research article the framework dedicated to three phase

Research ArticleThe Framework Dedicated to Three Phase FlowsWellbore Modelling

Bartlomiej Bielecki and Andrzej Krajka

Faculty of Mathematics Physics and Computer Science Maria Curie-Skłodowska University Poland

Correspondence should be addressed to Bartlomiej Bielecki bartbielgmailcom

Received 29 October 2014 Revised 12 January 2015 Accepted 19 January 2015

Academic Editor Lu Zhen

Copyright copy 2015 B Bielecki and A Krajka This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

To predict physical properties in a wellbore during oil and gas production scientists use empirical correlations or mechanisticapproach algorithms The typical research in this field is concentrated on a single property analysis as heat transfer pressuretemperature and so forth Here the most proper correlations regarding the subject are presented And the authors studied howto join all correlations into the full framework which returns all production parameters at every depth in a wellbore Additionallythe presented simulation results are studied here Based on presented algorithms the proper tool has been applied and the resultsshown in this paper are taken from this application

1 Introduction

All of well physical properties in wellbores are deeply studiedwith many approaches found in literature Oil and gas indus-try is a cutting edge domain with a lot of investment involvedThis is the reason why it is hard to find any descriptionof a framework which join correlations altogether into onesolution The solution can describe the wellbore physicalproperties at every depth at every stage of production evenfor the single whole oilfield It means that all calculationshave been performed every foot across the wellbore usingstandard oilfield unit system Once a few wellbores areconsidered the results object keeps all calculations stillperformed for every foot for all wells independently Thefinal results set is dedicated for oilfield engineers so pressuretemperature and production results are most desirable Thewhole model considered as full wellbore simulation waspresented as a PhD thesis at The University of Texas [1]The global simulation model is presented there but withoutany description of the framework Both models are based onthe same principles but with different approaches Our aimis to show the possibility of joining correlations altogetherusing fast and efficient algorithm Additionally our model isextended for gas lift which is a part of this framework andcan be used in wireless sensor network well monitoring [2]

Optimisation of gas lift is a well known problem [3 4] butwe are concentrating on the gas injection influence on thefinal results of simulation Unfortunately the authors do nothave any possibility to verify the simulation results by thecomparison to the real data This kind of data is protected bythe oil companies

Correlations presented in Section 2 are typical wellboresimulation models Three types of correlations described inthis section are identified empirical correlationsmechanisticapproach and homogeneous models Typical approachesfor the vertical wells are presented Three phase flows areconsidered The pressure model was developed based on[5] study with the consideration of flow regimes and theirdependence on the features of empirical correlations Thetemperature model is based on Sagar et al [6] model

Section 3 presents the framework with the pseudocodeand some chart to understand the complexity of wellboremodelling This is the model with the diversion on reservoirand wellbore calculation Reservoir calculations use layerpressure and geothermal temperature Wellbore correlationsuse the pressure and temperature models as the iterationswhere the pressure values from any previous calculation arepassed to the calculation object at current depth This modelwas developed based on well known principles [5] Its mainadvantage is a speed and effectiveness

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 183982 13 pageshttpdxdoiorg1011552015183982

2 Mathematical Problems in Engineering

In Section 4 typical case study can be found The toolwhich has been developed to present all data at everystage of simulation has been implemented We can presentvariables as production rate pressure temperature densitysolubility compressibility formation volume factor viscosityheat capacity surface tension gasoil ratio gaswater ratiomass rate bubble point pressure flow direction flow regimesvelocity hydrostatic and kinetic pressure drop Reynoldsnumber and others as function of mentioned It givesthe possibility to present different kind of data which arecomparable to other researches

To summarize the main contributions of this paperare to show the framework idea which groups wellborecorrelations altogether This framework gives the reliablesimulation results which is proved in this paper

2 Correlations

The authors would like to present the single correlation forevery property There is no clear algorithm that suggestswhich correlation should be taken under the considerationas the most proper one It depends on many factors as fluiddensity trajectory and so forth The best way to gain theaccurate modelling results is to compare obtained data withthe real one Good example can be found from Tulsa oilfieldsanalysis [1] where it has been proven by the comparison ofrelative performance factor that the mechanistic approachgives better results than empirical correlations Ansari modelpresented in Figure 1 is the best in this example It does notnecessarily means that the Ansari model is reliable for everywellbore simulation It is only regarding this case

Here the most important part of calculation is presentedThree major parameters as pressure temperature and inflowexist as a function of the following variables bubble pointpressure gas solubility gas compressibility formation volumefactors viscosities heat capacities surface tension comple-tion data and reservoir properties

21 Pressure Our method is based on solving the equationusing a finite difference approximation method with thecontinuity momentum and energy equation as follows

984858119897+1

119894minus 984858119897

119894

Δ119905+984858V119897+1119894+12

minus 984858V119897+1119894minus12

Δ119909= 0

984858V119897+1119894+12

minus 984858119897

119894+12

Δ119905+[984858V2119894+12

minus 984858V2119894minus12

]119897

Δ119909+119901119897+1

119894+1minus 119901119897+1

119894

Δ119909

+ 984858119897

119894+12119892 + [

[

119891 (984858V)2119894+12

2119863

]

]

119897

= 0

984858119894

119864119897+1

119894+1minus 119864119897+1

119894

Δ119905+ 984858V119894

119864119897

119894minus 119864119897

119894minus1

Δ119909+ 119875119897+1

119894

(V119894+12 minus V119894minus12)119897

Δ119909

+ 2119867(119879119903 minus 119879119908)

119897

119903119908

= 0

(1)

Mechanisticapproach

correlationsEmpirical

Hom

ogen

eous

0

1

2

3

4

5

6

Relat

ive p

erfo

rman

ce fa

ctor

for a

ll ve

rtic

al w

ell d

ata

Hag

edor

n

Dun

s

Begg

s

Ork

iszew

ski

Muk

herje

e

Has

an

Azi

z

Ans

ari

Figure 1 The different models comparison [1] The lower valuemeans the better results

Across the years the multiphase flow theory has beenimproved andmakes the calculation difficult to predict due toseveral reasons In the flow problem the number of variablesaffecting pressure drop is enormous The interfacial tensionbetween phases is taken into account but all kinds of availablemodels try to reduce the total number of variables with theintroduction of nondimensional parameter group Frictionalpressure losses are more difficult considering multiphaseflow so in addition new kind of energy losses has beenused slippage loss All these features make the model socomplicated that we cannot describe it in this paper It isimportant to emphasize that usually the correlations whichare used are based on empirical method

Pressure loss across the wellbore during the production isdivided into three components

(119889119901

119889119897)

total= (

119889119901

119889119897)

hydrostatic+ (

119889119901

119889119897)

kinetic+ (

119889119901

119889119897)

frictional

(2)

The cohesive forces (CF) and frictional forces (FF) havethe crucial influence on fluid drive especially regardingphases separation or bind The friccohesity model has beenconsidered as a CF and FF product Using the Mansinghequation [7]

120590 = 1205900 [(119905

1199050

plusmn119861

119905)(

119899

1198990

plusmn 00012 (1 minus 984858))] (3)

the friccohesity as a function of surface tension and cohesiveand frictional forces is calculated In this equation 1199050 and 119905

are the reference and sample flow times respectively 1199030 isreference friccohesity and 1198990 and 119899 are the pendant dropnumber of reference and sample respectively Referencefriccohesity is represented by the following formula

1205900 =1205830

1198781198790

(4)

Mansingh survisemeter is a device to measuring friccohesity[8] The friccohesity analysis is dependent on hydrocarboncomposition In this model we are concentrating on the

Mathematical Problems in Engineering 3

compressibility factor calculation with pseudo-reduced pres-sure and temperature solutions but the composition role isreplaced by the density value It does notmean that themodelis not predicted for hydrocarbon composition but from theoptimization point of view our solution is faster upon thissimplification The deeper friccohesity analysis is consider asa future work for this studyThat is the reason why the Papayas the best compressibility correlation in the comparisonto the others has been chosen [9] There are solutionswhich do not consider kinetic pressure losses [10] Inside thecorrelations which determine the flow regime liquid holdupfor more than single phase the pressure drop equation isknown under the different form presented as follows

(119889119901

119889119897)

total=

(119889119901119889119897)hydrostatic + (119889119901119889119897)frictional

1 minus 119864119896

(5)

Depending on the correlation this kinetic term has thedifferent form but usually this is a variation of the equation

119864119896 = 0000216119891V119898Vsg984858ns

119901 (6)

Velocity of each phase is calculated from

119881 =576

Π1198632

119899

sum

119894=1

119861119894119876119894 (7)

Oil formation volume factor algorithms are calculated usingthree different methods correlation based [5] artificial neu-ral networks based prediction [11] and hybrid soft computingbased techniques [12] The crucial issue of gas formation vol-ume factor correlation is compressibility Last study estimatedthis value from the experimental data taken as a functionof pressure and temperature [13] Water formation volumefactor is already described as function of two polynomialspressure and temperature dependent [14] Consider

119861119908 = (1 + Δ119881119908119875) (1 + Δ119881119908119879)

Δ119881119908119875 = minus 10001 times 10minus2+ 133391 times 10

minus4119879

+ 550654 times 10minus71198792

Δ119881119908119879 = minus 195 times 10minus9119901119879 minus 1728 times 10

minus131199012119879

minus 3589 times 10minus7119901 minus 2253 times 10

minus101199012

(8)

In (6) friction factor is changing due to flow regimes asa function of Reynolds number diameter and pressure asfollows [15]

119891 = 119891119871 (1 minus 120595119887)13

(1 minus 12059513

119887) + 119891119879120595

13

119887 (9)

where 120595 is intermittency factor

120595119887 =log (Re Re119871)log (Re119879Re119871)

(10)

Reynolds number is calculated from the following formula

Re =124 lowast 984858119898119881119898119863

120583119898

(11)

