Journal of Petroleum Science and Engineering 71 (2010) 77–86

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Journal of Petroleum Science and Engineering

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Research paper

Modeling two-phase fluid and heat flows in geothermal wells

A.R. Hasan a, C.S. Kabir b,⁎a Chemical Engineering Department, University of Minnesota-Duluth, Duluth, MN 55438, USAb Hess Corporation, One Allen Center, 500 Dallas Street, Houston, TX 77002, USA

⁎ Corresponding author.E-mail addresses: rhasan@d.umn.edu (A.R. Hasan), s

0920-4105/$ – see front matter © 2010 Elsevier B.V. Aldoi:10.1016/j.petrol.2010.01.008

a b s t r a c t

a r t i c l e i n f o

Article history:Received 5 August 2009Accepted 11 January 2010

Keywords:geothermal wellsfluid and heat flowsmodeling pressure-dropstwo-phase flow in wells

This study presents a robust model for a two-phase flow in geothermal wells using the drift-flux approach.For estimating the static head, we use a single expression for liquid holdup, with flow-pattern-dependentvalues for flow parameter and rise velocity that gradually changes near the transition boundaries to avoiddiscontinuity in the estimated gradients. Frictional and kinetic heads are estimated with the simplehomogeneous modeling approach.As the geothermal fluids ascend up the well, loss of both momentum and heat occurs. The consequentpressure loss often leads to flashing and increase in steam fraction (quality) despite heat loss. Accurateestimation of heat loss, which leads to significant changes in fluid properties influencing pressure-drop, is,therefore, important in modeling flow in geothermal wells. Heat transfer from the wellbore fluid to thesurrounding formation is rigorously modeled by treating the wellbore as a heat sink of finite radius in aninfinite-acting medium (formation) and accounting for the resistances to heat transfer by various elementsof the wellbore.We present a comparative study involving the new model and those that are often used for geothermalwells. These models include those of Ansari et al.'s, Orkiszewski's, Hagedorn and Brown's, and thehomogeneous model. The main ingredient of this study entails the use of a small but reliable dataset.Statistical analyses suggest that all the models behave similarly, although the proposed model offersmarginally greater accuracy and simplicity of use. Uncertainty of performance appears to depend upon thequality of data input, rather than the model characteristics.

kabir@hess.com (C.S. Kabir).

l rights reserved.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

Historically, pressure-traverse computation in geothermal well-bores followed a trend similar to that in wells producing hydro-carbons. This similarity is appreciated by noting that all the principalflow regimes, bubbly, slug, churn, and annular, are common to bothsystems. Some of the authors of early studies computed the slipbetween steam and water phases with a different model in each flowregime, which we term as the hybrid approach. One potentialdifficulty of this approach is that the transition between the flowregimes may not be smooth, thereby triggering discontinuity. Theearly studies of Gould (1974), Chierici et al. (1981), Ambastha andGudmundsson (1986a,b), and Chadha et al. (1993), among others, fallinto the hybrid-approach category. A few authors, such as Upadhyayet al. (1977), used different models in their entirety to find the mostreliable one. Orkiszewski's (1967) correlation appeared to have anedge in their study. In a more recent study, Acuña and Arcedera(2005) expressed a similar sentiment while discussing one fieldexample. However, this view is not universal as Tanaka and Nishi

(1988) showed in a study with 16 wells, containing varying amountsof CO2.

Despite reporting comparative studies, most authors dealt with avery few wells, thereby posing an important question: is the numberof field tests sufficient to draw statistically significant conclusionsregarding superiority of one model over any other? To compound theproblem, the dataset was incomplete; that is, both bottomholetemperature (BHT) and geothermal gradient often went unreported.Any comparative study involving models' relative performance is akinto the one used extensively in the petroleum literature. However, thenumber of wells in the petroleum industry dwarfs those that areavailable in geothermal production, thereby posing challenges inidentifying reliable models for pressure-traverse computation ingeothermal wells.

More recently, Garg et al. (2004a,b) reported successful applica-tion of a method that is based largely on Duns and Ros (1963) flow-pattern map and Hughmark's (1963) correlation for pressure-dropcalculations, with parameters optimized to best represent the fielddata. Their dataset (2004b) appears complete, unlike those presentedearlier; Gould (1974), for example. As alluded to earlier, absence ofbottomhole temperature and the geothermal gradientmake any heat-transfer computation a very challenging proposition. While Garg et al.model most two-phase flow based largely on previous models meant

78 A.R. Hasan, C.S. Kabir / Journal of Petroleum Science and Engineering 71 (2010) 77–86

for oil wells, they modeled heat transfer with a constant overall-heat-transfer coefficient (Ut). The underlying heat-transfer model owes toRamey's (1962) pioneering work. Our study shows that improvedheat-transfer modeling occurs with the variable-Ut approach, com-mensurate with variable-tubular diameter. Overall, statistical analy-ses of a reliable dataset comprising 36 wells show that no modelemerged as a clear winner, thereby suggesting that deficiency in dataquality cannot be made up by a model's sophistication.

