q ( t )e

dpe e i ( )d e J e d

d

(7)

Integration of the above equation leads to:

0 01

1

0 0

0

( t t ) ( t t )I

j j ij i ii

t t Iwf , ji

ij jit

j j

j j

q ( t ) q ( t )e i ( t ) e i ( t )

dpdi ( )e e J dd d

(8)

By assuming that during the productionintervals, the bottomhole pressure is varyinglinearly and the injection rates are constant,the describing equation would be as follows:

(5)

(6)

Injector

Producer



0

1 1

0

0

1

1 2

( t t )

j j

( t t ) t ( k )n Iwf , j( k )

ij i j jk i k

j

kj j

q ( t ) q ( t )e

pe ( e ) I J

t

j ( , ,..., I )

(9)

4. Oil production modelTo estimate the oil fractional flow in theproduction stream, an oil fractional flowmodel should be used in association withCapacitance Resistive Model. One of thesuggested models is a power law relationbetween instantaneous water/oil ratio, woF ,

and cumulative water injection rate, iW [13].

Hence, the estimated water/oil ratio can becalculated from eq. 10.

iwo WF (10)

For a balanced system, total injection andtotal production are equal; hence, iW is equal

to the total liquid production. This equationcan be used for each well separately. Byusing eq.10 as the oil fractional flow model,we have:

jjijjwo

wj

ojwjoj

ojoj WF

qqqq

qtf ,

,, 11

11

1

1)(

(11)

In this equation, cumulative water injection isas follows:

dssidssiWinjinj N

k

t

tkkj

t

t

N

kkkjji .)(.])([

11,

00

(12)

where is the connectivity between aninjector/producer well pair or weightcoefficient.On the other hand, oil production rate fromproducer j is equal to the oil fraction, ojf ,

multiplied by total production, )(tq j as

follows:

)()()( tqtftq jojoj (13)

The combination of eq. 11, eq. 12 & eq. 13leads to the oil production rate from eachproducer:

1 0

1

1 2

joj N t

j kj kk t

pro

injj

q ( t )q ( t )

( i ( s ).ds )

( j , ,...,N )

(14)

Hence, after estimating the CRM parameters(time constants & weight coefficients), theoil fractional flow parameters ( j , j )

should be estimated by minimizing thedifference between real and estimated valuesduring history matching.Since the direct application of eq. 10 in thisform for finding model parameters is verydifficult, a logarithmic form of this equationis suggested here.

)log()log()log( ,, jijjjwo WF (15)

By using a linear regression and minimizingthe difference between real and estimatedvalues, )log( j and j can be calculated.

This linear form of power law relation showsthe limitations of using this model in

(13)

(14)

(15)



predicting oil production rate. In other words,this model can be applied if the logarithmicplot of water/oil ratio versus cumulativewater injection is linear.

5. Optimization algorithmThe most important part of an optimizationproblem is determining the objectivefunction. Depending on the purpose of theoptimization, different objective functionscan be defined. Some of these objectivefunctions are as follows:

1. Maximizing cumulative oil productionduring a definite time interval

2. Minimizing cumulative waterproduction during a definite timeinterval

3. Maximizing net present value of theproject by considering injection costs

4. Maintaining the oil production ratefrom a specific field

Hence, the purpose of optimization can be tomaximize ultimate oil production rate or tomaximize the profit of the waterflooding in areservoir. To maximize the profit of awaterflooding project during a specific timeinterval ],[ 0 tt , the suggested objective

function is as follows:

injpro N

k

t

t kw

N

j

t

t ojo dssipdssqpR11 00

).(.).(. (16)

where, op and wp are oil and water price per

barrel, respectively.Furthermore, decision variables areproduction and injection rates. An upper anda lower boundary for injection throughinjectors can be defined according to theconditions of the reservoir.

The overall procedure of this optimization byusing capacitance resistive model and oilfractional flow model is illustrated in Fig. 3.In this paper, the Microsoft Excel Solver isused to determine the CRM and fractionalflow parameters. Moreover, it is used tosolve the nonlinear mathematical equation todetermine injection rates. Microsoft ExcelSolver uses the Generalized ReducedGradient (GRG2) Algorithm for optimizingnonlinear problems.