Hence

log(Re119871300

) = 17 ((119901

119863) minus 1)

log(Re119879105

) = 07 ((119901

119863) minus 1)

(12)

In the whole calculation density and viscosity are the mostimportant flow parameters Ideally they are determinedexperimentally in the laboratory on actual fluid samplestaken from the field under study In many cases correlationsare region dependent so every simulation should provideoptions to choose a place of production Dead saturatedand undersaturated are considered as oil types and identifiedusing fluid properties Every type has its own correlations set[16]

The last parameter in (6) is no slip density which isdescribed in the following formula [5]

984858ns =119881119897984858119897

119881119898

+ (1 minus119881119897

119881119898

) 984858119892 (13)

Hydrostatic pressure is calculated from

(119889119901

119889119897)

hydrostatic=

1

144

984858119898 (119879 120572 Fr 120601 119878119905) 119889119909119889119897

(14)

As it is shown basically 120588119898 is a function of temperatureinclination and flow regime Every correlation has the deepstudy of mixture density Pressure drop caused by friction hasbeen studied widely Usually the variations of the followingequation are considered [5]

(119889119901

119889119897)

119891

= 1294 times 10minus4119891984858ns984858119897119881

2

119897

119863 (15)

A flow in a pipe is turbulent if the Reynolds number isgreater than 4000 and laminar below 2100 For laminar flowthe friction factor is calculated by assumption as 64Re butfor the transition and turbulent flow Chen correlation is anexample of friction factor calculation [17] Consider

119891 = [(4 log 119877

37065119863minus50452 log(120575)

Re)

2

]

minus1

(16)

where

120575 =1

28257(119877

119863)

11098

+ (7149

Re)

08981

(17)

The equations which have been presented in this section areexamples of pressure loss correlation Flow regimes and pipetrajectory are the crucial factors in every calculation A lotof exceptions have been studied and analysed and during thesimulation process


In the pressure calculation mixture density depends onsurface tension which can be described as a function ofpressure and temperature [18] As an example the presentedcorrelation has been created for crudeoil gas systems [19]

119878119879 = 1205741 minus(119905 minus 74) (1205741 minus 1205742)

206 (18)

where

1205741 = 75 minus (11081199010349

)

1205742 = 53 minus (010481199010637

)

(19)

Pressure calculation is very complicated in this model andshould be presented in another paper It is worth to say thatparameters as superficial velocities Froude numbers volumefractions in different flow patterns and pipe propertieshaving the influence on frictional forces are considered Wewould like to focus on this part in next publication

Pressure correlations are good example of diversitybetween different studies and approaches In our tool fourdifferent pressure correlations return different results inthe bottom hole respectively [5] Beggs-Brill 1620 psi Ork-iszewski 1580 psi Aziz-Govier-Fogarasi 1616 psi and Duns-Ross 1507 psi

22 Temperature From the simulation point of view tem-perature is a parameter of pressure but from the engineeringpoint it is one of themost important factors regarding the pro-duction description and understanding In the calculationthe efficient correlation is the iteration based on the previousvalues [6]

119879119891 (119911) = 119866119879 (119911) minus119860 sin120572119869119862119898

+ 119860119865 + 119860119892119892 sin120572 + exp119864 (20)

where

119860 =Π119863119880120581

119872119872119862[120581 +

119863

2119862(log(

48radic119879119863119905

119863) minus 029)]

minus1

119864 = [(minus119911 minus 1199110

119860)

sdot (119879119891 (1199110) minus 119866119879 (1199110) +119860 sin120572119869119862119898

minus 119860119865 minus 119860119892119892 sin120572)]

(21)

119865 is a correction factor combined with hydrocarbon expan-sion from high pressure to low pressure during the tem-perature change known as a Joule-Thomson effect Heattransfer occurs between the wellbore fluid and the formationovercoming resistances offered by the tubing wall tubing-casing annulus casing wall and cement [19]

The mixture heat capacity calculated depends on howmany phases are considered Consider

119862119898 =sum119899

119894=1119872119894119862119894

sum119899

119894=1119872119894

(22)

Heat capacity of each phase is a function of temperatureand density usually estimated from the experimental dataGambill correlations are valid respectively for oil and gas asit is shown [20]

119862 = 984858minus12

(0338 + 000045 (119879 minus 460)) (23)

23 Inflow The derivation of Darcyrsquos law is used in inflowcalculation to determine the flow through the permeablegeological layer This equation is valid for liquid phase inflowat the perforation points but bubble point pressure has notbeen taken under the consideration [5]

119876 =000708119896ℎ (119901119903 minus 119901119908)

120583119861 [log (119903119890119903119908) minus 075 + 119878] (24)

Drainage radius may be multiplied by some coefficientsdepends on reservoir shape Skin factor is estimated basedon well test analysis A positive value indicates there is apressure decline in the near vicinity of well that is morethan expected based on the radial flow equation but does notnecessarily indicate formation damage [21] Considering gasinflow Darcy equation is adjusted in terms of gas properties

119876 =000708119896ℎ (119875

2

119903minus 1198752

119908)

120583120573119879 [log (119903119890119903119908) minus 075 + 119878] (25)

Inflow equations do not account for the phase change insolution gas reservoirs In a solution gas drive there isexpansion of the hydrocarbons below bubble point which isbeneficial because it adds energy to the system Gas liberationin a solution gas drive is also detrimental to oil productionbecause it lowers the effective permeability of oil [22]

3 The Framework

The framework represents an idea of how to organize themodelling part to get the complete information about thewhole physics during the production process Solving everycorrelation and calculation needs to be organized properlyEvery correlation is a function of data taken from produceror calculated by other correlations There are few approachesinvolved which usually meet in commercial solution andthey are protected by copyrights against publicityThe authorswould like to present optimised approach for productionsimulations divided in stages

31 Initialization Considering the well length and numberof physical data during the production it is very importantto establish the simulation points (meshing) in a project Allcorrelations should run in these points so the number of datafor one loop of simulation is significant Transient analysisis highly recommended as well especially during the gas liftor water injection procedures Correlations use different unitsystems and they create difficulties in the framework as wellThere is no standardization to keep the modelling resultsFlow area in the tubing affects the algorithms so three optionsare available tubing flow annular flow and tubing and annu-lar flow In some cases more than one tubing in a single well


provides the production There are wells where the tubing issplit into two independent items from any depth Sometimesonly one casing string without any tubing is involved It ishighly desirable to have the temperature or pressure datafrom gauges across the well especially at the wellhead andbottom hole Then every meshing point is calculated uponthis data Finally the complexity of modelling properties isrelatively high in this kind of the framework

The producer supports the data which has been called asinitial information (II) They are divided between sections asfollows Highly unlikely all of them are given very precisely(for every depth) but from the simulation point of view themore the data is given the better the results are obtainedHere they are presented as follows

(i) trajectory this set usually has four values true verticaldepth (TVD) measured depth (MD) inclination(Incl) and azimuth (Az) this is the well geometrydescription

(ii) geothermal temperature (119866119879) this is a trajectoryfunction some frameworksmay be based on geother-mal temperature especially in a very first part ofcalculation finally knowing this value is highly rec-ommended in terms of the comparison with thereal data in many cases the full understanding ofgeothermal temperature explains the well behaviour

(iii) completion items in the reality any additional itemchanges the physical data at this particular depth inparticular it is valid for packers and running electricsubmersible pumps gas lift valves inflow controldevices sand screens and so forth

(iv) completion data there are conductivity (Conn)roughness (119877) inner diameter (ID) outer diame-ter (OD) number of casing strings and cementproperties outer diameter conductivity and cementproperties are important regarding the heat transferand once the flow is observed in the annulus

(v) mixture properties itmay contain gasoil ratio (GOR)in surface condition information which affects theamount of gas out of the solution alternativelysome calculations use gaswater rate (GWR) it alsoincludesAPI gravity in surface condition as ameasureof how heavy or light a petroleum liquid is comparedto water and gas specified gravity (SG) in surface con-dition which is the ratio of the density of a substanceto the density of air and water density (WD)

(vi) layer data as depth (LD) permeability (119896) pressure(LP) and reservoir extend (RE) are values usuallygiven as constant for every layer layer permeabilityand pressure have the crucial impact on the produc-tion results

(vii) real data the subjects involved in industry have thereal data which can be used as a reference in themodelling process it may contain wellhead temper-ature (WHT) bottom hole temperature (BHT) well-head pressure (WHP) wellhead temperature (WHT)separator temperature (ST) separator pressure (SP)

production time (119905) anddormpressure in the annulus(DormP)

(viii) other information it is very flexible and cannot bespecified to the regular mandatory set it may containmany factors which has the important impact on thesimulation results that is gamma ray data and sur-rounding well information it is almost impossible topredict this information in the simulation model andusually this data is considered during the modellinginterpretation

32 Framework Steps Five steps of framework are identified

Step 1 Get the producer data mentioned as II previously

Step 2 Fill the mandatory data as geothermal temperatureflow diameter heat transfer geothermal gradient and TVDgradient

Step 3 Calculate correlations based on geothermal tempera-ture and layer pressure

Step 4 Calculate rates densities and viscosities for everyphase

Step 5 Find the final correlations based on temperature andpressure from the wellbore This level is different from Step3 because physical properties are calculated using the iter-ation algorithm instead of layers definition and geothermaltemperature data

We present the framework in pseudocode based on objectoriented programming

33 Correlations Once the literature review has been donethe most proper correlations regarding this framework havebeen chosen We are not interested in showing the corre-lations algorithm here but just in putting the emphasis onthe dependence Steps 3 and 5 use correlations with thedifferent entry parameters For Step 3 temperature (119879) is thegeothermal temperature (119866119879) from Step 2 pressure (119901) is thelayer pressure (LP) from Step 1 densities and other physicalproperties are taken from Steps 1 and 2 as well For Step 5temperature and pressure are iterated and density and otherphysical properties are taken from Steps 4 and 5 Here thecorrelations dependence is shown as follows

