research article a direct eulerian grp scheme for the

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 171732 7 pageshttpdxdoiorg1011552013171732

Research ArticleA Direct Eulerian GRP Scheme for the Prediction of Gas-LiquidTwo-Phase Flow in HTHP Transient Wells

Jiuping Xu1 Min Luo12 Jiancheng Hu12 Shize Wang3 Bin Qi3 and Zhiguo Qiao3

1 Uncertainty Decision-Making Laboratory Sichuan University Chengdu 610064 China2 College of Mathematics Chengdu University of Information Technology Chengdu 610225 China3 Research School of Engineering Technology China Petroleum and Chemical Corporation Deyang 618000 China

Correspondence should be addressed to Jiuping Xu xujiupingscueducn

Received 10 September 2013 Accepted 17 October 2013

Academic Editor Mohamed Fathy El-Amin

Copyright copy 2013 Jiuping Xu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A coupled systemmodel of partial differential equations is presented in this paper which concerns the variation of the pressure andtemperature velocity and density at different times and depths in high temperature-high pressure (HTHP) gas-liquid two-phaseflowwells A new dimensional splitting technique with Eulerian generalized riemann problem (GRP) scheme is applied to solve thisset of conservation equations where Riemann invariants are introduced as the main ingredient to resolve the generalized RiemannproblemThe basic data of ldquoX wellrdquo (HTHPwell) 7100m deep located in Southwest China is used for the case history calculationsCurve graphs of pressures and temperatures along the depth of the well are plotted at different times The comparison with theresults of Lax Friedrichs (LxF) method shows that the calculating results are more fitting to the values of real measurement and thenew method is of high accuracy

1 Introduction

The prediction of pressure and temperature of transient gas-liquid flow in a wellbore is important but difficult for wellcompletion test because they are characterized by the depen-dence of pressure density velocity and other flowparameterson both time and space As for pressure prediction researchthere exist empirical formulas such as those given by Beggsand Brill [1] Mukherjee and Brill [2 3] and so on Differentresearchers such as Hurlburt and Hanratty [4] and Cazarez-Candia and Vasquez-Cruz [5] have proposed mechanisticmodels assuming that flow is under steady-state conditionsother researchers such as Taitel et al [6] Ouyang and Aziz[7] have proposed unsteady-state gas-liquid two-phase flowmodels Fontanilla and Aziz [8] and Ali [9] presented twosimultaneous ordinary differential equations for estimatingthe steam pressure and quality and solved these equationsby using the fourth-order Runge-Kutta method Howeverthose models can only predict the pressure profiles but notthe temperature profiles and ignored their interdependence

Concerning both pressure and temperature in HTHPwells Wu et al have presented a coupled system model of

differential equations in [10] but this model only consideredthe single phase flow statement In this paper we builda set of coupled partial differential equations of pressuretemperature density and velocity in HTHP gas-liquid two-phase flow wells on the base of the model which was build byXu et al in [11]The numerical model which accords with theactual situation of the well allows for the change of obliqueangledifferent heat transfermedium in annular and the depthof the physical properties of the formation

We found an algorithm solving model with generalizedRiemann problem (GRP) scheme which is an analyticextension of the Godunov scheme in [12] and originallydesigned by Li and Chen in [13] for the shallow waterequations A direct and simple derivation of the Euleriangeneralized Riemann problem scheme is presented to getthe integration in time of the conservation laws Riemanninvariants are applied in order to resolve the singularityat the jump discontinuity The approach has the advantagethat the contact discontinuity in each local wave pattern isalways fixed with speed zero while the rarefaction and theshock waves are located on either side Since the extensionof this scheme to multidimensional cases is obtained using

2 Abstract and Applied Analysis

Liquid slugLiquid film

Gas bubbleTGF

VGFTi

VLF TLS TGS

V1

Vs 120579

Figure 1 The two-phase flow

dS

dS v

dv

s

Figure 2 Control volume 1

the dimensional splitting technique getting the integrationin time of the conservation laws is more direct and simple

In this paper we use GRP method for solving thisproblem and get more accurate prediction of pressure andtemperature compared with those obtained from the existingcorrelations such as LxF method in [11] The basic data forthe calculation are from Xwell 7100m of depth in SouthwestChina The curves of the gas pressure and temperature alongthe depth of the well are plotted The results can provide atechnical reliance for the process of designing well tests inHTHP gas-liquid wells and a dynamic analysis of productionfrom wells

2 Model Formulation

Considering the two-phase flow system shown in Figure 1 themixture density and velocity are related to the in situ liquidvolume fraction (holdup)119867 as follows

120588119898= 120588119897119867 + 120588

119892(1 minus 119867)

V119898= V119897119867 + V

119892(1 minus 119867)

119906119898= 119906119897119867 + 119906

119892(1 minus 119867)

(1)

21 Mass Balance Consider the flow model shown inFigure 2 According to the fluid moves through the fixedcontrol volume depicted by John and Anderson in [14] wehave

∬119878

120588119898V119898119889119904 = minus

120597

120597119905∭

]120588119898119889V119898 (2)

Control volume

120588

P

dz

V

e

120588 + dp

e + de

A + dA

A

P + dP

V + dV


Under transient conditions applied to the control volumein Figure 3 in the limit as 119889119911 becomes very small the volumeand surface integral in (2) becomes

120597

120597119905∭

]120588119898119889V =

120597

120597119905(120588119898119860119889119911)

∬119878

120588119898V119898119889119904 = 120588

119898V119898119889V119898+ 120588119898119860119889V119898+ 119860V119898119889120588119898

= 119889 (120588119898V119898119860)

(3)

Substituting (3) into (2) we get the mass balance equa-tion

120597120588119898

120597119905+120597 (120588119898V119898)

120597119911= 0 (4)

22 Momentum Balance As shown in Figure 4 the integralform of the 119911 component the momentum equation can bewritten as follows with the external forces

120597

120597119905∭

]120588119898119906119898119889119881 +∬

119878

(120588119898119906119898V119898) sdot 119889119878

= minus∬119878

(119875119889119878) 119889119911 minus 120588119898119892 cos 120579119860119889119911 minus

120582120588119898V2119898

2119889119860119889119911

(5)

where 120588119898119892 cos 120579119860119889119911 is the force of gravity (120582120588

119898V21198602)119889119911 is

the shear stress and120597

120597119905∭

]120588119898119906119898119889119881 =

120597

120597119905(120588119898V119898119860119889119911)

∬119878

(120588119898119906119898V119898) sdot 119889119878 = minus 120588

119898V2119898119860

+ (120588119898+ 119889120588119898) (V119898+ 119889V119898)2

(119860 + 119889119860)

minus∬119878

(119875119889119878)119911= minus119875119860 + (119875 + 119889119875) (119860 + 119889119860) minus 2119875(

119889119860

2)

(6)

Abstract and Applied Analysis 3

Positive z directionz

PdS

PdS

minuspA (pdS)z =(pdS)z =

(pdS)z =

(P + dP)(A + dA)