2. Mathematical model

2.1. Momentum transport

The total pressure gradient during fluid flow is the sum of thestatic, friction, and kinetic gradients, which is written as

− dpdz

= gρm sin θ +f v2m ρm2d

+ ρmvgdvmdz

: ð1Þ

For most geothermal wells, which are vertical or near-vertical, thestatic-head component dominates. Because the static head dependsdirectly on the volume average-mixture density, for geothermal wells,two-phase flowmodeling boils down to estimating density of the fluidmixture or steam-volume fraction.

We individually model each flow regime because the in-situ steam-volume fraction, fg, depends on the flow pattern; that is, bubbly, slug,churn, and annular.We use a simple, mechanistic model for delineatingflow regimes as explained in Appendix A. Note that the drift-fluxapproach has also been used successfully by Shi et al. (2005a,b) formodeling two- and three-phase flows involving hydrocarbons.

For all flow regimes the gas phase or steam moves faster than theliquid because of its buoyancy and its tendency to flow close to thechannel center. This flow behavior allows us to express the in-situsteam velocity as the sum of bubble-rise velocity, v∞, and the channel-center mixture velocity, Covm. However, in-situ velocity is the ratio ofsuperficial velocity to volume fraction. Therefore, the generalizedform of steam-volume fraction relationshipwithmeasured variables –superficial velocity of steam and liquid phases – can be written as(Hasan et al., 2007)

fg =vsg

Covm + v∞: ð2Þ

While Eq. (1) is general in its application, the values of the flowparameter Co, and rise velocity v∞, are dependent on the type of flowand flow pattern. Table 1 presents these values for cocurrent upflow.In the following, we briefly explain the rationale for these values.

In turbulent flow, prevalent in most geothermal wells, the velocityprofile across the channel is relatively flat and the velocity at thecenter is about 1.2 times the average-mixture velocity. In bubbly andslug flows, most of the bubbles move through this central portion,which explains the value of 1.2 for Co in these two flow regimes. Inchurn flow, the high mixture velocity that breaks up the large bubblesalso circulates the steam more thoroughly across the channel,resulting in a slightly lower value of Co. This is also the reason forusing Co=1.15 for dispersed bubbly flow. Our recent work strongly

Table 1Parameters for flow type and flow pattern.

Flow pattern Flow parameter Co Rise velocity v∞

Bubbly 1.2 v∞bSlug 1.2 v–∞Churn 1.15 v–∞Dispersed bubbly 1.15 v∞bAnnular 1.0 0

suggests no slippage between the two phases in annular flow regime;consequently, Co=1 and v∞=0 in this flow regime (Kabir and Hasan,2006; Hasan and Kabir, 2007) are well justified.

For rise velocity of small bubbles in bubbly flow, v∞b, we use theHarmathy (1960) correlation, which applies even to liquid/liquid flow

v∞b = 1:53 g ρL−ρg� �

σ =ρL2h i1

=4 : ð3Þ

The Taylor bubble-rise velocity, v∞T, has been investigated bymany researchers. We use the following relation for its estimation invertical cylindrical channels as detailed in Appendix A:

v∞T = 0:35ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigD ρl−ρg

� �= ρL

r: ð4Þ

Appendix A presents this model in details, including the procedurefor calculating the average-rise velocity, v ̄∞, of the small and Taylorbubble in slug and churn flows.

Overall, this model differs from previous studies in that (i) we showunified formulation of the model in various flow types, (ii) transitionfrom one flow regime into another is smoothed by a weighting scheme,and (iii) annular flow is modeled with the no-slip approach.

2.2. Energy transport

Production or injection of steam–watermixtures inevitably involvessignificant heat exchange between the wellbore fluid and its surround-ings. During production, the hot fluid continues to lose heat to theincreasingly cold surroundings, on its ascent up the wellbore. In con-trast, the injected fluid may either gain (cold water) or lose (steam orhot water) heat upon its descent.