(i) BubblePointPressure(T SepT API SG SepP GOR)(ii) GasSolubility(T API p SG)(iii) DeadOilViscosity(T API)(iv) SaturatedOilViscosity(GasSolubility

DeadOilViscosity)(v) UnderSaturatedOilViscosity(p BubblePointPressure

SaturatedOilViscosity)(vi) OilViscosity(DeadOilViscosity SaturatedOilViscosity

UnderSaturatedOilViscosity p BubblePointPressure)(vii) WaterViscosity(WD T)


Measureddepth

True verticaldepth

Inclination

Depth

Dormpressure

Orifice

P

T

Gauges

Wellhead

Bottom hole

Trajectory

GLV

GLV

Gas lift

Producer

Geology

Bore

Step 1

Density

Oil Water Gas

Layer

Layer

Casing

Casing

Tubing

Tubing

Insidediameter

diameterOutside

Roughness

Conductivity

Permeability

Pressure

Shape

Size

Position

Skin factor

Figure 2 Step 1 data flow

(viii) GasViscosity(T SG)(ix) GasCompressibility(T SG p)(x) OilFormationVolumeFactor(T API GOR SG)(xi) WaterFormationVolumeFactor(T p)(xii) GasFormationVolumeFactor(T GasCompressibility p)(xiii) OilHeatCapacity(API T)(xiv) WaterHeatCapacity(T WD)(xv) GasHeatCapacity(T SG)(xvi) OilInflow(RE k LP p OilFormationVolumeFactor S)(xvii) WaterInflow(RE k LP p WaterFormationVolumeFac-

tor S)(xviii) GasInflow(SG RE k LP p GasCompressibility

GasViscosity GasFormationVolumeFactor S)(xix) SurfaceTension(T p)(xx) EarthThermalDiffusivity(T)(xxi) FrictionFactor(ReNS R FlowDiameter)

The temperature and pressure correlations are considered asa part of Step 5

34 Initial Information Step 1 In Step 1 there is nothing to dowith the calculation Only the producer trajectory bore liftgeology and density data are loaded and stored in one objectStep 1 (see Figure 2)

35 Basic Properties Step 2 This is the very first stepwhen calculation is performed Geothermal temperature isinterpolated based on two crucial values WHT and BHT

temperatureStep=(BHTminusWHT)MD[wellBottom]TVDTemperature[0]=WHTfor i=0 To WellBottomTVDTemperature[i+1]=TVDTemperature[i]+temperatureStepfor i=0 To WellBottomGT[i]=TVDTemperature[TVD[i]]

Listing 1 Geothermal temperature algorithm

for i=0 To WellBottomGG[i]=(GT[i+1]minusGT[i])(MD[i+1]minusMD[i])

Listing 2 Geothermal gradient algorithm

with respect toMD and TVD data As the results geothermaltemperature is given in Listing 1

Then geothermal gradient (GG) is calculated as shown inListing 2

Heat transfer [HT] calculation is shown in Listing 3For the tubing flow the flow diameter is equal to tubing

ID

36 Reservoir Properties Step 3 Here the correlations areused based on reservoir properties Passing the parametersto proper correlations the algorithm returns results arraysHence we have Listing 4


for i=0 To WellBottomHT[i]=(log(OD[i]ID[i])Conn[i])+(log(boreDiameter[i]OD[i])Conn[i])

Listing 3 Heat transfer algorithm

for i=0 To WellBottom [(1) BubblePointPressure[i]=correlationBubblePointPressure(GT[i] SepT API SG SepP GOR)(2) GasSolubility[i]=correlationGasSolubility(GT[i] API LP[i] GOR)(3) GasViscosity[i]=correlationGasViscosity(GT[i] SG)(4) DeadOilViscosity[i]=correlationDeadOilViscosity(GT[i] API)(5) SaturatedOilViscosity[i]=correlationSaturatedOilViscosity(GasSolubility[i] DeadOilViscosity[i])(6) UnderSaturatedOilViscosity[i]=correlationUnderSaturatedOilViscosity(LP[i] BubblePointPressure[i]SaturatedOilViscosity[i](7)OilViscosity[i]=correlationOilViscosity(DeadOilViscosity[i] SaturatedOilViscosity[i]UnderSaturatedOilViscosity[i] LP[i] BubblePointPressure[i])(8) WaterViscosity[i]=correlationWaterViscosity(WD GT[i])(9) GasCompressibility[i]=correlationGasCompressibility(GT[i] SG LP[i])(10) GasFormationVolumeFactor[i]=correlationGasFormationVolumeFactor(GT[i] GasCompressibility[i] LP[i])(11) OilFormationVolumeFactor[i]=correlationOilFormationVolumeFactor(GT[i] API GOR SG)(12) WaterFormationVolumeFactor[i]=correlationWaterFormationVolumeFactor(GT[i] API GOR SG)(13) OilHeatCapacity[i]=correlationOilHeatCapacity(API GT[i])(14) WaterHeatCapacity[i]=correlationWaterHeatCapacity(GT[i] WD)(15) GasHeatCapacity[i]=correlationGasHeatCapacity(GT[i] SG)](16) return new objectStep 3(BubblePoint GasSolubility GasViscosity OilViscosity DeadOilViscosity SaturatedOilViscosityUnderSaturatedOilViscosity WaterViscosity GasCompressibility GasFormationVolumeFactor OilFormationVolumeFactorWaterFormationVolumeFactor GasHeatCapacity OilHeatCapacity WaterHeatCapacity)

Listing 4 Step 3 algorithm

As the results objectStep3 contains correlations whichhave been used in inflowproduction correlations in Step 4

37 Flow Properties Step 4 Here another portion of reservoircalculation is presented It contains flow ratesmass flow ratesheat capacities and densities for all phases and their mixturewith the separation for liquid and gas phase The algorithmhas to take into consideration the different unit systems toavoid any problems with the passing parameters to the nextstep (see Listing 5)

Lines 5 6 and 7 are dependent on producer type Becausethe gas lift in current case was involved only GasFlowRateis the function of GOR In line 8 the condition is created toadjust the amount of lifted gas by gas lift valve into the well-bore

38 Wellbore Properties Step 5 Based on the simply thermo-dynamic principles it is obvious that gaseous and liquid statesnot only are merged into each other in a continuous mannerbut also are in fact similar in nature Volumes of moleculesand the intermolecular forces are necessary in establishingthe relationship between pressure volume and temperature

of gases and liquids So in the foundation of this model Vander Waals equation is used

(119901 +119886

V2) (V minus 119887) = 119896119879 (26)

Here the final step is presentedThis algorithm is the iterationso the initial temperature and pressure values are taken fromthe bottom hole gauge In this step temperature and pressurecorrelations are dependent as follows

(i) T(T[i+1] GT[i+1] Step4MixtureHeatCapacity Step4MixtureMassFlowRate t EarthThermalDiffusivityGT[i] BoreRadius FlowDiameter HT Incl GG)

(ii) P(P[i+1] Step4OilFlowRate Step4WaterFlowRateStep4GasFlowRate Step3OilFormationVolumeFac-tor Step3WaterFormationVolumeFactor Step3Gas-FormationVolumeFactor Step5OilViscosity Step5WaterViscosity Step5SurfaceTension depth Flow-Direction Step5WaterDensity Step5OilDensityStep5GasDensity FlowDiameter Incl gradTVD R)

Pressure and temperature are calculated from the bottom tothe top of the well as the flow occurs so index 119894 + 1 means


for i=0 To WellBottom [(1) TemperatureInflow[i]=GT[i](2) OilInflow[i]=correlationOilInflow(RE[i] k[i] LP[i] OilFormationVolumeFactor[i] OilViscosity[i] S[i])(3)WaterInflow[i]=correlationWaterInflow(RE[i] k[i] LP[i]WaterFormationVolumeFactor[i] WaterViscosity[i] S[i])(4) GasInflow[i]=correlationGasInflow(SG RE[i] k[i] GasCompressibility[i] GasViscosity[i] GT[i] S[i] LP[i] P[i])(5) OilFlowRate[i]=OilInflow[i](6)WaterFlowRate[i]=WaterInflow[i](7) GasFlowRate[i]=GasInflow[i] lowastGOR(8) if (GLVExist == true) GasFlowRate[i]+=GasInflowFromGLV(9) LiquidFlowRate[i]=WaterFlowRate[i]+OilFlowRate[i](10) GasMassFlowRate[i]=GasFlowRate[i] lowastSG(11) OilMassFlowRate[i]=OilFlowRate[i] lowastAPI(12)WaterMassFlowRate[i]=WaterFlowRate[i] lowastWD(13) LiquidMassFlowRate[i]=OilMassFlowRate[i]+WaterMassFlowRate[i](14)MixtureMassFlowRate[i]=LiquidMassFlowRate[i]+GasMassFlowRate[i](15) LiquidViscosity[i]=(OilMassFlowRate[i] lowastStep 3OilViscosity[i]+WaterMassFlowRate[i] lowastStep 3WaterViscosity[i])(OilMassFlowRate[i]+WaterMassFlowRate[i])(16)MixtureViscosity[i]=(OilMassFlowRate[i] lowastStep 3OilViscosity[i]+WaterMassFlowRate[i] lowastStep 3WaterViscosity[i]+GasMassFlowRate[i] lowastStep 3GasViscosity[i])(OilMassFlowRate[i]+WaterMassFlowRate[i]+GasMassFlowRate[i])(17) LiquidHeatCapacity[i]=(OilMassFlowRate[i] lowastStep 3OilHeatCapacity[i]+WaterMassFlowRate[i] lowastStep 3WaterHeatCapacity[i])(OilMassFlowRate[i]+WaterMassFlowRate[i])(18)MixtureHeatCapacity[i]=(OilMassFlowRate[i] lowastStep 3OilHeatCapacity[i]+WaterMassFlowRate[i] lowastStep 3WaterHeatCapacity[i]+GasMassFlowRate[i] lowastStep 3GasHeatCapacity[i])(OilMassFlowRate[i]+WaterMassFlowRate[i]+GasMassFlowRate[i])(19) LiquidDensity[i]=(OilMassFlowRate[i] lowastAPI+WaterMassFlowRate[i] lowastWD)(OilMassFlowRate[i]+WaterMassFlowRate[i])(20)MixtureDensity[i]=(OilMassFlowRate[i] lowastAPI+WaterMassFlowRate[i] lowastWD+GasMassFlowRate[i] lowastSG)(OilMassFlowRate[i]+WaterMassFlowRate[i]+GasMassFlowRate[i])]return new objectStep 4( )


the previous correlations results Additionally the pressurecorrelation returns object which contains the following

return new objectPressure(P FlowDirection FlowTypeLiquidSuperficialVelocity GasSuperficialVelocityMix-tureVelocity LiquidVolumeFraction KineticPressure-Drop ReNS HydrostaticPressureLoss FrictionFactor)