120588gcos120579Adz

1205821205882

2dAdz

minuspdA

2

(pdS)z = minuspdA

2

dA

2

dA

2


Substituting (6) into (5) we obtain momentum balanceequation

120597

120597119905(120588119898V119898) +

120597

120597119911(119875 + 120588

119898V2119898) = minus120588

119898119892 cos 120579 minus

120582120588119898V2119898

2119889 (7)

23 Energy Balance For the transient flow it leads directlyto the energy equation in terms of temperature As shownin Figure 5 we should consider the heat transmission withinwellbore and from wellbore to formation as transient

According to the energy balance law the heat variationflowing on control volume that is equal to the combinationheat of inflow and outflow and the heat transferring to thesecond dimension we get the energy balance equation oftransient flow

(119908119862119901119879) 119911 minus (119908119862

119901119879) (119911 + 119889119911) minus 2120587119903

119905119900119880119905119900(119879 minus 119879

119903) 119889119911

=

120597 (120588119898119862119901119879)

120597119905119860119889119911

(8)

where 119879119903= (119870119890119879119890119894+ 119903119905119900119880119905119900119879119890119879wbD)(119870119890 + 119903119905119900119880119905119900119879wbD) and

119908 = 120588119898V119898119860 Equation (8) equals the following equation

120597 (120588119898V119898119879)

120597119911+120597 (120588119898119879)

120597119905=2120587119870119890119903119905119900119880119905119900(119879119890119894minus 119879)

119862119875(119870119890+ 119903119905119900119880119905119900119879wbD)

(9)

Finally we obtain the coupled system model of partialdifferential equations

120597120588119898

120597119905+120597 (120588119898V119898)

120597119911= 0

120597 (120588119898V119898)

120597119905+

120597 (119875 + 120588119898V2119898)

120597119911= minus120588119898119892 cos 120579 minus 120582

2119889

120597 (120588119898119879)

120597119905+120597 (120588119898V119898119879)

120597119911=2120587119870119890119903119905119900119880119905119900(119879119890119894minus 119879)

119862119875(119870119890+ 119903119905119900119880119905119900119879wbD)

120588119898=

119872119875120574119892

119885119877119879

(119875 119879 120588119898 V119898) = (119875

0 1198790 1205881198980 V1198980) 119905 = 0 119911 = 0

(10)

3 Format Construction

We unify the conservation equations (4) (7) and (9) whichare also included in (10) into the following formation

120597119860119898

120597119905+120597119861119898

120597119911= 119862119898 119898 = 1 2 3

119880 =

1198601= 120588119898

1198602= 120588119898V119898

1198603= 0

119865 (119880) =

1198611= 120588119898V119898

1198612= 119875 + 120588

119898V2119898

1198613= 120588119898V119898119879

119878 (119880) =

1198621= 0

1198622= minus120588119898119892 cos 120579 minus 120582

2119889

1198623=2120587119870119890119903119905119900119880119905119900(119879119890119894minus 119879)

119862119875(119870119890+ 119903119905119900119880119905119900119879wbD)

(11)

We define the equally spaced grid points the interfacepoints and the cells as

119911119895= 119895Δ119911 119911

119895+12=

119911119895+ 119911119895+1

2 119862

119895= [119911119895minus12

119911119895+12

]

(12)

We assume that the data at time 119905 = 119905119899are piecewise

linear with a slope 120590119899119895and we have 119880(119911 119905

119899) = 119880

119899

119895+ 120590119899

119895(119911 minus

119911119895) 119911 isin (119911

119895minus12 119911119895+12

)The second-order Godunov scheme for (11) takes the

following form119880119899+1119895

= 119880119899

119895minus(Δ119905Δ119909)(119865(119880

119899+12

119895+12)minus119865(119880

119899+12

119895minus12))

where 119880119899+12119895+12

is the midpoint value or the value of 119880 atthe cell interface (119911

119895+12 119905119899) with accuracy of second order

More specifically the mid-point value 119880119899+12119895+12

is computedwith the formulas 119880119899+1

119895+12= 119880119899

119895+12+ (Δ1199052) (120597119880120597119905)

119899

119895+12and

119880119899

119895+12= 119877119860

(0 119880119899

119895+12minus 119880119899

119895+12+) Also 119877119860 ((119911 minus 119911

119895+12)(119905 minus

119905119899) 119880119899

119895+12minus 119880119899

119895+12+) is the solution of the Riemann problem

centered at (119911119895+12

119905119899) Moreover 119880119899

119895+12minusand 119880119899

119895+12+are

the limiting values of initial data 119880(119911 119905119899) on both sides of

(119911119895+12

119905119899) We present a direct and simple derivation of

the Eulerian geralized Riemann problem (GRP) scheme andapply Riemann invariants in order to resolve the singularityat the jump discontinuity

The local wave configuration is usually piecewise smoothand consists of rarefactionwaves shocks and contact discon-tinuities As the general rarefaction waves are considered theinitial data can be regarded as a perturbation of the Riemanninitial data 119880

119871and 119880

119877


FormationFormationCasingCasing TubingTubing AnnulusAnnulus CementCement

rci

rc0

rcem

rti

rto

Figure 5 The radial transfer of heat

The GRP scheme assumes piecewise linear data for theflow variables which leads to the generalized Riemannproblem for (11) subject to the initial data

119880 (119911 0) =

119880119871+ 1199111198801015840

119871 119911 lt 0

119880119877+ 1199111198801015840

119877 119911 gt 0

(13)

where 119880119871 119880119877 1198801015840

119871 and 1198801015840

119877are constant vectors

The initial structure of the solution is determined by theassociated Riemann solution denoted by lim

119905rarr0119880(120582119905 119905) =

119877119860

(120582 119880119871 119880119877) 120582 = 119909119905

4 Solving Process

Step 1 Set the step length In this paper

ℎ = 1 (m) 120591 = 60 (s) (14)

Step 2 Obtain each pointrsquos inclination 120579119895= 120579119895minus1+ (1205791198951015840 minus

1205791198951015840minus1)ℎΔ119904

1198951015840minus1

Step 3 The in situ liquid volume fraction (holdup) in (1) canbe calculated from

119867(120579119895) =

098119864119897

04846

11986511990300868

times 1 + (1 minus 119864119897) ln[

4711987301244

V119897

119864119897

08692

11986511990305056]

times [sin (18120579119895) minus

1

3sin3 (18120579

119895)]

(15)

Step 4 Calculate the following parameters by Liao and Fengin [15]

119880119905119900= (

119903119905119900

119903119905119894ℎ119903

+119903119905119900ln (119903119905119900119903119905119894)

119903119905119894ℎ119903

+1

ℎ119888+ ℎ119903

+119903119905119900ln (1199031198880119903119888119894)

119903cas+119903119905119900ln (119903ℎ1199031198880)

119903cem)

minus1

119879wbD =2120587119870119890(119879119890119894minus 119879wb)

sum119898

119895=1(119876119895minus1minus 119876119895)

(16)

Step 5 For piecewise given initial data 119880119899(119911) = 119880119899119895+ 120590119899