Wellbore heat transfer impacts fluid properties and, in turn, thedynamics offluidflow. Consequently, the coupled nature ofmomentumand energy transport may require simultaneous solutions for bothprocesses. Significant research efforts have been spent on modelingwellbore heat transfer in the last few decades, Hasan et al. (2009) beingthemost recent example.A general energybalance for thewellborefluidgives

dhdz

− g sinαJgc

+vJgc

dvdz

= ∓Qw: ð5Þ

In Eq. (5), Q represents the rate of heat transfer between the fluidand the surrounding formation per unit depth of the wellbore; thenegative sign applies to fluid production and the positive sign appliesto fluid injection. Heat loss from the fluid to the formation is modeledin two steps: that of loss from fluid to the wellbore/formationinterface and from this interface to the formation. The details of theenergy transport model are in Appendix B. Briefly,

Q≡−LRwcp Tf−Tei� �

: ð6Þ

For a wellbore surrounded by earth, LR is given by

LR≡2πcPw

rtoUtokeke + rtoUtoTDð Þ

� �: ð7Þ

For oil and/or gas production when the fluid undergoes nosignificant phase change, enthalpy is expressed as a function ofpressure and temperature and Eqs. (4)–(6) are solved to obtain anexpression for fluid temperature as a function of well depth. However,for geothermal wells where water evaporates, application of such aprocedure becomes infeasible. Consequently, Eq. (4) must bediscretized and solved numerically with a known boundary condition.In the simplest case, when pressure and enthalpy are known at a

Table 3Correlation with measured wellbore pressure profile, Garg et al. data.

Orki H–B Homo Ansari H–K

r 0.9879 0.9889 0.9875 0.9714 0.9848t-test 150.712 157.363 148.438 96.817 134.305

79A.R. Hasan, C.S. Kabir / Journal of Petroleum Science and Engineering 71 (2010) 77–86

position, Eqs. (1) and (5) are solved numerically to obtain pressureand enthalpy of the fluid in the next position in the well. Fluidenthalpy and pressure are then used to estimate steam mass fractionor quality, x, with the following expression:

x =h−hsatL

hLg: ð8Þ

In Eq. (8), Hlsat and Hlg are the liquid saturation enthalpy and heat

of vaporization, respectively, and are depended on fluid pressure.Therefore, the computed quality at the new location depends on thefluid pressure at that location. However, as Eq. (1) shows, fluidpressure, in turn, depends on vapor velocity and, hence, the quality.This coupled nature of the transport processes demands a robustiterative-solution technique.

3. Analysis of field data

Most studies in the past provided a very limited dataset and oftenwith incomplete information. Absence of both the geothermalgradient and reservoir temperature falls into this category; thismissing information becomes a serious impediment to performing anobjective analysis. In this regard, the data provided by Garg et al.(2004b) is exemplary. We used their quality data from 36 wells forthis study.

3.1. Pressure-drop and pressure-traverse calculations

Two approaches to the statistical analysis with five independenttwo-phase flow models formed the cornerstone of this investigation.These models included those of Orkiszewski (1967), Hagedorn andBrown or H–B (1965), homogeneous, Ansari et al. (1994), and our own(H–K), as presented here. For comparison, we also included the resultsof Garg et al. (2004a,b) in a limited form. The two statistical approachesinvolved the total pressure-drop consideration (pwf−pwh) and eachmeasured pressure point or the pressure profile in the wellbore.Measured pressure profiles enriched this dataset immensely and wetreated each depth-dependent value as an independent measurement.

Table 2 summarizes the statistical results involving the totalwellbore pressure-drop. Garg et al.'s results are based on their reportedpwh values for their best fit correlation; no attempt was made toreproduce their results. AsTable 2 shows, theGarg et al. results appear tobe an improvement over others in some statistical measures. However,we note that their empirical model reflects a fit of certain parameterswith this dataset. Theyalsoused a roughness ratio, ε/d, of zero, assumingthe geothermal wells that are subject to scaling, to be smooth.

InTable 2 theaveragepercent error reflects the trend inperformancerelative to the measurement, the absolute average error indicates howlarge the errors are, the percent standard deviation suggests scatteringof errors about their average value, the average error is a measure ofoverall trend independent of measurement, the average absolute errormirrors the magnitude of average error, and the standard deviation issimply scattering of results, independent of measured pressure-drop.

Following the analysis with the traditional measured overallpressure-drops, we embarked on analyzing the flow profiles. Each

Table 2Statistical analysis of wellbore total pressure-drop, ε=2.5×10−5, Garg et al. data.

Orki H–B Homo Ansari H–K Garg

Avg. %error −6.983 −0.710 −12.852 −12.301 −1.294 −1.437Abs. avg. %error 16.119 13.007 15.079 19.846 10.175 2.343%Standard deviation 42.067 4.280 77.423 74.107 7.793 1.463Avg. error −0.318 0.036 −0.966 −0.865 0.185 −0.155Abs. avg. error 1.687 1.389 1.464 2.356 1.245 0.297Standard deviation 10.630 8.427 7.615 12.878 7.513 1.916

well had about 17 independent measured pressure points along thewellbore, digitized from the charts. Our analysis omitted the knownbottom most measurement, simply because that happens to be thestarting point for the bottom-up computational approach. Overall, weended up having 576 data points for the 36 wells analyzed.