As a FlowType we consider segregated intermittent dis-tributed and transient Every type has its own flow regimeswhich is the part of the pressure correlation The importantpart of this step is to find the amount of gas which is outof the solution The relation between solubility of liquidcomponents and the heat of solution is given as

[120575 ln119909119894120575 (1119879)

]

119875

= minusΔ 119904119897119899ℎ119894

119877 (27)

where 119909119894 is the mole fraction of 119894th alkane in water andΔ 119904119897119899ℎ119894 is the difference between the partial enthalpy of the119894th hydrocarbon at infinite dilution and the molar enthalpyof the pure hydrocarbon The heat of solution includes twoeffects positive heat of cavity formation and negative heat ofhydrophobic interaction between the hydrocarbon andwater

These two effects cancel each other at 119879119898 So the descriptionof solubility of hydrocarbons in water may be presented as

ln119909119894 (119879) = ln119909119894 (119879119898) + (Δ 119904119897119899119862119875119894

119877)[ln( 119879

119879119898

) +119879119898

119879minus 1]

(28)

In this step phase intermixing of viscosity heat capacity andrates is also considered

Finally Step 5 is presenetd as shown in Listing 6

39 Final Join All these steps have to be joined in the finalcalculationwhich solves allmodelling stages Considering thewhole oilfield this simulation does not take into consider-ation limited amount of wells Every well can be treated asa single thread calculation until the gas lift is considered asan optimisation problem for the oilfieldThen the proper gasdistribution is dependent on everywell simulationThis studyis planned as a future work

Hence we have Listing 7Function InitP is the initial pressure calculation based

on WHP i BHP data with the consideration of trajectorycurvature


(1) P[wellBottom]=BHP(2) T[wellBottom]=BHT(3) for WellBottom-1 To i=0 [(4) GasDensity[i]=Step 4GassMassFlowRate[i] lowastSG(5) OilDensity[i]=Step 4OilMassFlowRate[i] lowastAPI(6)WaterDensity[i]=Step 4WaterMassFlowRate[i] lowastWD(7) SumGasFlowRate[i]=GasFlowRate[i]+SumGasFlowRate[i+1](8) SumOilFlowRate[i]=OilFlowRate[i]+SumOilFlowRate[i+1](9) SumWaterFlowRate[i]=WaterFlowRate[i]+SumWaterFlowRate[i+1](10) SumLiquidFlowRate[i]=SumOilFlowRate[i]+SumWaterFlowRate[i](11) GOR[i]=GasFlowRate[i]OilFlowRate[i](12) GasSolubility[i]=correlationGasSolubility(T[i+1] OilDensity[i] P[i+1] GasDensity[i+1])(13) GasCompressibility[i]=correlationgasCompressibility(T[i+1] GasDensity[i] P[i+1])(14) GasFormationVolumeFactor[i]=correlationGasFormationVolumeFactor(T[i+1] GasCompressibility[i] P[i+1])(15) OilFormationVolumeFactor[i]=correlationOilFormationVolumeFactor(T[i+1] OilDensity[i] GOR[i] GasDensity[i])(16)WaterFormationVolumeFactor[i]=correlationWaterFormationVolumeFactor(T[i+1] P[i+1])(17) EarthThermalDiffusivity[i]=correlationEarthThermalDiffusivity(T[i+1])(18) SumGasFlowRate[i] lowast=GasFormationVolumeFactor[i](19) SumOilFlowRate[i] lowast=OilFormationVolumeFactor[i](20) SumWaterFlowRate[i] lowast=WaterFormationVolumeFactor[i](21) SumLiquidFlowRate[i]=SumOilFlowRate[i]+SumWaterFlowRate[i](22) if (GasSolubility[i] lt GOR[i])FreeGasFlowRate[i]=GasFlowRate[i]minus(GasSolubility[i] lowastOilFlowRate[i])else FreeGasFlowRate[i]=0(23) GasInSolutionFlowRates[i]=Step 4GasFlowRate[i]minusFreeGasFlowRate[i](24) BubblePointPressure[i]=correlationBubblePointPressure(T[i] SepT OilDensity[i] GasDensity[i] SepP GOR[i])(25) GasViscosity[i]=correlationGasViscosity(T[i+1] GasDensity[i])(26) DeadOilViscosity[i]=correlationDeadOilViscosity(T[i+1] OilDesnity[i])(27) SaturatedOilViscosity[i]=correlationSaturatedOilViscosity(GasSolubility[i] DeadOilViscosity[i])(28) UnderSaturatedOilViscosity[i]=correlationUnderSaturatedOilViscosity(P[i+1] BubblePointPressure[i]SaturatedOilViscosity[i](29) OilViscosity[i]=correlationOilViscosity(DeadOilViscosity[i] SaturatedOilViscosity[i] UnderSaturatedOilViscosity[i]P[i+1] BubblePointPressure[i])(30)WaterViscosity[i]=correlationWaterViscosity(WaterDensity[i] T[i+1])(31) OilHeatCapacity[i]=correlationOilHeatCapacity(OilDensity[i] T[i+1])(32)WaterHeatCapacity[i]=correlationWaterHeatCapacity(T[i+1] WaterDensity[i])(33) GasHeatCapacity[i]=correlationGasHeatCapacity(T[i+1] GasDensity[i])(34) LiquidViscosity[i]=(Step 4OilMassFlowRate[i] lowastStep 5OilViscosity[i]+Step 4WaterMassFlowRate[i] lowastStep 5WaterViscosity[i])(Step 4OilMassFlowRate[i]+Step 4WaterMassFlowRate[i])(35)MixtureViscosity[i]=(Step 4OilMassFlowRate[i] lowastStep 5OilViscosity[i]+Step 4WaterMassFlowRate[i] lowastStep 5WaterViscosity[i]+Step 4GasMassFlowRate[i] lowastStep 5GasViscosity[i])(Step 4OilMassFlowRate[i]+Step 4WaterMassFlowRate[i]+Step 4GasMassFlowRate[i])(36) LiquidHeatCapacity[i]=(Step 4OilMassFlowRate[i] lowastStep 5OilHeatCapacity[i]+Step 4WaterMassFlowRate[i] lowastStep 5WaterHeatCapacity[i])(Step 4OilMassFlowRate[i]+Step 4WaterMassFlowRate[i])(37)MixtureHeatCapacity[i]=(Step 4OilMassFlowRate[i] lowastStep 5OilHeatCapacity[i]+Step 4WaterMassFlowRate[i] lowastStep 5WaterHeatCapacity[i]+Step 4GasMassFlowRate[i] lowastStep 5GasHeatCapacity[i])(Step 4OilMassFlowRate[i]+Step 4WaterMassFlowRate[i]+Step 4GasMassFlowRate[i])(38) LiquidDensity[i]=(Step 4OilMassFlowRate[i] lowastOilDensity[i]+Step 4WaterMassFlowRate[i] lowastWaterDensity[i])(Step 4OilMassFlowRate[i]+Step 4WaterMassFlowRate[i])(39)MixtureDensity[i]=(Step 4OilMassFlowRate[i] lowastOilDensity[i]+Step 4WaterMassFlowRate[i] lowastWaterDensity+Step 4GasMassFlowRate[i] lowastGasDensity[i])(Step 4OilMassFlowRate[i]+Step 4WaterMassFlowRate[i]+Step 4GasMassFlowRate[i])(40) SumGasMassFlowRate[i]+=Step 4GasMassFlowRate[i]+SumGasMassFlowRate[i+1](41) SumOilMassFlowRate[i]+=Step 4OilMassFlowRate[i]+SumOilMassFlowRate[i+1](42) SumWaterMassFlowRate[i]+=Step 4WaterMassFlowRate[i]+SumWaterMassFlowRate[i+1](43)MixtureMassFlowRate[i]=SumGasMassFlowRate[i]+SumOilMassFlowRate[i]+SumWaterMassFlowRate[i](44) T[i]=correlationTemperature( )(45) SurfaceTension[i]=correlationSurfaceTension(T[i] P[i+1])(46) P[i]=correlationPressure( )

Listing 6 Continued


]return new objectStep 5(accumulatedGasMassFlowRate accumulatedOilMassFlowRate accumulatedWaterMassFlowRate gasDoilD watD gasSolubility gor gasCompressibility GFVF OFVF WFVF bubblePoint gasViscosity oilViscosity liquidViscositymixtureViscosity waterViscosity oilHeatCapacity waterHeatCapacity gasHeatCapacity T SurfaceTension P)


for (i=0 To numberOfWells) [(1) Step 2Add(Step 2Run(Step 1)(2) Step 3Add(Step 2Run(oilFiledData Step 2))(3) InitP=InitP(Step 1)(4) Step 4Add(Step 4Run(Step 3 Step 2 InitP GasLift)(5) Step 5Add(Step 5Run(Step 1 Step 2 Step 4 SG API WD))]