119895(119911 minus

119911119895) 119911 isin (119911

119895minus12 119911119895+12

) we solve the Riemann problem for(11) to define the Riemann solution 119880119899

119895+12= 119877(0 119880

119899

119895+

(Δ1199112)120590119899

119895 119880119899

119895+1minus (Δ1199112)120590

119899

119895+1) which is the same as the

classical Godunov scheme and the Riemann solver in [16] isused in the solution

Step 6 Determine (120597119880120597119905)119899119895+12

and evaluate the new cellaverages 119880119899+1

119895 We apply monotonic algorithm slope limiters

to suppress the local oscillations near discontinuities Weuse parameter 120572 = 19 in 120590119899+1

119895= minmod (120572(119880119899+1

119895minus

119880119899+1

119895minus1)Δ119911 120590

119899+1minus

119895 120572(119880119899+1

119895+1minus 119880119899+1

119895)Δ119911) where 119880119899+1minus

119895+12=

119880119899

119895+12+ Δ119911(120597119880120597119911)

119899

119895+12and 120590119899+1minus

119895= (1Δ119911) (Δ119880)

119899+1minus

119895=

(1Δ119911) (119880119899+1minus

119895+12minus 119880119899+1minus

119895minus12)

5 Results and Discussion

In this simulation we study a pipe inXwell located in SichuanBasin Southwest China All the needed parameters are givenin [17] as follows fluid density is 1000 kgm3 depth of thewell is 7100m friction coefficient is 12 ground temperatureis 160∘C ground thermal conductivity parameter is 206ground temperature gradient is 00218∘Cm Parameters ofpipes are given in Table 1 Inclination azimuth and verticaldepth are given in Table 2

Through the simulation we use GRPmethod to calculatethe prediction of pressure and temperature of the oil in thepipe and draw a sensitive analysis for the resultsWe comparethe results of pressure and temperature calculated for the wellhead at 1200 s byGRP and LxF schemewith themeasurementresults which also shows that GRP scheme is more accurate


Table 1 Parameters of pipes

Diameter Thickness Weight Expansion Coefficient Young Modulus889 953 189 00000115 215 03 1400889 734 1518 00000115 215 03 750889 645 1369 00000115 215 03 420073 782 128 00000115 215 03 60073 551 952 00000115 215 03 150

Table 2 Parameters of azimuth inclination and vertical depth

Number Measured Inclination Azimuth Verticaldepth

1 0 0 12033 02 303 197 1212 302873 600 193 12028 599734 899 075 12657 898595 1206 125 1249 1205456 1505 104 12462 1504327 1800 049 12375 1799188 2105 249 12527 2104049 2401 127 12313 23999110 2669 244 12012 26677911 3021 014 12739 30196312 3299 118 12260 32975013 3605 205 12325 36033614 3901 016 12145 38992215 4183 292 12124 41810916 4492 273 12922 44899517 481607 198 12161 48138718 509907 274 12993 50967419 539407 013 12046 53916120 570607 063 12959 57034721 598307 209 12014 59803422 630207 269 12291 62991923 659707 245 12941 65940624 691112 015 12488 690796

in the real calculation We obtain series of results containedin tables and figures and analyze these results as follows

When the bottom pressure is 70MPa temperatures areplotted in Figure 6 at different depths and shown in detailin Table 3 When the output keeps constant the temperatureincreases with the increasing depth of the well and when thedepth fixed the temperature increases with the increasingtime In addition it can be seen from the figure that thetemperature changes quickly in the early stage but stabilizesover time especially after 1200 s

It is established that when depth is constant the pressureshown in Figure 7 and Table 4 increased with an increaseof the time When the output keeps constant the pressureincreased with the increasing depth of the well This isbecause with time increasing the flow increases and then the

2000 3000 4000Depth (M)

5000 6000 7000 8000100008090

100110120130140150160170180

Tem

pera

ture

(∘C)

300 s1200 s

900 s3600 s

Figure 6 Temperature distribution at different depths

20253035404550556065707580

Pres

sure

(Mpa

)

2000 3000 4000Depth (M)

5000 6000 7000 800010000

300 s1200 s

900 s3600 s

Figure 7 Pressure distribution at different depths

frictional heat leads to an increase in the pressure It can alsobe seen that the pressure changes quickly in the early stagebut stabilizes over time

As shown in Table 5 for the comparative results of thewell head temperature at 1200 s the relative error betweenthe calculation results and the measurement results of GRPscheme method is 512 and by LxF method is 670 whilethe relative error between the results in pressure predition at


Table 3 Temperature at different depths on 300 s 900 s 1200 s and3600 s

Depth Time300 s 900 s 1200 s 3600 s

0 8122 11529 12419 13227300 8545 12138 12755 13334600 9267 12554 13176 13656900 9554 12948 13496 138871200 10116 13377 13758 141941500 10649 13666 14077 143151800 11176 14067 14323 145172100 11698 14354 14629 147392400 12186 14566 14849 149142700 12689 14845 15078 151933000 13155 15178 15299 153853300 13685 15374 15486 155913600 14088 15402 15446 157873900 14467 15712 15873 159454200 14825 15934 16087 161884500 15274 16153 16165 162654800 15577 16255 16272 163455100 15975 16342 16349 164565400 16236 16456 16487 165245700 16432 16574 16545 166576000 16636 16656 16767 167976300 16791 16877 16887 169656600 16823 16945 16957 169816900 17024 17056 17178 17152

the same time calculated byGRP schememethod is 881 andby LxF method is 973 which shows that the distributionprediction of the two-phase flow is more accurate in actualcalculation by GRP scheme method

6 Conclusion

In this paper considering the variation of pressure tempera-ture velocity and density at different times and depths in gas-liquid two-phase flow we present a system model of partialdifferential equations according to mass momentum andenergy We establish an algorithm solving model with a newdifference method with a direct Eulerian GRP scheme whichis proven to be efficient for the numerical implementation inthis paper The basic data of the X well (HTHP well) 7100mdeep in Sichuan Basin Southwest China was used for casehistory calculations and a sensitivity analysis is completedfor the model The gas-liquidrsquos pressure and temperaturecurves along the depth of the well are plotted and thecurves intuitively reflect the flow law and the characteristicsof heat transfer in formation The results can provide thetechnical reliance for the process of designing well tests inhigh temperature-high pressure gas-liquid two-phase flowwells and dynamic analysis of production Furthermore theworks in this paper can raise safety and reliability of deepcompletion test and will yield notable economic and social

Table 4 Pressure at different depths on 300 s 900 s 1200 s and3600 s

Depth Time300 s 900 s 1200 s 3600 s

0 4255 4662 5024 5134300 4323 4753 5064 5267600 4486 4871 5046 5347900 4587 4913 5141 54691200 4673 5043 5232 54791500 4846 5124 5336 55531800 4943 5243 5447 56122100 5034 5383 5542 57372400 5196 5692 5437 57852700 5353 5722 5545 58973000 5444 5846 5678 59343300 5524 5997 5747 60953600 5676 5994 5895 61223900 5733 6098 5904 62294200 5893 6122 6029 63334500 5934 6245 6124 64484800 6089 6343 6223 64335100 6156 6419 6322 64785400 6335 6524 6418 65345700 6469 6545 6512 66346000 6545 6679 6615 67566300 6699 6749 6711 67476600 6746 6852 6822 68586900 6928 6946 6992 6955