Table 3 showing correlation coefficients of 576 test points suggeststhat all the models are highly correlated with data. In fact, thesignificance of the correlation coefficient, r, is presented in terms ofthe t values, which is given by the following expression (Blalock,1972, p. 400):

t = rffiffiffiffiffiffiffiffiffiffiffiffiffiffin−2ð Þ

pn o=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1−r2� �q

ð9Þ

where n indicates the number of data points. We point out that for anyof themodels to assure existence of correlationwith data, we needed a tvalue of only 2.6 within the 99% confidence interval. Obviously, eachmodel met this criterion with ease. As with the previous analysis, noclear performance winner is apparent. No measures of the Garg et al.model are presented because their model was not part of thisevaluation.

Table 4 reports the repeat of some of the statistical measures,presented earlier in Table 2, with the 576 data points. Here, pressuredifference at each depth is simply the difference of the measured valueand the corresponding model-generated value. We think this largerdataset is a better indicator of amodel's performance because the entirepressure traverse is treated rather than the overall-pressure-drop itselfin each well. The inherent advantage of comparing and contrasting theentire pressure profile is that the heat-transfer considerations areimplicitly accounted for. As indicated elsewhere, we think that thevariable-U modeling, as espoused here, is superior to the constant-Uapproach used in the past.

Figs. 1–3 compare the results of the five models with field data.Good correlation of all models with field data is readily apparent. Thesuccess of the homogeneous model should not be too surprisingbecause many wells exhibited annular flow. As we showed recently(Hasan and Kabir, 2007), the homogeneous model is quite appropri-ate for annular flow. Overall, the analyses lead to the inevitableconclusion; that is, all the models behave very similarly and nonereally stand out; this outcome is not surprising given the data quality.Our recent experiences with oil wells (Hasan et al., 2007, 2009)suggest that modeling two-phase flow and the attendant heat transferprobably boils down to using a reasonable approach and payingattention to the dataset. Put another way, lack of data quality cannotbe made up by a model, regardless of its sophistication.

3.2. Heat-transfer calculations

As stated earlier, heat-transfer calculations are implicit in anypressure-traverse algorithm. In particular, such computation becomesmore important in a geothermal well because of much higher energydissipation than its oil counterpart. Most studies in the past reported

Table 4Statistical analysis of measured wellbore pressure profile, Garg et al. data.

Orki H–B Homo Ansari H–K

Avg. error −0.624 −0.274 −0.819 −0.736 −0.091Abs. avg. error 1.041 0.873 1.110 1.528 0.927Standard deviation 22.093 20.656 22.091 33.273 21.595

Fig. 1. Model pressures (H–K and Orkiszewski) correlate well with field data.

80 A.R. Hasan, C.S. Kabir / Journal of Petroleum Science and Engineering 71 (2010) 77–86

the use of constant-Ut approach with Ramey's (1962) heat-transfermodel. Although the intrinsic approach of the use of Ramey's model iscorrect, we think that the lack of granularity in Ut may have adverselycontributed to the match quality of pressure-drop calculations.

Two issues are important while evaluating Uto in any flow situation.First, we need to honor the convective-heat-transfer coefficient in thetubing/casing annulus, the controlling resistanceover thedepth intervalof interest, and second, any change in flow cross-sectional area shouldbe accounted for. Let us illustrate both pointswith example calculations.

Fig. 4 shows the pressure profile calculated by the five models forthe KE1-22 well. This particular example was chosen because itexhibited multiple flow regimes, starting with single-phase waterflow at the bottomhole and ending with annular two-phase flow atthewellhead. Therefore, the homogeneousmodel was not expected toperform well in estimating pressures in the entire wellbore. Forinstance, in slug flow, the homogeneous model significantly under-estimates liquid holdup, and consequently the static head, leading to26% lower pressure-gradient estimation compared to the proposedmodel. In the accompanied heat-transfer calculations, Uto varied from7.4 Btu/h ft2 °F at the bottom to about 10 Btu/h ft2 °F at the top. Fig. 5presents the Ut variation along with the estimated fluid temperature.Stepwise calculations for this well are shown in Appendix C for twostations, depicting two flow regimes.

Although, the Ansari et al. model's results in this flow regimenearly match that of the proposed model, the cumulative effect of thisslight (∼4%) difference results in a significant difference in theestimated WHP. We point out that the calculated pressure-drop forannular flow with the Ansari et al. model is very sensitive to theestimation of the liquid-film thickness, thereby explaining thedifference in the estimated WHP. Just like the Ansari et al. model,the Hagedorn–Brown and Orkiszewski methods estimate somewhatlower liquid holdup. Consequently, the cumulative effect of thebottom-up computational approach translates into a significantlylower pressure loss, with a consequent higher WHP than the reportedvalue. That the proposedmodel outperformed others in this particularcase is probably fortuitous and should not be construed as the norm.