Listing 7 Final join algorithm

0 500 1000 1500 2000 2500 3000

MD (ft)

0

200

400

600

800

1000

1200

1400

1600

TVD

(ft)

Figure 3 Well trajectory

4 Case Study Simulation

Based on the authors experience in oil and gas industry thesmall oilfieldmodel was createdThe application tool has alsobeen developed It gives the possibility to look through allthe results We emphasize on the single well and the fullinterpretation of simulated properties is not a part of thisstudy A few examples of the simulated results are presentedbelow

The well trajectory (Figure 3) is very characteristic forrelatively shallow wells commonly found in the middleeast Geothermal temperature profile (Figure 4) meets thetrajectory This is the only model so the surface temperatureof about 350K is relatively high here This wellbore isa single casing and tubing string with the ID 12 inchesand 9 inches respectively Total water and oil inflow chart(Figure 5) oil inflow (Figure 6) and gas inflow (Figure 7)charts confirm that two productive layers are given at thisoilfield Total oil production from this simulation is estimatedon 2588 BPD which is very accurate value in comparison toreal production data for this kind of wells Gas inflow chartdoes not consider gas lift from the annulus and it is part

0 500 1000 1500 2000 2500 3000

MD (ft)

345

350

355

360

365

370

375

Geo

ther

mal

tem

p (K

)

Figure 4 Geothermal temperature profile

0

500

1000

1500

2000

2500

3000

3500

0 500 1000 1500 2000 2500 3000 3500

4000

4500

Inflo

w (B

PD)

MD (ft)

Water inflowOil inflow

Figure 5 Inflow

of gas accumulated in this reservoir The most importantphysical properties in thewellbore are pressure (Figure 9) andtemperature (Figure 8) These values meet the criteria very


0 500 1000 1500 2000 2500 3000

MD (ft)

0

05

1

15

2

25

3

35

4

45

Gas

inflo

w (s

cfD

ft)

Figure 6 Gas inflow per feet

0 500 1000 1500 2000 2500 3000

MD (ft)

0

5

1

6

2

7

3

8

4

9

Oil

inflo

w (B

PDft

)

Figure 7 Oil inflow per feet

0 500 1000 1500 2000 2500 3000

MD (ft)

345

350

340

380

355

360

365

370

375

Tem

pera

ture

(K)

Figure 8 Wellbore temperature

closely but the difference between the simulated temperatureand the geothermal temperature is low (Figure 10) Theseare the results where this simulation is performed under theearly stage of production Below the reservoir the coefficientbetween temperature and pressure is equal so it means thatthis is to prove that the simulation is correct In Table 1 someresults from the simulation are presented

Presented values are the confirmation that the frameworkand its implementation work properly These values behaveas a real one Flow rates at the bottom are almost 0 and thesurface values are highly reliable 2588 BPD of oil 4052 BPDof water and 13MMScfD of gas Temperature and pressurevalues are highly accurate even if the wellhead temperatureis very high Formation volume factors heat capacities andviscosities are accurate as well The small disadvantage isobserved for the Reynolds number at the bottom This value

0 500 1000 1500 2000 2500 3000

MD (ft)

1500

1600

1700

1800

1900

2000

2100

2200

Well

bore

pre

ssur

e (ps

i)

Figure 9 Wellbore pressure

0 500 1000 1500 2000 2500 3000

MD (ft)

0149

01495

015

01505

0151

Tem

pera

ture

(K)

Figure 10 The difference between wellbore temperature andgeothermal temperature

is almost 0 and it is little bit unexpected according to Moodydiagram and its properties At the surface the Reynoldsnumber gets the proper value in terms of pipe roughnesswhich is equal to 00006 inches

5 Conclusions

The literature study regarding the single property simula-tions gives the possibility to create the framework whichallows obtaining the simulation for physical properties dur-ing the oil production The authors choose the correlationswhich meet the criteria in terms of accuracy and efficiencyThe obtained results can be classified as proper ones in thecomparison to the real data Unfortunately upon the legalprocedures the comparison results could not be publishedBut based on the authors experienced in oil and gas industrythe presented correlation results are very reliable The pres-sure and temperature profiles as the results of all physicalproperties simulations meet the expectations In this paperthe authors does not analyse physiochemical properties orchemical coordinates but this analysis is considered as asubject of future paper This paper has been created toshow the possibility of wellbore simulation which has beenproved as an accurate comparison to the real data at oilfieldsIt is important fact that the whole simulation is runningfast because in some cases multithreading technology isused here The flexibility of the framework idea gives theopportunity to adjust every correlation according to the latest


Table 1 Simulation values

Property Wellhead value Bottom hole valueWater heat capacity [BtulblowastR] 0982555750110051 0960586678396243Water formation volume factor 100467452445733 102484739136526Water density [lbft2] 8324 835165343126831Water viscosity [cP] 0382879865231827 025322317446799Water rate [BPD] 405262648 024196459483204MD [ft] 0 3188TVD [ft] 0 1160Wellbore temperature [K] 350150423209288 36909601487285Surface tension [dynmm] 518881443443093 415637927992043Reynolds number 179714829320058 000218243682336947Wellbore pressure [psi] 150801890934205 200011967522Oil viscosity [cp] 139027568317487 183235372068421Oil rate [BPD] 2588381657 00145917902247552Oil density [API] 30 300205298689915Oil heat capacity [BtulblowastR] 0442969196896299 0486093081722976Oil formation volume factor 125236622654504 10949263698297Liquid volume fraction 0999655206045957 0999999596771036Heat transfer [BtuDft2F] 355147698582206 207748085888011Gas heat capacity [BtulblowastR] 049558307261851 0543829017247149Gas formation volume factor 0010147402881213 000924193736098171Geothermal temperature [K] 350015683814304 368130489335027Gas viscosity [cP] 0016290648370466 00177988938761987Gas solubility 236687450637186 257906526460697Gas rate [MMScfD] 1294190828 00000072958951123Gas density [SG] 07 0700000001749332Gas compressibility 0859040738142422 0908102165721585Flow type Segregated DistributedBubble point pressure [psi] 239864226965175 111431088951275

studiesThis idea may be extended on optimisation problemsfor gas lift managing and distribution for the oilfield

Symbol Description

119864119896 Dimensionless kinetic termV119898 Mixture velocity [fts]Vsg Gas superficial velocity [fts]V119897 Liquid superficial velocity [fts]984858ns No slip density [lbft2]984858119897 Liquid density [lbft2]984858119892 Gas density [lbft2]984858119898 Mixture density [lbft2]119901 Pressure [psi]119901119903 Reservoir pressure [psi]119901119908 Wellbore pressure [psi]119863 Pipe diameter where the flow occurs [in]119903ti Tubing inner diameter [in]119903to Tubing outer diameter [in]119861 Volume factor [bblSTD]119876 Inflow [BPD]119891 Friction factor119891119871 Friction factor of laminar flow region

119891119879 Friction factor of turbulent flow regionRe Reynolds numberRe119871 Laminar to transition boundary Reynolds numberRe119879 Transition to turbulent boundary Reynolds number119899 Flow behaviour index119877 Roughness [in]119879 Temperature [K] [C] [F]119879119891 Inflow temperature [K] [C] [F]120572 Pipe inclination [deg]120573 Gas compressibility factorFr Froude number120601 Volume fraction119878119879 Surface tension [dynmm]120583119898 Mixture viscosity [cP]119872119898 Mass of mixture [kg]119862 Heat capacity [Btulb lowast R]119862119898 Mixture heat capacity [Btulb lowast R]119880 Heat transfer [BtuDayft2F]120581 Layer thermal conductivity [BtuDayft2F]119892119892 Geothermal gradient [Fft]119866119879 Geothermal temperature [K] [C] [F]119879119863 Thermal diffusivity of Earth119905 Production time [h]


119869 Mechanical equivalent of heat [ft-lbBtu]119898 Inflow mass rate [kg]119903119908 Wellbore radius [in]119903119890 Drainage radius [ft]119896 Permeability [mD]119878 Skin factor

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to thank the referees for their valuablecomments which helped to improve this paper

References

[1] P Pourafshary A coupled wellborereservoir simulator to modelmultiphase flow and temperature distribution [PhD thesis]2007

[2] B Bielecki B Ksiezopolski A Krajka and A Wierzbicki ldquoTheconcept and securityanalysis of wireless sensor network for gaslift in oilwellsrdquo Annales UMCS Informatica vol 14 no 2 pp76ndash85 2014

[3] R Sharma K Fjalestad and B Glemmestad ldquoOptimizationof lift gas allocation in a gas lifted oil field as non-linearoptimization problemrdquo Modeling Identification and Controlvol 33 no 1 pp 13ndash25 2012

[4] K Rashid W Bailey and B Couet ldquoA survey of methods forgas-lift optimizationrdquoModelling and Simulation in Engineeringvol 2012 Article ID 516807 16 pages 2012

[5] Gas Lift Manual Gabor Takacs PennWell 2005[6] R Sagar D R Doty and Z Schmidt ldquoPredicting temperature

profiles in a flowing wellrdquo SPE Production Engineering vol 6no 4 pp 441ndash448 1991

[7] A Chandra V Patidar M Singh and R K Kale ldquoPhysic-ochemical and friccohesity study of glycine l-alanine andl-phenylalanine with aqueous methyltrioctylammonium andcetylpyridinium chloride fromT = (29315 to 30815) Krdquo Journalof Chemical Thermodynamics vol 65 pp 18ndash28 2013