Table 5 Comparative results of the well head at 1200 s

Well-head Temperature PressureMeasurement results 18065 7610Results by GRP method(relative error) 17178 (512) 6992 (881)

Results by LxF method(relative error) 16930 (670) 6936 (973)

benefits and avoid or lessen accidents caused by impropertechnical design

Nomenclature

119860 A total length of conduit (m2)119862119869 Joule-Thompson coefficient (Kpa)

119862119901 Heat capacity (JkgsdotK)

119863 A hydraulic diameter (m)119866 Acceleration constant of gravity (ms2)119870119890 Formation conductivity (JmsdotK)

119875 Pressure (KPa)119903119863 Dimensionless radius

119903119905119900 Outer radius of conduit (m)

119879 Temperature (K)119905119863 Dimensionless time


119879119890 Temperature of the stratum (K)

119879wb Wellbore temperature (K)119879wbD Dimensionless wellbore temperature (K)119879119903 Temperature of the second surface (K)

119879119890119894 Initial temperature of formation (K)

119880119905119900 Overall-heat-transfer coefficient (WmsdotK)

119881 Velocity (ms)119885 A total length of conduit (m)119911 The distance coordinate in the direction

along the conduitℎ119888 Heat transfer coefficient for natural

convection based on outside tubing surfaceand the temperature difference betweenoutside tubing and inside casing surface

ℎ119903 Heat transfer coefficient for radiation based

on the outside tubing surface and thetemperature difference between the outsidetubing and inside casing surface

119870cas Thermal conductivity of the casing materialat the average casing temperature

119870cem Thermal conductivity of the cement at theaverage cement temperature and pressure

120582 The friction coefficient dimensionless120574119892 Euler constant 1781

120588 Density (kgm3)120579 Inclination angle flow conduit

Conflict of Interests

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

Acknowledgments

This research was supported by the Key Program of NSFC(Grant no 70831005) and the Key Project of China Petroleumand Chemical Corporation (Grant no GJ-73-0706)

References

[1] HD Beggs and J R Brill ldquoA study of two-phase flow in inclinedpipesrdquo Journal of Petroleum Technology vol 25 pp 607ndash6171973

[2] H Mukherjee and J P Brill ldquoLiquid holdup correlations forinclined two-phase flowrdquo Journal of Petroleum Technology vol35 no 5 pp 1003ndash1008 1983

[3] H Mukherjee and J P Brill ldquoEmpirical equations to predictflow patterns in two-phase inclined flowrdquo International Journalof Multiphase Flow vol 11 no 3 pp 299ndash315 1985

[4] E T Hurlburt and T J Hanratty ldquoPrediction of the transitionfrom stratified to slug and plug flow for long pipesrdquo Interna-tional Journal of Multiphase Flow vol 28 no 5 pp 707ndash7292002

[5] O Cazarez-Candia and M A Vasquez-Cruz ldquoPrediction ofpressure temperature and velocity distribution of two-phaseflow in oil wellsrdquo Journal of Petroleum Science and Engineeringvol 46 no 3 pp 195ndash208 2005

[6] Y Taitel O Shoham and J P Brill ldquoSimplified transient solutionand simulation of two-phase flow in pipelinesrdquo ChemicalEngineering Science vol 44 no 6 pp 1353ndash1359 1989

[7] L-B Ouyang and K Aziz ldquoTransient gas-liquid two-phase flowin pipes with radial influx or effluxrdquo Journal of Petroleum Scienceand Engineering vol 30 no 3-4 pp 167ndash179 2001

[8] J P Fontanilla and K Aziz ldquoPrediction of bottom-hole con-ditions for wet steam injection wellsrdquo Journal of CanadianPetroleum Technology vol 21 no 2 8 pages 1982

[9] S M F Ali ldquoA comprehensive wellbore streamwater flowmodel for steam injection and geothermal applicationsrdquo Societyof Petroleum Engineers Journal vol 21 no 5 pp 527ndash534 1981

[10] Z Wu J Xu X Wang K Chen X Li and X Zhao ldquoPredictingtemperature and pressure in high-temperature-high-pressuregas wellsrdquo Petroleum Science and Technology vol 29 no 2 pp132ndash148 2011

[11] J Xu M Luo S Wang B Qi and Z Qiao ldquoPressure and tem-perature prediction of transient flow in HTHP injection wellsby Lax-Friedrichs methodrdquo Petroleum Science and Technologyvol 31 no 9 pp 960ndash976 2013

[12] S K Godunov ldquoA difference method for numerical calculationof discontinuous solutions of the equations of hydrodynamicsrdquoMatematicheskii Sbornik vol 47 no 89 pp 271ndash306 1959

[13] J Li and G Chen ldquoThe generalized Riemann problem methodfor the shallow water equations with bottom topographyrdquoInternational Journal for Numerical Methods in Engineering vol65 no 6 pp 834ndash862 2006

[14] D John and J R Anderson Computational Fluid DynamicsmdashTheBasics withApplicationsMcGraw-Hill NewYorkNYUSA1995

[15] X-W Liao and J-L Feng ldquoPressure-temperature coupling cal-culation of transientwellbore heat transfer in deep geopressuredgas reservoirrdquo Petroleum Exploration and Development vol 32no 1 pp 67ndash69 2005

[16] E F Toro Riemann Solvers and Numerical Methods for FluidDynamics A Practical Introduction Springer Berlin Germany1997

[17] J Xu J Hu M Luo S Wang B Qi and Z Qiao ldquoOptimisationof perforation distribution in HTHP vertical wellsrdquo CanadianJournal of Chemical Engineering vol 91 pp 332ndash343 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Liquid slugLiquid film

Gas bubbleTGF

VGFTi

VLF TLS TGS

V1

Vs 120579

Figure 1 The two-phase flow

dS

dS v

dv

s


the dimensional splitting technique getting the integrationin time of the conservation laws is more direct and simple

In this paper we use GRP method for solving thisproblem and get more accurate prediction of pressure andtemperature compared with those obtained from the existingcorrelations such as LxF method in [11] The basic data forthe calculation are from Xwell 7100m of depth in SouthwestChina The curves of the gas pressure and temperature alongthe depth of the well are plotted The results can provide atechnical reliance for the process of designing well tests inHTHP gas-liquid wells and a dynamic analysis of productionfrom wells

2 Model Formulation

Considering the two-phase flow system shown in Figure 1 themixture density and velocity are related to the in situ liquidvolume fraction (holdup)119867 as follows

120588119898= 120588119897119867 + 120588

119892(1 minus 119867)

V119898= V119897119867 + V

119892(1 minus 119867)

119906119898= 119906119897119867 + 119906

119892(1 minus 119867)

(1)

21 Mass Balance Consider the flow model shown inFigure 2 According to the fluid moves through the fixedcontrol volume depicted by John and Anderson in [14] wehave