Fig. 2. Model pressures (H–B and homoge

As indicated earlier, investigators often used a constant value for theoverall-heat-transfer coefficient for the entire wellbore. Althoughconvenient, this approach can lead to errors in both pressure-dropand heat-loss estimation, simply because a well configuration (diam-eter, annular cross-sectional area, etc.) can change significantly withdepth. To illustrate this point, let us recalculate the pressure profile fortheKE1-22well for a larger casingdiameter (33 in compared to 26.8 in),and allow the full effect of natural-convective heat transfer in theannulus (Hasan and Kabir, 2002). Changes in tubing diameter areconsistentwith deposition of scale, as pointed out byGoyal et al. (1980),among others.

Fig. 6 shows the increased pressure loss with the proposed model,leading to a loweredWHP. The increased heat transfer, as indicated bythe increased Ut in Fig. 7, is directly responsible for the decreasedWHP. As one would expect, increased heat transfer leads to a lowerwellhead enthalpy and the consequent lower steam quality, 0.072compared to 0.09. Lower steam quality, in turn, leads to a higher fluiddensity and higher pressure loss resulting in 13.7 psi or 3.9% lowerWHP for the proposed model. Note that the greater heat loss may notnecessarily lead to a higher pressure loss. In cases where the frictionalpressure loss predominates, such as when the steam quality is veryhigh, greater heat loss leading to lowering of quality can reduce thefrictional loss much more than would be compensated by anincreased static-head loss. Therefore, rigorous heat-transfer compu-tation, along with proper pressure loss estimation, is essential formodeling geothermal wells.

4. Discussion

Presence of noncondensible gas and/or solids can seriously compli-cate any pressure-drop computations, particularly those in geothermalwells. For instance, Barelli et al. (1982) have shown that matchingpressure and temperature becomes infeasible if the presence of CO2 isneglected. The assumptions of Henry's law to account for solubility ofCO2 and the attainment of thermodynamic equilibrium at each grid cellin the wellbore appear to be reasonable for the field cases they studied.

neous) correlate well with field data.

Fig. 3. Model pressure (Ansari et al.) correlate well with field data.Fig. 5. Variable-Ut and fluid temperature profiles, KE1-22 well.

81A.R. Hasan, C.S. Kabir / Journal of Petroleum Science and Engineering 71 (2010) 77–86

In the same study, Barelli et al. assumed that NaClwas the only source ofsalt and that the activity coefficients were unity. Tanaka and Nishi(1988) also presented an approach to handle CO2 in both the liquid andvapor phases. In contrast, Satman and Alkan (1989) proposed a calcitedeposition model. Presence of both CO2 and salt was neglected in thisstudy. However, we recommend their inclusion when the situationdemands.

Just as in oil wells (Hasan et al., 2007), the two-phase flowmodelsare nondiscriminating in a statistical sense, when applied togeothermal wells. This finding is rather reassuring in that a potentialuser may direct his or her attention elsewhere; that is, collecting flowrate data that are in alignment with pressure and temperaturemeasurements. For instance, Goyal et al. (1980) reported that errorsin the mass flow rate, pressure, and enthalpy at the wellhead mayeasily cause BHP estimation errors well in excess of ±10%. In addition,Ortiz-Ramirez (1983) reported that the success of pressure-dropcalculations lay largely on inputting the correct downhole-fluidtemperature in liquid-dominated systems. Put another way, modelsophistication cannot overcome incoherent rate/pressure and otherinput data. Note that even simple homogeneous model was at parwith other models. However, the homogeneous model owes itssuccess largely to the dominance of flow homogeneity in this annular-flow-dominated dataset.

5. Conclusions

1. This study presents an easy-to-use, two-phase flow model forgeothermal wells that applies to a variety of well orientation andgeometry, with minimal changes in model's parameters. Theaccompanied heat-transfer model accounts for variable-pipe diam-eter with emphasis on convective-heat transport in the annulus,thereby ensuring appropriate energy balance throughout thewellbore.

2. Statistical analyses with a high-quality dataset suggest that all thefive models studied in this paper perform similarly, meaning nosingle approach emerged as a clear winner. Nonetheless, the model

Fig. 4. Pressure-traverse computation with five models, KE1-22 well.

proposed here offers a simpler rendition of many complicatedones. Put another way, the model ensures transparency and offerssmooth transition boundaries between the flow regimes. Inaddition, this modeling approach returns marginally superiorstatistical measures.