Figure 3. Workflow for the CRM application inhistory-matching and optimization [14]

6. Case study6-1. 1*1 ModelIn this section, Capacitance-Resistive Modelfor a synthetic field with 1 injector-1producer is applied. These historical injectionand production rates are for a real field andthis well pair is supposed to be separatedfrom the other part of the field. In Fig. 4, theinjection rate versus time is illustrated.By applying CRM, the interwell parametersare calculated as it is shown in Table 1.



0100020003000400050006000700080009000

0 500 1000 1500 2000 2500 3000 3500

Rate

(bbl

/day

)

Time (day)

Injection Rate

Injection

Fig. 4. The injection rate versus time

Table 1. CRM parameters

τ λq0,

RB/Tqt Error,

RB/Tqt Correlation

ratioP1

0.59

1 15657 267 0.97

In Fig. 5, the real production rate is comparedto the estimated production rate which iscalculated by CRM. Hence, CRM is able toestimate liquid (water + oil) production rateaccurately. In addition, by using CRMparameters we are able to forecast theproduction rate by accounting any change inthe injection rate. In Fig. 6, the calculationprocedure to find fractional flow modelparameters (power low model) is illustrated.

0100020003000400050006000700080009000

0 500 1000 1500 2000 2500 3000 3500

Rate

(bbl

/day

)

Time (day)

Production Rate

Real

Estimated

Figure 5. The real production rate in comparison withestimated production rate

Using power low model parameters andselecting the desired objective functionwhich is discussed in the previous section,injection rate optimization is possible. The

objective function in this study is tomaximize the profit by considering the waterinjection cost equal to 1-3 $/bbl and differentoil prices. In addition, the minimum andmaximum limits of injection rate aresupposed to be 0, 10000 bbl/day,respectively.

Figure 6. The graph of fractional flow modelcalculation

Table 2. Power low parametersα β qo Correlation ratio

6.412E-13 1.857 0.91

According to these results (Table 3, 4 & 5), ifwater injection costs 1$/bbl, for oil pricesless than 18 $/bbl, water injection is notfeasible. Also, if each oil barrel is worth $19,optimized water injection rate is equal to5100 bbl/day. Finally, if each oil barrel isworth $20, optimized water injection rate isequal to 10000 bbl/day. These calculationsfor different water injection costs and oilprices are presented in the tables below.

Table 3. Optimized injection rate dependent on oilprice, and water injection cost equal to 1 $/bbl

Oil Price($)

Water Price($)

Optimized Injection rate(RB/D)

18 1 0

19 1 5100

20 1 10000




Oil Price($)

Water Price($)


36 2 0

37 2 774

38 2 5100

39 2 9329

40 2 10000


Oil Price($)

Water Price($)


55 3 0

56 3 2227

57 3 5100

58 3 7930

59 3 10000

6-2. 5*4 Model6-2-1. IntroductionIn this case, a synfield with 5 injectors, 4producers and two high permeable streaks isconsidered. It is assumed that all the wells arevertical and they are perforated in all layers.The most important rock and fluid propertiesare presented in Table 6 and a schematicview of this reservoir is illustrated in Fig. 7.This model consists of 31*31*5 grids in X, Yand Z direction, respectively.

6-2-2. History matchingIn this study, Capacitance Resistive Modelparameters are used to forecast theproduction rate and to estimate thewaterflooding performance quickly. In thiscase, 3051 days of operation in the form of120 injection/production data are put into themodel, and the model parameters, aspresented in Table 7, are obtained afterhistory matching.