∬119878

120588119898V119898119889119904 = minus

120597

120597119905∭

]120588119898119889V119898 (2)

Control volume

120588

P

dz

V

e

120588 + dp

e + de

A + dA

A

P + dP

V + dV


Under transient conditions applied to the control volumein Figure 3 in the limit as 119889119911 becomes very small the volumeand surface integral in (2) becomes

120597

120597119905∭

]120588119898119889V =

120597

120597119905(120588119898119860119889119911)

∬119878

120588119898V119898119889119904 = 120588

119898V119898119889V119898+ 120588119898119860119889V119898+ 119860V119898119889120588119898

= 119889 (120588119898V119898119860)

(3)

Substituting (3) into (2) we get the mass balance equa-tion

120597120588119898

120597119905+120597 (120588119898V119898)

120597119911= 0 (4)

22 Momentum Balance As shown in Figure 4 the integralform of the 119911 component the momentum equation can bewritten as follows with the external forces

120597

120597119905∭

]120588119898119906119898119889119881 +∬

119878

(120588119898119906119898V119898) sdot 119889119878

= minus∬119878

(119875119889119878) 119889119911 minus 120588119898119892 cos 120579119860119889119911 minus

120582120588119898V2119898

2119889119860119889119911

(5)

where 120588119898119892 cos 120579119860119889119911 is the force of gravity (120582120588

119898V21198602)119889119911 is

the shear stress and120597

120597119905∭

]120588119898119906119898119889119881 =

120597

120597119905(120588119898V119898119860119889119911)

∬119878

(120588119898119906119898V119898) sdot 119889119878 = minus 120588

119898V2119898119860

+ (120588119898+ 119889120588119898) (V119898+ 119889V119898)2

(119860 + 119889119860)

minus∬119878

(119875119889119878)119911= minus119875119860 + (119875 + 119889119875) (119860 + 119889119860) minus 2119875(

119889119860

2)

(6)



PdS

PdS


(pdS)z =

(P + dP)(A + dA)

120588gcos120579Adz

1205821205882

2dAdz

minuspdA

2

(pdS)z = minuspdA

2

dA

2

dA

2



120597

120597119905(120588119898V119898) +

120597

120597119911(119875 + 120588

119898V2119898) = minus120588

119898119892 cos 120579 minus

120582120588119898V2119898

2119889 (7)



(119908119862119901119879) 119911 minus (119908119862

119901119879) (119911 + 119889119911) minus 2120587119903

119905119900119880119905119900(119879 minus 119879

119903) 119889119911

=

120597 (120588119898119862119901119879)

120597119905119860119889119911

(8)

where 119879119903= (119870119890119879119890119894+ 119903119905119900119880119905119900119879119890119879wbD)(119870119890 + 119903119905119900119880119905119900119879wbD) and


120597 (120588119898V119898119879)

120597119911+120597 (120588119898119879)

120597119905=2120587119870119890119903119905119900119880119905119900(119879119890119894minus 119879)

119862119875(119870119890+ 119903119905119900119880119905119900119879wbD)

(9)


120597120588119898

120597119905+120597 (120588119898V119898)

120597119911= 0

120597 (120588119898V119898)

120597119905+

120597 (119875 + 120588119898V2119898)

120597119911= minus120588119898119892 cos 120579 minus 120582

2119889

120597 (120588119898119879)

120597119905+120597 (120588119898V119898119879)

120597119911=2120587119870119890119903119905119900119880119905119900(119879119890119894minus 119879)

119862119875(119870119890+ 119903119905119900119880119905119900119879wbD)

120588119898=

119872119875120574119892

119885119877119879

(119875 119879 120588119898 V119898) = (119875

0 1198790 1205881198980 V1198980) 119905 = 0 119911 = 0

(10)



120597119860119898

120597119905+120597119861119898

120597119911= 119862119898 119898 = 1 2 3

119880 =

1198601= 120588119898

1198602= 120588119898V119898

1198603= 0

119865 (119880) =

1198611= 120588119898V119898

1198612= 119875 + 120588

119898V2119898

1198613= 120588119898V119898119879

119878 (119880) =

1198621= 0

1198622= minus120588119898119892 cos 120579 minus 120582

2119889

1198623=2120587119870119890119903119905119900119880119905119900(119879119890119894minus 119879)

119862119875(119870119890+ 119903119905119900119880119905119900119879wbD)

(11)


119911119895= 119895Δ119911 119911

119895+12=

119911119895+ 119911119895+1

2 119862

119895= [119911119895minus12

119911119895+12

]

(12)



119899) = 119880

119899

119895+ 120590119899

119895(119911 minus

119911119895) 119911 isin (119911

119895minus12 119911119895+12


following form119880119899+1119895

= 119880119899

119895minus(Δ119905Δ119909)(119865(119880

119899+12

119895+12)minus119865(119880

119899+12

119895minus12))

where 119880119899+12119895+12





119895+12= 119880119899

119895+12+ (Δ1199052) (120597119880120597119905)

119899

119895+12and

119880119899

119895+12= 119877119860

(0 119880119899

119895+12minus 119880119899

119895+12+) Also 119877119860 ((119911 minus 119911

119895+12)(119905 minus

119905119899) 119880119899

119895+12minus 119880119899


centered at (119911119895+12

119905119899) Moreover 119880119899

119895+12minusand 119880119899

119895+12+are


(119911119895+12




119871and 119880

119877



rci

rc0

rcem

rti

rto



119880 (119911 0) =

119880119871+ 1199111198801015840

119871 119911 lt 0

119880119877+ 1199111198801015840

119877 119911 gt 0

(13)

where 119880119871 119880119877 1198801015840

119871 and 1198801015840



119905rarr0119880(120582119905 119905) =

119877119860

(120582 119880119871 119880119877) 120582 = 119909119905

4 Solving Process


ℎ = 1 (m) 120591 = 60 (s) (14)


1205791198951015840minus1)ℎΔ119904

1198951015840minus1


119867(120579119895) =

098119864119897

04846

11986511990300868

times 1 + (1 minus 119864119897) ln[

4711987301244

V119897

119864119897

08692

11986511990305056]

times [sin (18120579119895) minus

1

3sin3 (18120579

119895)]

(15)


119880119905119900= (

119903119905119900

119903119905119894ℎ119903

+119903119905119900ln (119903119905119900119903119905119894)

119903119905119894ℎ119903

+1

ℎ119888+ ℎ119903

+119903119905119900ln (1199031198880119903119888119894)

119903cas+119903119905119900ln (119903ℎ1199031198880)

119903cem)

minus1

119879wbD =2120587119870119890(119879119890119894minus 119879wb)

sum119898

119895=1(119876119895minus1minus 119876119895)

(16)


119895(119911 minus

119911119895) 119911 isin (119911

119895minus12 119911119895+12


119895+12= 119877(0 119880

119899

119895+

(Δ1199112)120590119899

119895 119880119899

119895+1minus (Δ1199112)120590

119899



Step 6 Determine (120597119880120597119905)119899119895+12




119895= minmod (120572(119880119899+1

119895minus

119880119899+1

119895minus1)Δ119911 120590

119899+1minus

119895 120572(119880119899+1

119895+1minus 119880119899+1

119895)Δ119911) where 119880119899+1minus

119895+12=

119880119899

119895+12+ Δ119911(120597119880120597119911)