[8] M Singh ldquoCombined device formeasuring of osmotic pressureconductance surface tension and viscosityrdquo Russian Journal ofPhysical Chemistry A vol 84 no 13 pp 2345ndash2350 2010

[9] J Obuba S Ikiesnkimama C E Ubani and I C EkekeldquoNatural gas compressibility factor correlation evaluation forniger delta gas fieldsrdquo IOSR Journal of Electrical and ElectronicsEngineering vol 6 no 4 pp 1ndash10 2013

[10] X Fang Y Xu X Su and R Shi ldquoPressure drop and frictionfactor correlations of supercritical flowrdquo Nuclear Engineeringand Design vol 242 pp 323ndash330 2012

[11] R P Sutton ldquoAn accurate method for determining oil PVTproperties using the Standing-Katz gas Z-factor chartrdquo SPEReservoir Evaluation and Engineering vol 11 no 2 Article ID246266 2008

[12] E AOsman andMAAl-Marhoun ldquoArtificial neural networksmodels for predicting PVT properties of oil field brinesrdquo inProceedings of the 14th SPE Middle East Oil and Gas Show andConference SPE 93765 Bahrain 2005

[13] A Kamari A Hemmati-Sarapardeh S-M Mirabbasi NNikookar and A HMohammadi ldquoPrediction of sour gas com-pressibility factor using an intelligent approachrdquo Fuel ProcessingTechnology vol 116 pp 209ndash216 2013

[14] W D McCain Jr The Properties of Petroleum Fluids PennWellBooks Tulsa Okla USA 2nd edition 1990

[15] S K Chen R Petroski and N E Todreas ldquoNumericalimplementation of the Cheng and Todreas correlation for wirewrapped bundle friction factors-desirable improvements in thetransition flow regionrdquo Nuclear Engineering and Design vol263 pp 406ndash410 2013

[16] WH SommertonThermal Properties and Temperature-RelatedBehavior of RockFluid Systems Developments in PetroleumScience 1992

[17] N H Chen ldquoAn explicit equation for friction factor in piperdquoIndustrial and Engineering Chemistry Fundamentals vol 18 no3 pp 296ndash297 1979

[18] N Matubayasi Surface Tension and Related ThermodynamicQuantities of Aqueous Electrolyte Solutions CRC Press NewYork NY USA 2013

[19] A Firoozabadi and D L Katz Surface Tension of ReservoirCrudeoilGas Systems Recognizing The Asphalt in The HeavyFraction Society of Petroleum Engineers 1988

[20] A R Hasan and C S Kabir ldquoWellbore heat-transfer modelingand applicationsrdquo Journal of Petroleum Science and Engineeringvol 86-87 pp 127ndash136 2012

[21] J Lee J B Rollins and J P Spivey Pressure Transient TestingSociety of Petroleum Engineers Richardson Tex USA 2003

[22] K Brown and D Beggs ldquoInflow performancerdquo in The Technol-ogy of Artificial Lift chapter 1 p 13 1977

Submit your manuscripts athttpwwwhindawicom

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Algebra

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


In Section 4 typical case study can be found The toolwhich has been developed to present all data at everystage of simulation has been implemented We can presentvariables as production rate pressure temperature densitysolubility compressibility formation volume factor viscosityheat capacity surface tension gasoil ratio gaswater ratiomass rate bubble point pressure flow direction flow regimesvelocity hydrostatic and kinetic pressure drop Reynoldsnumber and others as function of mentioned It givesthe possibility to present different kind of data which arecomparable to other researches

To summarize the main contributions of this paperare to show the framework idea which groups wellborecorrelations altogether This framework gives the reliablesimulation results which is proved in this paper

2 Correlations

The authors would like to present the single correlation forevery property There is no clear algorithm that suggestswhich correlation should be taken under the considerationas the most proper one It depends on many factors as fluiddensity trajectory and so forth The best way to gain theaccurate modelling results is to compare obtained data withthe real one Good example can be found from Tulsa oilfieldsanalysis [1] where it has been proven by the comparison ofrelative performance factor that the mechanistic approachgives better results than empirical correlations Ansari modelpresented in Figure 1 is the best in this example It does notnecessarily means that the Ansari model is reliable for everywellbore simulation It is only regarding this case

Here the most important part of calculation is presentedThree major parameters as pressure temperature and inflowexist as a function of the following variables bubble pointpressure gas solubility gas compressibility formation volumefactors viscosities heat capacities surface tension comple-tion data and reservoir properties

21 Pressure Our method is based on solving the equationusing a finite difference approximation method with thecontinuity momentum and energy equation as follows

984858119897+1

119894minus 984858119897

119894

Δ119905+984858V119897+1119894+12

minus 984858V119897+1119894minus12

Δ119909= 0

984858V119897+1119894+12

minus 984858119897

119894+12

Δ119905+[984858V2119894+12

minus 984858V2119894minus12

]119897

Δ119909+119901119897+1

119894+1minus 119901119897+1

119894

Δ119909

+ 984858119897

119894+12119892 + [

[

119891 (984858V)2119894+12

2119863

]

]

119897

= 0

984858119894

119864119897+1

119894+1minus 119864119897+1

119894

Δ119905+ 984858V119894

119864119897

119894minus 119864119897

119894minus1

Δ119909+ 119875119897+1

119894

(V119894+12 minus V119894minus12)119897

Δ119909

+ 2119867(119879119903 minus 119879119908)

119897

119903119908

= 0

(1)

Mechanisticapproach

correlationsEmpirical

Hom

ogen

eous

0

1

2

3

4

5

6

Relat

ive p

erfo

rman

ce fa

ctor

for a

ll ve

rtic

al w

ell d

ata

Hag

edor

n

Dun

s

Begg

s

Ork

iszew

ski

Muk

herje

e

Has

an

Azi

z

Ans

ari

Figure 1 The different models comparison [1] The lower valuemeans the better results

Across the years the multiphase flow theory has beenimproved andmakes the calculation difficult to predict due toseveral reasons In the flow problem the number of variablesaffecting pressure drop is enormous The interfacial tensionbetween phases is taken into account but all kinds of availablemodels try to reduce the total number of variables with theintroduction of nondimensional parameter group Frictionalpressure losses are more difficult considering multiphaseflow so in addition new kind of energy losses has beenused slippage loss All these features make the model socomplicated that we cannot describe it in this paper It isimportant to emphasize that usually the correlations whichare used are based on empirical method

Pressure loss across the wellbore during the production isdivided into three components

(119889119901

119889119897)

total= (

119889119901

119889119897)

hydrostatic+ (

119889119901

119889119897)

kinetic+ (

119889119901

119889119897)

frictional

(2)

The cohesive forces (CF) and frictional forces (FF) havethe crucial influence on fluid drive especially regardingphases separation or bind The friccohesity model has beenconsidered as a CF and FF product Using the Mansinghequation [7]

120590 = 1205900 [(119905

1199050

plusmn119861

119905)(

119899

1198990

plusmn 00012 (1 minus 984858))] (3)

the friccohesity as a function of surface tension and cohesiveand frictional forces is calculated In this equation 1199050 and 119905

are the reference and sample flow times respectively 1199030 isreference friccohesity and 1198990 and 119899 are the pendant dropnumber of reference and sample respectively Referencefriccohesity is represented by the following formula

1205900 =1205830

1198781198790

(4)

Mansingh survisemeter is a device to measuring friccohesity[8] The friccohesity analysis is dependent on hydrocarboncomposition In this model we are concentrating on the



(119889119901

119889119897)

total=


1 minus 119864119896

(5)


119864119896 = 0000216119891V119898Vsg984858ns

119901 (6)


119881 =576

Π1198632

119899

sum

119894=1

119861119894119876119894 (7)


119861119908 = (1 + Δ119881119908119875) (1 + Δ119881119908119879)


minus4119879

+ 550654 times 10minus71198792


minus131199012119879


minus101199012

(8)


119891 = 119891119871 (1 minus 120595119887)13

(1 minus 12059513

119887) + 119891119879120595

13

119887 (9)


120595119887 =log (Re Re119871)log (Re119879Re119871)

(10)


Re =124 lowast 984858119898119881119898119863

120583119898

(11)

Hence

log(Re119871300

) = 17 ((119901

119863) minus 1)

log(Re119879105

) = 07 ((119901

119863) minus 1)

(12)



984858ns =119881119897984858119897

119881119898

+ (1 minus119881119897

119881119898

) 984858119892 (13)


(119889119901

119889119897)

hydrostatic=

1

144

984858119898 (119879 120572 Fr 120601 119878119905) 119889119909119889119897

(14)


(119889119901

119889119897)

119891

= 1294 times 10minus4119891984858ns984858119897119881

2

119897

119863 (15)


119891 = [(4 log 119877

37065119863minus50452 log(120575)

Re)

2

]

minus1

(16)

where

120575 =1

28257(119877

119863)

11098

+ (7149

Re)

08981

(17)





206 (18)

where

1205741 = 75 minus (11081199010349

)

1205742 = 53 minus (010481199010637

)

(19)




119879119891 (119911) = 119866119879 (119911) minus119860 sin120572119869119862119898

+ 119860119865 + 119860119892119892 sin120572 + exp119864 (20)

where

119860 =Π119863119880120581

119872119872119862[120581 +

119863

2119862(log(

48radic119879119863119905

119863) minus 029)]

minus1

119864 = [(minus119911 minus 1199110

119860)

sdot (119879119891 (1199110) minus 119866119879 (1199110) +119860 sin120572119869119862119898

minus 119860119865 minus 119860119892119892 sin120572)]

(21)



119862119898 =sum119899

119894=1119872119894119862119894

sum119899

119894=1119872119894

(22)


119862 = 984858minus12

(0338 + 000045 (119879 minus 460)) (23)


119876 =000708119896ℎ (119901119903 minus 119901119908)

120583119861 [log (119903119890119903119908) minus 075 + 119878] (24)