3. Experience with modeling field data suggests that data qualitysupersedes any other considerations. In fact, the use of any of thetwo-phase flow models used here, and perhaps a few others notconsidered here, can do a good job in modeling wellbore pressure-drops.

NomenclatureAx cross-sectional area for flow, ft2

cp heat capacity, Btu/lbm °FCo flow parameter, dimensionlessCob flow parameter for bubbly flow, dimensionlessCos flow parameter for slug flow, dimensionlessd tubing ID, ftf Moody friction factor, dimensionlessfm Moody friction factor for two-phase mixtures, dimensionlessfg in-situ steam-volume fraction, dimensionlessfL liquid holdup, dimensionlessFθ well-deviation factor given by Eq. (A-10), dimensionlessg gravitational acceleration, ft/s2

gc conversion factor, 32.17(lbm ft)/lbf/s2

h enthalpy of steam/water mixture, Btu/lbmhc convective heat-transfer coefficient, Btu/h ft2 °Fhg enthalpy of steam, Btu/lbmhL enthalpy of water, Btu/lbmJ conversion factor, 778*32.2 lbm/lbfkc casing thermal conductivity, Btu/hft °Fkcem cement thermal conductivity, Btu/hft °Fke earth thermal conductivity, Btu/h ft °Fkins insulation thermal conductivity, Btu/h ft °F

Fig. 6. Pressure-traverse computation with 33 in casing, KE1-22 well.

Fig. 7. Constant-Ut and fluid temperature profiles, KE1-22 well.

82 A.R. Hasan, C.S. Kabir / Journal of Petroleum Science and Engineering 71 (2010) 77–86

kt tubing thermal conductivity, Btu/h ft °FLR relaxation length parameter (Eq. (7)), 1/ftn number of data pointsp pressure, psia−dp/dz total pressure gradient, psi/ft−(dp/dz)Aaccelerational pressure gradient, psi/ft−(dp/dz)Ffrictional pressure gradient, psi/ft−(dp/dz)Hhydrostatic pressure gradient, psi/ftqL liquid volumetric flow rate, ft3/hqg vapor (steam) volumetric flow rate, ft3/hqm two-phase mixture volumetric flow rate, ft3/hQ heat-transfer rate per unit length of wellbore, Btu/h ftr correlation coefficient, dimensionlessr radius, ftrcem cement inside radius, ftrci casing inside radius, ftrco casing outside radius, ftrto tubing-outside radius, ftrwb wellbore radius, ftRem Reynolds number (=Dvm ρm /µm)), dimensionlesst producing time, ht statistical parameter defined by Eq. (9), dimensionlesstD dimensionless producing time (=ket /ρecerwb

2 )Tei undisturbed earth or formation temperature at any depth,

°F (T)TD dimensionless temperature (=2πke(Twb−Tei)/Q)Tf fluid temperature, °FTwb wellbore/earth interface temperature, °FUto overall-heat-transfer coefficient, Btu/h ft2 °Fvgb superficial-gas velocity needed for transition from bubbly to

slug flow given by Eq. (A-12), ft/svgc superficial-gas velocity needed for transition from churn to

annular flow given by Eq. (A-15), ft/svsL superficial velocity of liquid, ft/svsg superficial velocity of steam, ft/svm velocity of steam/liquid mixture, ft/svms minimummixture velocity needed for dispersed bubbly, ft/sv∞ rise velocity of steam bubbles, ft/sv–∞ average rise velocity of steam bubbles, ft/sv∞b rise velocity of small steam bubbles (in bubbly flow), ft/sv∞T rise velocity of Taylor bubbles, ft/sw mass flow rate, lbm/hdv/dz velocity gradient, ft/sx mass fraction [=vsgρg/(vsgρg+vsLρL)] or steam quality

{=(h−hLsat)/(hg−hL)}

z vertical well depth, ft

Z gas-law deviation factor, dimensionlessε pipe-roughness factor, ftΛ roughness parameter in Eq. (A-4), dimensionlessρg steam density, lbm/ft3

ρL liquid density, lbm/ft3

ρm mixture density, lbm/ft3

δ liquid-film thickness, ftδ dimensionless thickness of the liquid film (=δ/d)µg steam viscosity, cpµL liquid viscosity, cpσ surface tension, lbm/s2

θ well inclination with horizontal, degrees

Acknowledgments

We thank Hess management for permission to publish this work.This paper has been abstracted from SPE 121351.