119899

119895+12and 120590119899+1minus

119895= (1Δ119911) (Δ119880)

119899+1minus

119895=

(1Δ119911) (119880119899+1minus

119895+12minus 119880119899+1minus

119895minus12)









1 0 0 12033 02 303 197 1212 302873 600 193 12028 599734 899 075 12657 898595 1206 125 1249 1205456 1505 104 12462 1504327 1800 049 12375 1799188 2105 249 12527 2104049 2401 127 12313 23999110 2669 244 12012 26677911 3021 014 12739 30196312 3299 118 12260 32975013 3605 205 12325 36033614 3901 016 12145 38992215 4183 292 12124 41810916 4492 273 12922 44899517 481607 198 12161 48138718 509907 274 12993 50967419 539407 013 12046 53916120 570607 063 12959 57034721 598307 209 12014 59803422 630207 269 12291 62991923 659707 245 12941 65940624 691112 015 12488 690796




2000 3000 4000Depth (M)

5000 6000 7000 8000100008090

100110120130140150160170180

Tem

pera

ture

(∘C)

300 s1200 s

900 s3600 s


20253035404550556065707580

Pres

sure

(Mpa

)

2000 3000 4000Depth (M)

5000 6000 7000 800010000

300 s1200 s

900 s3600 s






Depth Time300 s 900 s 1200 s 3600 s

0 8122 11529 12419 13227300 8545 12138 12755 13334600 9267 12554 13176 13656900 9554 12948 13496 138871200 10116 13377 13758 141941500 10649 13666 14077 143151800 11176 14067 14323 145172100 11698 14354 14629 147392400 12186 14566 14849 149142700 12689 14845 15078 151933000 13155 15178 15299 153853300 13685 15374 15486 155913600 14088 15402 15446 157873900 14467 15712 15873 159454200 14825 15934 16087 161884500 15274 16153 16165 162654800 15577 16255 16272 163455100 15975 16342 16349 164565400 16236 16456 16487 165245700 16432 16574 16545 166576000 16636 16656 16767 167976300 16791 16877 16887 169656600 16823 16945 16957 169816900 17024 17056 17178 17152


6 Conclusion



Depth Time300 s 900 s 1200 s 3600 s

0 4255 4662 5024 5134300 4323 4753 5064 5267600 4486 4871 5046 5347900 4587 4913 5141 54691200 4673 5043 5232 54791500 4846 5124 5336 55531800 4943 5243 5447 56122100 5034 5383 5542 57372400 5196 5692 5437 57852700 5353 5722 5545 58973000 5444 5846 5678 59343300 5524 5997 5747 60953600 5676 5994 5895 61223900 5733 6098 5904 62294200 5893 6122 6029 63334500 5934 6245 6124 64484800 6089 6343 6223 64335100 6156 6419 6322 64785400 6335 6524 6418 65345700 6469 6545 6512 66346000 6545 6679 6615 67566300 6699 6749 6711 67476600 6746 6852 6822 68586900 6928 6946 6992 6955





Nomenclature























Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











PdS

PdS


(pdS)z =

(P + dP)(A + dA)

120588gcos120579Adz

1205821205882

2dAdz

minuspdA

2

(pdS)z = minuspdA

2

dA

2

dA

2



120597

120597119905(120588119898V119898) +

120597

120597119911(119875 + 120588

119898V2119898) = minus120588

119898119892 cos 120579 minus

120582120588119898V2119898

2119889 (7)



(119908119862119901119879) 119911 minus (119908119862

119901119879) (119911 + 119889119911) minus 2120587119903

119905119900119880119905119900(119879 minus 119879

119903) 119889119911

=

120597 (120588119898119862119901119879)

120597119905119860119889119911

(8)

where 119879119903= (119870119890119879119890119894+ 119903119905119900119880119905119900119879119890119879wbD)(119870119890 + 119903119905119900119880119905119900119879wbD) and


120597 (120588119898V119898119879)

120597119911+120597 (120588119898119879)

120597119905=2120587119870119890119903119905119900119880119905119900(119879119890119894minus 119879)

119862119875(119870119890+ 119903119905119900119880119905119900119879wbD)

(9)


120597120588119898

120597119905+120597 (120588119898V119898)

120597119911= 0

120597 (120588119898V119898)

120597119905+

120597 (119875 + 120588119898V2119898)

120597119911= minus120588119898119892 cos 120579 minus 120582

2119889

120597 (120588119898119879)

120597119905+120597 (120588119898V119898119879)

120597119911=2120587119870119890119903119905119900119880119905119900(119879119890119894minus 119879)

119862119875(119870119890+ 119903119905119900119880119905119900119879wbD)

120588119898=

119872119875120574119892

119885119877119879

(119875 119879 120588119898 V119898) = (119875

0 1198790 1205881198980 V1198980) 119905 = 0 119911 = 0

(10)



120597119860119898

120597119905+120597119861119898

120597119911= 119862119898 119898 = 1 2 3

119880 =

1198601= 120588119898

1198602= 120588119898V119898

1198603= 0

119865 (119880) =

1198611= 120588119898V119898

1198612= 119875 + 120588

119898V2119898

1198613= 120588119898V119898119879

119878 (119880) =

1198621= 0

1198622= minus120588119898119892 cos 120579 minus 120582

2119889

1198623=2120587119870119890119903119905119900119880119905119900(119879119890119894minus 119879)

119862119875(119870119890+ 119903119905119900119880119905119900119879wbD)

(11)


119911119895= 119895Δ119911 119911

119895+12=

119911119895+ 119911119895+1

2 119862

119895= [119911119895minus12

119911119895+12

]

(12)



119899) = 119880

119899

119895+ 120590119899

119895(119911 minus

119911119895) 119911 isin (119911

119895minus12 119911119895+12


following form119880119899+1119895

= 119880119899

119895minus(Δ119905Δ119909)(119865(119880

119899+12

119895+12)minus119865(119880

119899+12

119895minus12))

where 119880119899+12119895+12





119895+12= 119880119899

119895+12+ (Δ1199052) (120597119880120597119905)

119899

119895+12and

119880119899

119895+12= 119877119860

(0 119880119899

119895+12minus 119880119899

119895+12+) Also 119877119860 ((119911 minus 119911

119895+12)(119905 minus

119905119899) 119880119899

119895+12minus 119880119899


centered at (119911119895+12

119905119899) Moreover 119880119899

119895+12minusand 119880119899

119895+12+are


(119911119895+12




119871and 119880

119877



rci

rc0

rcem

rti

rto



119880 (119911 0) =

119880119871+ 1199111198801015840

119871 119911 lt 0

119880119877+ 1199111198801015840

119877 119911 gt 0

(13)

where 119880119871 119880119877 1198801015840

119871 and 1198801015840



119905rarr0119880(120582119905 119905) =

119877119860

(120582 119880119871 119880119877) 120582 = 119909119905

4 Solving Process


ℎ = 1 (m) 120591 = 60 (s) (14)