119876 =000708119896ℎ (119875

2

119903minus 1198752

119908)

120583120573119879 [log (119903119890119903119908) minus 075 + 119878] (25)


3 The Framework




























Measureddepth

True verticaldepth

Inclination

Depth

Dormpressure

Orifice

P

T

Gauges

Wellhead

Bottom hole

Trajectory

GLV

GLV

Gas lift

Producer

Geology

Bore

Step 1

Density

Oil Water Gas

Layer

Layer

Casing

Casing

Tubing

Tubing

Insidediameter

diameterOutside

Roughness

Conductivity

Permeability

Pressure

Shape

Size

Position

Skin factor















ID












(119901 +119886

V2) (V minus 119887) = 119896119879 (26)











[120575 ln119909119894120575 (1119879)

]

119875

= minusΔ 119904119897119899ℎ119894

119877 (27)



ln119909119894 (119879) = ln119909119894 (119879119898) + (Δ 119904119897119899119862119875119894

119877)[ln( 119879

119879119898

) +119879119898

119879minus 1]

(28)








Listing 6 Continued






0 500 1000 1500 2000 2500 3000

MD (ft)

0

200

400

600

800

1000

1200

1400

1600

TVD

(ft)





0 500 1000 1500 2000 2500 3000

MD (ft)

345

350

355

360

365

370

375

Geo

ther

mal

tem

p (K

)


0

500

1000

1500

2000

2500

3000

3500

0 500 1000 1500 2000 2500 3000 3500

4000

4500

Inflo

w (B

PD)

MD (ft)


Figure 5 Inflow



0 500 1000 1500 2000 2500 3000

MD (ft)

0

05

1

15

2

25

3

35

4

45

Gas

inflo

w (s

cfD

ft)


0 500 1000 1500 2000 2500 3000

MD (ft)

0

5

1

6

2

7

3

8

4

9

Oil

inflo

w (B

PDft

)


0 500 1000 1500 2000 2500 3000

MD (ft)

345

350

340

380

355

360

365

370

375

Tem

pera

ture

(K)




0 500 1000 1500 2000 2500 3000

MD (ft)

1500

1600

1700

1800

1900

2000

2100

2200

Well

bore

pre

ssur

e (ps

i)


0 500 1000 1500 2000 2500 3000

MD (ft)

0149

01495

015

01505

0151

Tem

pera

ture

(K)



5 Conclusions






Symbol Description







Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(119889119901

119889119897)

total=


1 minus 119864119896

(5)


119864119896 = 0000216119891V119898Vsg984858ns

119901 (6)


119881 =576

Π1198632

119899

sum

119894=1

119861119894119876119894 (7)


119861119908 = (1 + Δ119881119908119875) (1 + Δ119881119908119879)


minus4119879

+ 550654 times 10minus71198792


minus131199012119879


minus101199012

(8)


119891 = 119891119871 (1 minus 120595119887)13

(1 minus 12059513

119887) + 119891119879120595

13

119887 (9)


120595119887 =log (Re Re119871)log (Re119879Re119871)

(10)


Re =124 lowast 984858119898119881119898119863

120583119898

(11)

Hence

log(Re119871300

) = 17 ((119901

119863) minus 1)

log(Re119879105

) = 07 ((119901

119863) minus 1)

(12)



984858ns =119881119897984858119897

119881119898

+ (1 minus119881119897

119881119898

) 984858119892 (13)


(119889119901

119889119897)

hydrostatic=

1

144

984858119898 (119879 120572 Fr 120601 119878119905) 119889119909119889119897

(14)


(119889119901

119889119897)

119891

= 1294 times 10minus4119891984858ns984858119897119881

2

119897

119863 (15)


119891 = [(4 log 119877

37065119863minus50452 log(120575)

Re)

2

]

minus1

(16)

where

120575 =1

28257(119877

119863)

11098

+ (7149

Re)

08981

(17)





206 (18)

where

1205741 = 75 minus (11081199010349

)

1205742 = 53 minus (010481199010637

)

(19)




119879119891 (119911) = 119866119879 (119911) minus119860 sin120572119869119862119898

+ 119860119865 + 119860119892119892 sin120572 + exp119864 (20)

where

119860 =Π119863119880120581

119872119872119862[120581 +

119863

2119862(log(

48radic119879119863119905

119863) minus 029)]

minus1

119864 = [(minus119911 minus 1199110

119860)

sdot (119879119891 (1199110) minus 119866119879 (1199110) +119860 sin120572119869119862119898

minus 119860119865 minus 119860119892119892 sin120572)]

(21)



119862119898 =sum119899

119894=1119872119894119862119894

sum119899

119894=1119872119894

(22)


119862 = 984858minus12

(0338 + 000045 (119879 minus 460)) (23)


119876 =000708119896ℎ (119901119903 minus 119901119908)

120583119861 [log (119903119890119903119908) minus 075 + 119878] (24)


119876 =000708119896ℎ (119875

2

119903minus 1198752

119908)

120583120573119879 [log (119903119890119903119908) minus 075 + 119878] (25)


3 The Framework




























Measureddepth

True verticaldepth

Inclination

Depth

Dormpressure

Orifice

P

T

Gauges

Wellhead

Bottom hole

Trajectory

GLV

GLV

Gas lift

Producer

Geology

Bore

Step 1

Density

Oil Water Gas

Layer

Layer

Casing

Casing

Tubing

Tubing

Insidediameter

diameterOutside

Roughness

Conductivity

Permeability

Pressure

Shape

Size

Position

Skin factor















ID












(119901 +119886

V2) (V minus 119887) = 119896119879 (26)











[120575 ln119909119894120575 (1119879)

]

119875

= minusΔ 119904119897119899ℎ119894

119877 (27)



ln119909119894 (119879) = ln119909119894 (119879119898) + (Δ 119904119897119899119862119875119894

119877)[ln( 119879

119879119898

) +119879119898

119879minus 1]

(28)








Listing 6 Continued






0 500 1000 1500 2000 2500 3000

MD (ft)

0

200

400

600

800

1000

1200

1400

1600

TVD

(ft)





0 500 1000 1500 2000 2500 3000

MD (ft)

345

350

355

360

365

370

375

Geo

ther

mal

tem

p (K

)


0

500

1000

1500

2000

2500

3000

3500

0 500 1000 1500 2000 2500 3000 3500

4000

4500

Inflo

w (B

PD)

MD (ft)


Figure 5 Inflow



0 500 1000 1500 2000 2500 3000

MD (ft)

0

05

1

15

2

25

3

35

4

45

Gas

inflo

w (s

cfD

ft)


0 500 1000 1500 2000 2500 3000

MD (ft)

0

5

1

6

2

7

3

8

4

9

Oil

inflo

w (B

PDft

)


0 500 1000 1500 2000 2500 3000

MD (ft)

345

350

340

380

355

360

365

370

375

Tem

pera

ture

(K)




0 500 1000 1500 2000 2500 3000

MD (ft)

1500

1600

1700

1800

1900

2000

2100

2200

Well

bore

pre

ssur

e (ps

i)


0 500 1000 1500 2000 2500 3000

MD (ft)

0149

01495

015

01505

0151

Tem

pera

ture

(K)



5 Conclusions






Symbol Description







Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












206 (18)

where

1205741 = 75 minus (11081199010349

)

1205742 = 53 minus (010481199010637

)

(19)




119879119891 (119911) = 119866119879 (119911) minus119860 sin120572119869119862119898

+ 119860119865 + 119860119892119892 sin120572 + exp119864 (20)

where

119860 =Π119863119880120581

119872119872119862[120581 +

119863

2119862(log(

48radic119879119863119905

119863) minus 029)]

minus1

119864 = [(minus119911 minus 1199110

119860)

sdot (119879119891 (1199110) minus 119866119879 (1199110) +119860 sin120572119869119862119898

minus 119860119865 minus 119860119892119892 sin120572)]

(21)



119862119898 =sum119899

119894=1119872119894119862119894

sum119899

119894=1119872119894

(22)


119862 = 984858minus12

(0338 + 000045 (119879 minus 460)) (23)


119876 =000708119896ℎ (119901119903 minus 119901119908)

120583119861 [log (119903119890119903119908) minus 075 + 119878] (24)


119876 =000708119896ℎ (119875

2

119903minus 1198752

119908)

120583120573119879 [log (119903119890119903119908) minus 075 + 119878] (25)


3 The Framework




























Measureddepth

True verticaldepth

Inclination

Depth

Dormpressure

Orifice

P

T

Gauges

Wellhead

Bottom hole

Trajectory

GLV

GLV

Gas lift

Producer

Geology

Bore

Step 1

Density

Oil Water Gas

Layer

Layer

Casing

Casing

Tubing

Tubing

Insidediameter

diameterOutside

Roughness

Conductivity

Permeability

Pressure

Shape

Size

Position

Skin factor















ID












(119901 +119886

V2) (V minus 119887) = 119896119879 (26)











[120575 ln119909119894120575 (1119879)

]

119875

= minusΔ 119904119897119899ℎ119894

119877 (27)



ln119909119894 (119879) = ln119909119894 (119879119898) + (Δ 119904119897119899119862119875119894

119877)[ln( 119879

119879119898

) +119879119898

119879minus 1]

(28)








Listing 6 Continued






0 500 1000 1500 2000 2500 3000

MD (ft)

0

200

400

600

800

1000

1200

1400

1600

TVD

(ft)





0 500 1000 1500 2000 2500 3000

MD (ft)

345

350

355

360

365

370

375

Geo

ther

mal

tem

p (K

)


0

500

1000

1500

2000

2500

3000

3500

0 500 1000 1500 2000 2500 3000 3500

4000

4500

Inflo

w (B

PD)

MD (ft)


Figure 5 Inflow



0 500 1000 1500 2000 2500 3000

MD (ft)