Appendix A. The drift-flux modeling approach

As Eq. (A-1) shows, the total pressure gradient during fluid flow isthe sum of static, friction, and kinetic gradients,

− dpdz

= gρm +fmv

2mρm2d

+ ρmvmdvmdz

: ðA� 1Þ

Themixture density, ρm, is the volumetric-weighted average of thetwo phases, ρL and ρg

ρm = ρgfg + ρL 1−fg� �

: ðA� 2Þ

In Eq. (A-2), fg is the in-situ volume fraction of the gas phase. Inaddition to estimating fg, a two-phase flow model also requires calcu-lating the two-phase friction factor, fm. We use the homogeneousmodelto estimate frictional pressure gradient.We estimate friction factor in allflow regimes frommixture Reynolds number, Rem=Dvm ρm/µm, using amass-average-mixture viscosity

μm = μgx + μ l 1−xð Þ ðA� 3Þ

and the Chen (1979) correlation for friction factor in rough pipes

fm =1

4 log ε=d3:7065− 5:0452

RemlogΛ

� �h i2 : ðA� 4Þ

In Eq. (A-4), ε is pipe roughness, and the dimensionless parameterΛ is given by

Λ =ε=dð Þ1:10982:8257

+7:149Rem

0:8981: ðA� 5Þ

To estimate volume fraction needed to calculate mixture density,we model behavior of each flow regime individually. However, for allflow regimes the gas (or lighter) phasemoves faster than the liquid (orheavier) because of its buoyancy and its tendency to flow close to thechannel center, where the velocity is higher than the average-mixturevelocity. Therefore, we express the in-situ gas velocity, vg, as the sumof the bubble-rise velocity and Co times the average-mixture velocity

vg = Covm + v∞: ðA� 6Þ

For cocurrent upflow, the mixture velocity, vm, is the sum of thegas and liquid superficial velocities, vm=vsg+vsL. Noting that in-situvelocity is the ratio of superficial velocity vsg to volume fraction

83A.R. Hasan, C.S. Kabir / Journal of Petroleum Science and Engineering 71 (2010) 77–86

fg (vg≡vsg/fg), we have a simple relation between volume fractionand phase velocities in these flow regimes

fg =vsg

Covm + v∞: ðA� 7Þ

Eq. (A-7) is the same as Eq. (2) in the text.

A.1. Model parameters

The values of the flow parameter, Co, and the rise velocity, v∞,depend upon the flow pattern, well deviation, flow direction. In thefollowing, we present the model parameters that we have eitherdeveloped or adopted for various flow regimes. For brevity and clarity,we have not discussed alternative values/expressions for some ofthese parameters that others may have found more acceptable. How-ever, much of that discussion appears in the text of Hasan and Kabir(2002).

The rise velocity for small bubbles in all flow directions and welldeviations is best represented by the Harmathy (1960) equation,which is same as Eq. (2) in the text

v∞b = 1:53 g ρl−ρg� �

σ =ρ2lh i1

=4 : ðA� 8Þ

The rise velocity of Taylor bubbles, characteristics of slug flow, isinfluenced by well deviation and annular geometry. We estimate theTaylor bubble-rise velocity from the following expression

v∞T = 0:35ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigD ρl−ρg

� �= ρL

rFθð Þ: ðA� 9Þ

In Eq. (A-9), the well-deviation factor Fθ, in terms of deviationangle from vertical θ, is given by

Fθ =ffiffiffiffiffiffiffiffiffifficosθ

p1 + sinθð Þ1:2: ðA� 10Þ

In slug flow, liquid slugs that separate the Taylor bubbles alsocontain small bubbles whose rise velocities are best estimated withEq. (A-8). Therefore, for slug flow we need a rise velocity that is anaverage of bubble-rise velocities, represented by Eqs. (A-8) and (A-9).We use the following averaging process that depends on themagnitude of the superficial-gas velocity compared to that neededfor transition from bubbly to slug flow, vgb

�v∞ = v∞b 1−e−0:1vgb = vsg−vgbð Þ� �+ v∞T e−0:1vgb = vsg−vgbð Þ� �

: ðA� 11Þ

We use Eq. (A-11) for rise velocity in churn flow as well. Note thatEq. (A-11) is completely empirical in nature. Many researchers haveshown that at the high velocity associated with annular flow, theeffects of well orientation, geometry, and flow direction are negligible.The parameter values for annular flow in Table 1 reflect that fact.

The values of Co depend on the flow pattern, well deviation, andflow direction. Table 1 in the text presents the values of Co for fullydeveloped flow patterns. Although transition from one flow regime toanother could be sharp, it is rarely abrupt. The actual values of Co usedwith Eq. (A-7) are modified along the line of Eq. (A-11) to account forsmooth transition from one flow regime to another.