1205791198951015840minus1)ℎΔ119904

1198951015840minus1


119867(120579119895) =

098119864119897

04846

11986511990300868

times 1 + (1 minus 119864119897) ln[

4711987301244

V119897

119864119897

08692

11986511990305056]

times [sin (18120579119895) minus

1

3sin3 (18120579

119895)]

(15)


119880119905119900= (

119903119905119900

119903119905119894ℎ119903

+119903119905119900ln (119903119905119900119903119905119894)

119903119905119894ℎ119903

+1

ℎ119888+ ℎ119903

+119903119905119900ln (1199031198880119903119888119894)

119903cas+119903119905119900ln (119903ℎ1199031198880)

119903cem)

minus1

119879wbD =2120587119870119890(119879119890119894minus 119879wb)

sum119898

119895=1(119876119895minus1minus 119876119895)

(16)


119895(119911 minus

119911119895) 119911 isin (119911

119895minus12 119911119895+12


119895+12= 119877(0 119880

119899

119895+

(Δ1199112)120590119899

119895 119880119899

119895+1minus (Δ1199112)120590

119899



Step 6 Determine (120597119880120597119905)119899119895+12




119895= minmod (120572(119880119899+1

119895minus

119880119899+1

119895minus1)Δ119911 120590

119899+1minus

119895 120572(119880119899+1

119895+1minus 119880119899+1

119895)Δ119911) where 119880119899+1minus

119895+12=

119880119899

119895+12+ Δ119911(120597119880120597119911)

119899

119895+12and 120590119899+1minus

119895= (1Δ119911) (Δ119880)

119899+1minus

119895=

(1Δ119911) (119880119899+1minus

119895+12minus 119880119899+1minus

119895minus12)









1 0 0 12033 02 303 197 1212 302873 600 193 12028 599734 899 075 12657 898595 1206 125 1249 1205456 1505 104 12462 1504327 1800 049 12375 1799188 2105 249 12527 2104049 2401 127 12313 23999110 2669 244 12012 26677911 3021 014 12739 30196312 3299 118 12260 32975013 3605 205 12325 36033614 3901 016 12145 38992215 4183 292 12124 41810916 4492 273 12922 44899517 481607 198 12161 48138718 509907 274 12993 50967419 539407 013 12046 53916120 570607 063 12959 57034721 598307 209 12014 59803422 630207 269 12291 62991923 659707 245 12941 65940624 691112 015 12488 690796




2000 3000 4000Depth (M)

5000 6000 7000 8000100008090

100110120130140150160170180

Tem

pera

ture

(∘C)

300 s1200 s

900 s3600 s


20253035404550556065707580

Pres

sure

(Mpa

)

2000 3000 4000Depth (M)

5000 6000 7000 800010000

300 s1200 s

900 s3600 s






Depth Time300 s 900 s 1200 s 3600 s

0 8122 11529 12419 13227300 8545 12138 12755 13334600 9267 12554 13176 13656900 9554 12948 13496 138871200 10116 13377 13758 141941500 10649 13666 14077 143151800 11176 14067 14323 145172100 11698 14354 14629 147392400 12186 14566 14849 149142700 12689 14845 15078 151933000 13155 15178 15299 153853300 13685 15374 15486 155913600 14088 15402 15446 157873900 14467 15712 15873 159454200 14825 15934 16087 161884500 15274 16153 16165 162654800 15577 16255 16272 163455100 15975 16342 16349 164565400 16236 16456 16487 165245700 16432 16574 16545 166576000 16636 16656 16767 167976300 16791 16877 16887 169656600 16823 16945 16957 169816900 17024 17056 17178 17152


6 Conclusion



Depth Time300 s 900 s 1200 s 3600 s

0 4255 4662 5024 5134300 4323 4753 5064 5267600 4486 4871 5046 5347900 4587 4913 5141 54691200 4673 5043 5232 54791500 4846 5124 5336 55531800 4943 5243 5447 56122100 5034 5383 5542 57372400 5196 5692 5437 57852700 5353 5722 5545 58973000 5444 5846 5678 59343300 5524 5997 5747 60953600 5676 5994 5895 61223900 5733 6098 5904 62294200 5893 6122 6029 63334500 5934 6245 6124 64484800 6089 6343 6223 64335100 6156 6419 6322 64785400 6335 6524 6418 65345700 6469 6545 6512 66346000 6545 6679 6615 67566300 6699 6749 6711 67476600 6746 6852 6822 68586900 6928 6946 6992 6955





Nomenclature























Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











rci

rc0

rcem

rti

rto



119880 (119911 0) =

119880119871+ 1199111198801015840

119871 119911 lt 0

119880119877+ 1199111198801015840

119877 119911 gt 0

(13)

where 119880119871 119880119877 1198801015840

119871 and 1198801015840



119905rarr0119880(120582119905 119905) =

119877119860

(120582 119880119871 119880119877) 120582 = 119909119905

4 Solving Process


ℎ = 1 (m) 120591 = 60 (s) (14)


1205791198951015840minus1)ℎΔ119904

1198951015840minus1


119867(120579119895) =

098119864119897

04846

11986511990300868

times 1 + (1 minus 119864119897) ln[

4711987301244

V119897

119864119897

08692

11986511990305056]

times [sin (18120579119895) minus

1

3sin3 (18120579

119895)]

(15)


119880119905119900= (

119903119905119900

119903119905119894ℎ119903

+119903119905119900ln (119903119905119900119903119905119894)

119903119905119894ℎ119903

+1

ℎ119888+ ℎ119903

+119903119905119900ln (1199031198880119903119888119894)

119903cas+119903119905119900ln (119903ℎ1199031198880)

119903cem)

minus1

119879wbD =2120587119870119890(119879119890119894minus 119879wb)

sum119898

119895=1(119876119895minus1minus 119876119895)

(16)


119895(119911 minus

119911119895) 119911 isin (119911

119895minus12 119911119895+12


119895+12= 119877(0 119880

119899

119895+

(Δ1199112)120590119899

119895 119880119899

119895+1minus (Δ1199112)120590

119899



Step 6 Determine (120597119880120597119905)119899119895+12




119895= minmod (120572(119880119899+1

119895minus

119880119899+1

119895minus1)Δ119911 120590

119899+1minus

119895 120572(119880119899+1

119895+1minus 119880119899+1

119895)Δ119911) where 119880119899+1minus

119895+12=

119880119899

119895+12+ Δ119911(120597119880120597119911)

119899

119895+12and 120590119899+1minus

119895= (1Δ119911) (Δ119880)

119899+1minus

119895=

(1Δ119911) (119880119899+1minus

119895+12minus 119880119899+1minus

119895minus12)