0

05

1

15

2

25

3

35

4

45

Gas

inflo

w (s

cfD

ft)


0 500 1000 1500 2000 2500 3000

MD (ft)

0

5

1

6

2

7

3

8

4

9

Oil

inflo

w (B

PDft

)


0 500 1000 1500 2000 2500 3000

MD (ft)

345

350

340

380

355

360

365

370

375

Tem

pera

ture

(K)




0 500 1000 1500 2000 2500 3000

MD (ft)

1500

1600

1700

1800

1900

2000

2100

2200

Well

bore

pre

ssur

e (ps

i)


0 500 1000 1500 2000 2500 3000

MD (ft)

0149

01495

015

01505

0151

Tem

pera

ture

(K)



5 Conclusions






Symbol Description







Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra


































Measureddepth

True verticaldepth

Inclination

Depth

Dormpressure

Orifice

P

T

Gauges

Wellhead

Bottom hole

Trajectory

GLV

GLV

Gas lift

Producer

Geology

Bore

Step 1

Density

Oil Water Gas

Layer

Layer

Casing

Casing

Tubing

Tubing

Insidediameter

diameterOutside

Roughness

Conductivity

Permeability

Pressure

Shape

Size

Position

Skin factor















ID












(119901 +119886

V2) (V minus 119887) = 119896119879 (26)











[120575 ln119909119894120575 (1119879)

]

119875

= minusΔ 119904119897119899ℎ119894

119877 (27)



ln119909119894 (119879) = ln119909119894 (119879119898) + (Δ 119904119897119899119862119875119894

119877)[ln( 119879

119879119898

) +119879119898

119879minus 1]

(28)








Listing 6 Continued






0 500 1000 1500 2000 2500 3000

MD (ft)

0

200

400

600

800

1000

1200

1400

1600

TVD

(ft)





0 500 1000 1500 2000 2500 3000

MD (ft)

345

350

355

360

365

370

375

Geo

ther

mal

tem

p (K

)


0

500

1000

1500

2000

2500

3000

3500

0 500 1000 1500 2000 2500 3000 3500

4000

4500

Inflo

w (B

PD)

MD (ft)


Figure 5 Inflow



0 500 1000 1500 2000 2500 3000

MD (ft)

0

05

1

15

2

25

3

35

4

45

Gas

inflo

w (s

cfD

ft)


0 500 1000 1500 2000 2500 3000

MD (ft)

0

5

1

6

2

7

3

8

4

9

Oil

inflo

w (B

PDft

)


0 500 1000 1500 2000 2500 3000

MD (ft)

345

350

340

380

355

360

365

370

375

Tem

pera

ture

(K)




0 500 1000 1500 2000 2500 3000

MD (ft)

1500

1600

1700

1800

1900

2000

2100

2200

Well

bore

pre

ssur

e (ps

i)


0 500 1000 1500 2000 2500 3000

MD (ft)

0149

01495

015

01505

0151

Tem

pera

ture

(K)



5 Conclusions






Symbol Description







Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















(119901 +119886

V2) (V minus 119887) = 119896119879 (26)











[120575 ln119909119894120575 (1119879)

]

119875

= minusΔ 119904119897119899ℎ119894

119877 (27)



ln119909119894 (119879) = ln119909119894 (119879119898) + (Δ 119904119897119899119862119875119894

119877)[ln( 119879

119879119898

) +119879119898

119879minus 1]

(28)








Listing 6 Continued






0 500 1000 1500 2000 2500 3000

MD (ft)

0

200

400

600

800

1000

1200

1400

1600

TVD

(ft)





0 500 1000 1500 2000 2500 3000

MD (ft)

345

350

355

360

365

370

375

Geo

ther

mal

tem

p (K

)


0

500

1000

1500

2000

2500

3000

3500

0 500 1000 1500 2000 2500 3000 3500

4000

4500

Inflo

w (B

PD)

MD (ft)


Figure 5 Inflow



0 500 1000 1500 2000 2500 3000

MD (ft)

0

05

1

15

2

25

3

35

4

45

Gas

inflo

w (s

cfD

ft)


0 500 1000 1500 2000 2500 3000

MD (ft)

0

5

1

6

2

7

3

8

4

9

Oil

inflo

w (B

PDft

)


0 500 1000 1500 2000 2500 3000

MD (ft)

345

350

340

380

355

360

365

370

375

Tem

pera

ture

(K)




0 500 1000 1500 2000 2500 3000

MD (ft)

1500

1600

1700

1800

1900

2000

2100

2200

Well

bore

pre

ssur

e (ps

i)


0 500 1000 1500 2000 2500 3000

MD (ft)

0149

01495

015

01505

0151

Tem

pera

ture

(K)



5 Conclusions






Symbol Description







Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra















[120575 ln119909119894120575 (1119879)

]

119875

= minusΔ 119904119897119899ℎ119894

119877 (27)



ln119909119894 (119879) = ln119909119894 (119879119898) + (Δ 119904119897119899119862119875119894

119877)[ln( 119879

119879119898

) +119879119898

119879minus 1]

(28)








Listing 6 Continued






0 500 1000 1500 2000 2500 3000

MD (ft)

0

200

400

600

800

1000

1200

1400

1600

TVD

(ft)





0 500 1000 1500 2000 2500 3000

MD (ft)

345

350

355

360

365

370

375

Geo

ther

mal

tem

p (K

)


0

500

1000

1500

2000

2500

3000

3500

0 500 1000 1500 2000 2500 3000 3500

4000

4500

Inflo

w (B

PD)

MD (ft)


Figure 5 Inflow



0 500 1000 1500 2000 2500 3000

MD (ft)

0

05

1

15

2

25

3

35

4

45

Gas

inflo

w (s

cfD

ft)


0 500 1000 1500 2000 2500 3000

MD (ft)

0

5

1

6

2

7

3

8

4

9

Oil

inflo

w (B

PDft

)


0 500 1000 1500 2000 2500 3000

MD (ft)

345

350

340

380

355

360

365

370

375

Tem

pera

ture

(K)




0 500 1000 1500 2000 2500 3000

MD (ft)

1500

1600

1700

1800

1900

2000

2100

2200

Well

bore

pre

ssur

e (ps

i)


0 500 1000 1500 2000 2500 3000

MD (ft)

0149

01495

015

01505

0151

Tem

pera

ture

(K)



5 Conclusions






Symbol Description







Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Listing 6 Continued






0 500 1000 1500 2000 2500 3000

MD (ft)

0

200

400

600

800

1000

1200

1400

1600

TVD

(ft)





0 500 1000 1500 2000 2500 3000

MD (ft)

345

350

355

360

365

370

375

Geo

ther

mal

tem

p (K

)


0

500

1000

1500

2000

2500

3000

3500

0 500 1000 1500 2000 2500 3000 3500

4000

4500

Inflo

w (B

PD)

MD (ft)


Figure 5 Inflow



0 500 1000 1500 2000 2500 3000

MD (ft)

0

05

1

15

2

25

3

35

4

45

Gas

inflo

w (s

cfD

ft)


0 500 1000 1500 2000 2500 3000

MD (ft)

0

5

1

6

2

7

3

8

4

9

Oil

inflo

w (B

PDft

)


0 500 1000 1500 2000 2500 3000

MD (ft)

345

350

340

380

355

360

365

370

375

Tem

pera

ture

(K)




0 500 1000 1500 2000 2500 3000

MD (ft)

1500

1600

1700

1800

1900

2000

2100

2200

Well

bore

pre

ssur

e (ps

i)


0 500 1000 1500 2000 2500 3000

MD (ft)

0149

01495

015

01505

0151

Tem

pera

ture

(K)



5 Conclusions






Symbol Description







Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra














0 500 1000 1500 2000 2500 3000

MD (ft)

0

200

400

600

800

1000

1200

1400

1600

TVD

(ft)





0 500 1000 1500 2000 2500 3000

MD (ft)

345

350

355

360

365

370

375

Geo

ther

mal

tem

p (K

)


0

500

1000

1500

2000

2500

3000

3500

0 500 1000 1500 2000 2500 3000 3500

4000

4500

Inflo

w (B

PD)

MD (ft)


Figure 5 Inflow



0 500 1000 1500 2000 2500 3000

MD (ft)

0

05

1

15

2

25

3

35

4

45

Gas

inflo

w (s

cfD

ft)


0 500 1000 1500 2000 2500 3000

MD (ft)

0

5

1

6

2

7

3

8

4

9

Oil

inflo

w (B

PDft

)


0 500 1000 1500 2000 2500 3000

MD (ft)

345

350

340

380

355

360

365

370

375

Tem

pera

ture

(K)




0 500 1000 1500 2000 2500 3000

MD (ft)

1500

1600

1700

1800

1900

2000

2100

2200

Well

bore

pre

ssur

e (ps

i)


0 500 1000 1500 2000 2500 3000

MD (ft)

0149

01495

015

01505

0151

Tem

pera

ture

(K)



5 Conclusions






Symbol Description







Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 500 1000 1500 2000 2500 3000

MD (ft)

0

05

1

15

2

25

3

35

4

45

Gas

inflo

w (s

cfD

ft)


0 500 1000 1500 2000 2500 3000

MD (ft)

0

5

1

6

2

7

3

8

4

9

Oil

inflo

w (B

PDft

)


0 500 1000 1500 2000 2500 3000

MD (ft)

345

350

340

380

355

360

365

370

375

Tem

pera

ture

(K)




0 500 1000 1500 2000 2500 3000

MD (ft)

1500

1600

1700

1800

1900

2000

2100

2200

Well

bore

pre

ssur

e (ps

i)


0 500 1000 1500 2000 2500 3000

MD (ft)

0149

01495

015

01505

0151

Tem

pera

ture

(K)



5 Conclusions






Symbol Description







Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Symbol Description







Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article the framework dedicated to three phase

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