Figure 7. A schematic view of a synfield with 5injectors, 4 producers and two high permeable streaks

Table 6. Rock and fluid properties of 5*4 ModelValueParameter

80X direction

Grid Size, ft 80Y direction

40Z direction

0.18Porosity, fraction

50Permeability, md

2.0OilViscosity, cP

0.5Water

5×10-6Oil

Compressibility, psi-1 1×10-6Water

1×10-6Rock

Table 7. CRM parameters for 5*4 ModelSumP4P3P2P1Parameters

0.5662.5175.3240.503τ

10.1000.0240.0690.807λ1j

10.2460.1980.0340.522λ2j

10.5610.0880.0740.277λ3j

10.5200.0620.2020.217λ4j

10.5670.1940.0430.196λ5j



9.9610.6010.096.20Error %

6-2-3. Production forecastBy using CRM parameters which areobtained through history matching,production has been predicted for 700 days.Hence, it is possible to study the capability ofcapacitance resistive model in predictingfuture production of the reservoir bycomparing the predicted production rateswith rates calculated by fine grid simulators(Eclipse). Fig. 8 shows the comparisonbetween the values obtained by CRM andEclipse for each producer. In this figure,“Real” represents the Eclipse results and theCRM results are shown as “Estimated”.The results show the estimated values ofCRM are in agreement with the calculatedvalues by Eclipse. Hence, CRM is capable offorecasting the future production even forreservoirs with more complexity andheterogeneities.

6-2-4. Optimization of the injection ratesIn this section, the calculation of injectionrate optimization by using CRM is presented.Fig. 9 shows the oil fractional flow model(Power law) graphs in which parameters ofmodels can be calculated for each producerand the interval that this model can beapplied. Minimizing the error betweenestimated values and real values (like historymatching) results in oil fractional flow modelparameters, as it is presented in Table 8 foreach producer.

Table 8. Estimated values for oil fractional flow modelparameters

Wells

1.7719.65E-12P1

3.7711.09E-22P2

2.3978.59E-15P3

2.3771.78E-15P4



Figure 8. Comparison between estimated values of CRM and the calculated values by Eclipse for each producer



Figure 9. The graphs based on power law model for estimating the fractional flow model parameters

To optimize the injection rate, it is assumedthat the rate of all injectors is constant andequal to the last value of the history for 30months. Moreover, it is assumed that theminimum and maximum injection rates are10 and 4000 STB/D, respectively. Thus,using CRM and fractional flow modelparameters, with an assumption of 2 and 70dollars cost per barrel for water and oil,respectively, the optimization based onmaximizing the ultimate profit can beaccomplished. Table 9 compares the rates ofinjectors before and after optimization.Using fine grid simulators provides us with a

measuring tool to compare the differencebetween production rates either by the initialinjection rates or the optimized rates. Thiscomparison for total production rate andprofit is illustrated in Fig. 10 and Fig. 11,respectively.

Table 9. Comparison between the rates of injectors(bbl/day) before and after optimization

I1 I2 I3 I4 I5Total

InjectionInitialrates

3001 1540 1291 963 1097 7892

Optimizedrates

10 10 3862 10 4000 7892



Figure 10. Comparison between the total production before and after optimization

Figure 11. Comparison between the profit before and after optimization

7. ConclusionsNowadays, waterflooding is known as one ofthe most common EOR methods and also asa method to maintain the reservoir pressurein order to increase the ultimate oil recoveryin the oil industry. Therefore, optimizedwater injection rate as an operational factor

and also as an economical factor is veryimportant. It has a considerable effect on theultimate performance of the enhanced oilrecovery project. In this paper, a new methodfor a rapid and continuous operational andeconomical calculation in an injection projectis presented. In addition, it has been shown



that it is possible to calculate optimizedinjection rate by using this model fordifferent reservoir and well geometries.

Nomenclature Model coefficient Model coefficient Weight coefficient Integrating variable Time constant

tc L2/F Total compressibility

woF Water/oil ratio)(ti L3/day Total injection rate

J L5/F-t Productivity index

P F/L2 Average pressure in the porevolume

wfP F/L2 Bottomhole pressure)(tq L3/day Total production rate

pV L3 Pore volume

iW L3 Cumulative water injection

AppendixTo derive the analytical solution of eq. 3, wefirst express the differential equation in ageneral form, or:

trqdtdq

1

A-1

where,

dt

dpJti

tr

wf

A-2

This first-order differential equation can begenerally solved by using the integratingfactor technique. For this equation, the

integrating factor is given by:

tdt

eetf 1 A-3

Then, by multiplying eq. A-1 by theintegrating factor, or:

treqdtdqe

tt

1 A-4

it immediately follows that

treqedtd tt

A-5

We now separate and integrate this equationwith respect to t,

dttrecqett A-6

where c is the constant of the integration. By

dividing both sides by t

e ,

dttreeceqttt A-7

The constant of the integration c can beestimated by using the initial condition

00 tqtqtt . Then Eq. A-7

becomes,

dttreeetqqtttt

0

0A-8

By substituting for tr in the aboveequation, we get

0

0

0

t t

tt

wf

t

q q t e

dpe e i J dd

or



0

0

0 0

t t

tt tt

wf

t t

q t q t e

dpe e i d Je e dd

A-9

where, i is a variable of integration. The thirdterm on the right of Eq. A-9 can be simplifiedby using integration by parts. The finalanalytical solution for Eq. 3 is,

0

0

0

0

0

0

tt t t

t

tt t t

wf wf wft

eq t q t e e i d

eJ p t e p t e p d

References[1] Latil, M., Enhanced oil recovery, 1st

edition, Technip, Paris, France, p. 1-2(1980).

[2] Green, D. W. and Willhite, G. P.,Enhanced oil recovery, SPE TextbookSeries VOL. 6, 2nd edition, Texas, USA,p. 1 (2003).

[3] Bruce, W. A., "An electrical devices foranalyzing oil reservoir behavior",Trans. AIME, p. 113-124, (1943).

[4] Liang, X., Weber, D. B., Edgar, T. F.,Lake, L. W., Sayarpour, M., andYousef, A. A., "Optimization of oilproduction based on a capacitancemodel of production and injectionrates", SPE 107713, HydrocarbonEconomics and Evaluation Symposium,Dallas, Texas, U.S.A (2007).

[5] Albertoni, A., and Lake L. W.,

"Inferring connectivity only from well-rate fluctuations in waterfloods", SPEReservoir Evaluation and EngineeringJournal, 6(1), 6 (2003).

[6] Yousef, A. A., Gentil, P., Jensen, J. L.,and Lake, L. W., "A Capacitance modelto infer interwell connectivity fromproduction and injection ratefluctuations", SPE 95322, SPE AnnualTechnical Conference and Exhibition,Dallas, Texas, U.S.A. (2005).

[7] Yousef, A. A., Jensen, J. L., and Lake,L. W., "Analysis and interpretation ofinterwell connectivity from productionand injection rate fluctuations using acapacitance model", SPE 99998,SPE/DOE Symposium on Improved OilRecovery, Tulsa, Oklahoma, U.S.A.(2006).

[8] Sayarpour, M., Zuluaga, E., Kabir, C. S.and Lake, L. W., "The use ofcapacitance-resistive models for rapidestimation of waterflood performance",SPE 110081, SPE Annual TechnicalConference and Exhibition, Anaheim,California, U.S.A. (2007).

[9] Sayarpour, M., Kabir, C. S., and Lake,L. W. "Field applications ofcapacitance-resistive models inwaterfloods", SPE 114983, SPE AnnualTechnical Conference and Exhibition,Denver, Colorado, U.S.A (2008).

[10] Weber, D. B., Edgar, T. F., Lake, L. W.,Lasdon, L., Kawas, S., and Sayarpour,M., "Improvements in capacitance-resistive modeling and optimization oflarge scale reservoirs", SPE 121299,SPE Western Regional Meeting, SanJose, California, U.S.A (2009).

[11] Delshad, M., Bastami, A. andPourafshary, P., "The use of



capacitance-resistive model forestimation of fracture distribution in thehydrocarbon reservoir", SPE 126076,SPE Saudi Arabia Section TechnicalSymposium, Khobar, Saudi Arabia,(2009).

[12] Bastami, A., The use of capacitance-resistive method for rapid estimation ofimmiscible flooding in enhanced oilrecovery, M.Sc. thesis in petroleumengineering, University of Tehran, Iran,(2009).

[13] Gentil, P. H., The use of multilinearregression models in patternedwaterfloods: Physical meaning of theregression coefficients, M.Sc. thesis inpetroleum engineering, The Universityof Texas at Austin, Texas, (2007).

[14] Sayarpour, M., Development andapplication of capacitance-resistivemodels to water/CO2 floods, Ph.D.dissertation in petroleum engineering,The University of Texas at Austin,Texas, (2008).

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