1 0 0 12033 02 303 197 1212 302873 600 193 12028 599734 899 075 12657 898595 1206 125 1249 1205456 1505 104 12462 1504327 1800 049 12375 1799188 2105 249 12527 2104049 2401 127 12313 23999110 2669 244 12012 26677911 3021 014 12739 30196312 3299 118 12260 32975013 3605 205 12325 36033614 3901 016 12145 38992215 4183 292 12124 41810916 4492 273 12922 44899517 481607 198 12161 48138718 509907 274 12993 50967419 539407 013 12046 53916120 570607 063 12959 57034721 598307 209 12014 59803422 630207 269 12291 62991923 659707 245 12941 65940624 691112 015 12488 690796




2000 3000 4000Depth (M)

5000 6000 7000 8000100008090

100110120130140150160170180

Tem

pera

ture

(∘C)

300 s1200 s

900 s3600 s


20253035404550556065707580

Pres

sure

(Mpa

)

2000 3000 4000Depth (M)

5000 6000 7000 800010000

300 s1200 s

900 s3600 s






Depth Time300 s 900 s 1200 s 3600 s

0 8122 11529 12419 13227300 8545 12138 12755 13334600 9267 12554 13176 13656900 9554 12948 13496 138871200 10116 13377 13758 141941500 10649 13666 14077 143151800 11176 14067 14323 145172100 11698 14354 14629 147392400 12186 14566 14849 149142700 12689 14845 15078 151933000 13155 15178 15299 153853300 13685 15374 15486 155913600 14088 15402 15446 157873900 14467 15712 15873 159454200 14825 15934 16087 161884500 15274 16153 16165 162654800 15577 16255 16272 163455100 15975 16342 16349 164565400 16236 16456 16487 165245700 16432 16574 16545 166576000 16636 16656 16767 167976300 16791 16877 16887 169656600 16823 16945 16957 169816900 17024 17056 17178 17152


6 Conclusion



Depth Time300 s 900 s 1200 s 3600 s

0 4255 4662 5024 5134300 4323 4753 5064 5267600 4486 4871 5046 5347900 4587 4913 5141 54691200 4673 5043 5232 54791500 4846 5124 5336 55531800 4943 5243 5447 56122100 5034 5383 5542 57372400 5196 5692 5437 57852700 5353 5722 5545 58973000 5444 5846 5678 59343300 5524 5997 5747 60953600 5676 5994 5895 61223900 5733 6098 5904 62294200 5893 6122 6029 63334500 5934 6245 6124 64484800 6089 6343 6223 64335100 6156 6419 6322 64785400 6335 6524 6418 65345700 6469 6545 6512 66346000 6545 6679 6615 67566300 6699 6749 6711 67476600 6746 6852 6822 68586900 6928 6946 6992 6955





Nomenclature























Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra














1 0 0 12033 02 303 197 1212 302873 600 193 12028 599734 899 075 12657 898595 1206 125 1249 1205456 1505 104 12462 1504327 1800 049 12375 1799188 2105 249 12527 2104049 2401 127 12313 23999110 2669 244 12012 26677911 3021 014 12739 30196312 3299 118 12260 32975013 3605 205 12325 36033614 3901 016 12145 38992215 4183 292 12124 41810916 4492 273 12922 44899517 481607 198 12161 48138718 509907 274 12993 50967419 539407 013 12046 53916120 570607 063 12959 57034721 598307 209 12014 59803422 630207 269 12291 62991923 659707 245 12941 65940624 691112 015 12488 690796




2000 3000 4000Depth (M)

5000 6000 7000 8000100008090

100110120130140150160170180

Tem

pera

ture

(∘C)

300 s1200 s

900 s3600 s


20253035404550556065707580

Pres

sure

(Mpa

)

2000 3000 4000Depth (M)

5000 6000 7000 800010000

300 s1200 s

900 s3600 s






Depth Time300 s 900 s 1200 s 3600 s

0 8122 11529 12419 13227300 8545 12138 12755 13334600 9267 12554 13176 13656900 9554 12948 13496 138871200 10116 13377 13758 141941500 10649 13666 14077 143151800 11176 14067 14323 145172100 11698 14354 14629 147392400 12186 14566 14849 149142700 12689 14845 15078 151933000 13155 15178 15299 153853300 13685 15374 15486 155913600 14088 15402 15446 157873900 14467 15712 15873 159454200 14825 15934 16087 161884500 15274 16153 16165 162654800 15577 16255 16272 163455100 15975 16342 16349 164565400 16236 16456 16487 165245700 16432 16574 16545 166576000 16636 16656 16767 167976300 16791 16877 16887 169656600 16823 16945 16957 169816900 17024 17056 17178 17152


6 Conclusion



Depth Time300 s 900 s 1200 s 3600 s

0 4255 4662 5024 5134300 4323 4753 5064 5267600 4486 4871 5046 5347900 4587 4913 5141 54691200 4673 5043 5232 54791500 4846 5124 5336 55531800 4943 5243 5447 56122100 5034 5383 5542 57372400 5196 5692 5437 57852700 5353 5722 5545 58973000 5444 5846 5678 59343300 5524 5997 5747 60953600 5676 5994 5895 61223900 5733 6098 5904 62294200 5893 6122 6029 63334500 5934 6245 6124 64484800 6089 6343 6223 64335100 6156 6419 6322 64785400 6335 6524 6418 65345700 6469 6545 6512 66346000 6545 6679 6615 67566300 6699 6749 6711 67476600 6746 6852 6822 68586900 6928 6946 6992 6955





Nomenclature























Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Depth Time300 s 900 s 1200 s 3600 s

0 8122 11529 12419 13227300 8545 12138 12755 13334600 9267 12554 13176 13656900 9554 12948 13496 138871200 10116 13377 13758 141941500 10649 13666 14077 143151800 11176 14067 14323 145172100 11698 14354 14629 147392400 12186 14566 14849 149142700 12689 14845 15078 151933000 13155 15178 15299 153853300 13685 15374 15486 155913600 14088 15402 15446 157873900 14467 15712 15873 159454200 14825 15934 16087 161884500 15274 16153 16165 162654800 15577 16255 16272 163455100 15975 16342 16349 164565400 16236 16456 16487 165245700 16432 16574 16545 166576000 16636 16656 16767 167976300 16791 16877 16887 169656600 16823 16945 16957 169816900 17024 17056 17178 17152


6 Conclusion



Depth Time300 s 900 s 1200 s 3600 s

0 4255 4662 5024 5134300 4323 4753 5064 5267600 4486 4871 5046 5347900 4587 4913 5141 54691200 4673 5043 5232 54791500 4846 5124 5336 55531800 4943 5243 5447 56122100 5034 5383 5542 57372400 5196 5692 5437 57852700 5353 5722 5545 58973000 5444 5846 5678 59343300 5524 5997 5747 60953600 5676 5994 5895 61223900 5733 6098 5904 62294200 5893 6122 6029 63334500 5934 6245 6124 64484800 6089 6343 6223 64335100 6156 6419 6322 64785400 6335 6524 6418 65345700 6469 6545 6512 66346000 6545 6679 6615 67566300 6699 6749 6711 67476600 6746 6852 6822 68586900 6928 6946 6992 6955





Nomenclature























Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra

























Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article a direct eulerian grp scheme for the

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