A.2. Flow-pattern transition criteria

The use of our unified approach with separate values ofparameters for estimating volume fraction requires establishing theexisting flow pattern. Transition from bubbly to slug flow occurs at thevolume fraction exceeding 0.25 in vertical systems. Experimentaldata of Zuber and Findlay (1965), and Hasan and Kabir (2002, p. 21),

among others, support this contention. For inclined systems, buoy-ancy concentrates the gas phase near the upper wall, causing localvolume fraction to exceed the needed value of 0.25 before the averagevolume fraction, leading to an earlier transition from bubbly flow.Using a geometrical approach Hasan and Kabir (2002, p. 42) were ableto obtain

vgb = 0:43vsL � 0:36v∞ð Þ cos θ: ðA� 12Þ

When v∞>v∞T, as would be the case for small-diameter channels,bubbly flow cannot exist. Also, when mixture velocity is higher thanvms, given by the following expression (Taitel et al., 1980; Shoham,1982):

2v1:2msf2d

0:4 ρLσ

� �0:6ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

0:4σ

g ρL−ρg� �

vuut = 0:725 + 4:15

ffiffiffiffiffiffivsgvm

s: ðA� 13Þ

Dispersed bubbly flow persists until the gas velocity is highenough for transition to churn flow occurs. Transition to churn flowfrom dispersed bubbly flow occurs when

vsg > 1:08vsL ðA� 14Þ

Transition from slug to churn flow occurs when vm exceeds vms

(given by Eq. (A-13)) and vsg exceeds that given by Eq. (A-14).Transition from churn to annular flow occurs when vsg exceeds thevalue given by the following expression (Taitel et al., 1980)

vgc = 3:1 gσ ρL−ρg� �

=ρ2gh i1

=4 : ðA� 15Þ

The presence of ρg in Eq. (A-15) suggests that at very highpressures, annular flow can exist at low-gas velocities. However, alittle reflection about the characteristic of this flow regime – a gas coresurrounded by a thin liquid film – suggests that enough gas mustremain in the core for the liquid not to be able to bridge the channelcross-section. To allow for this reality, we arbitrarily use a minimum fgof 0.7 as an additional requirement for the existence of annular flow.

A.3. Transition smoothing

For Co, we use the values reported in Table 1 for bubbly flow. For allother flow patterns we use an exponential weighted-average value ofCo, derived from the fully developed Co values of the adjoining flowregimes. Therefore, for slug flow

Co = Cob 1−e−0:1vbs = vsg−vbsð Þ� �+ Cos e−0:1vbs = vsg−vbsð Þ� �

: ðA� 16Þ

For upward flow, the parameter values are the same for bubbly andfully developed slug flows; that is, Co equals 1.2 for cocurrent slugflow. Similarly, for churn flow, Co is given by the following expression

Co = 1:2 1−e−0:1vms = vm−vmsð Þ� �+ 1:15 e−0:1vms = vm−vmsð Þ� �

: ðA� 17Þ

Appendix B. Energy transport model

Heat transfer between the wellbore fluid and the surroundingformation is modeled by equating heat loss from the fluid to thewellbore/formation interface to heat loss from this interface tosurrounding earth. Heat loss from the tubing fluid (at a temperatureof Tf) to the wellbore/formation interface (at Twb) can be written as:

Q = −2πrtoUto Tf−Twb

� �: ðB� 1Þ

84 A.R. Hasan, C.S. Kabir / Journal of Petroleum Science and Engineering 71 (2010) 77–86

Heat loss from wellbore/formation interface to earth will dependon the temperature difference, Twb−Tei, and the dimensionlesstemperature TD (Hasan and Kabir, 2002) and is given by

Q≡−2πkeTD

Twb−Teið Þ ðB� 2Þ

where TD is given by

TD = ln e −0:2tDð Þ + 1:5−0:3719e−tD� �h i ffiffiffiffiffi

tDp

: ðB� 3Þ

Combining Eqs. (B-1) and (B-2), we get

Q = −LR T1−Teið Þ ðB� 4Þ

where

LR =2πwcp

rtoUtokeke + rtoUtoTD

� �: ðB� 5Þ

In Eq. (B-5), Ute represents the overall coefficient for heat transferfrom the tubing fluid to the wellbore/formation interface. Because heattransfer is more dominant in the radial direction than its verticalcounterpart, one can treat the resistances to heat flow in series. For heattransfer from the fluid to the wellbore/formation interface, theseresistances include those offered by (1) tubing fluid, (2) any insulationaround the tubing, (3) tubing material, (4) annulus-fluid, (5) casingmaterial, and (6) the cement sheath, assuming no other annuli.Therefore, we estimate Uto by inverting the sum of the resistancesoffered by all the elements, which is given by

1Uto

=rtortiht

+rto ln rins = rtoð Þ

kins+

rto ln rto = rtið Þkt

+rto ln rco = rcið Þ

kc

+rto ln rco = rcið Þ

hc+

rto ln rcemo = rcoð Þkcem

: ðB� 6Þ

Details of estimation procedure for some of these terms (such as,annulus natural-convective-heat-transfer coefficient, hc) can be foundin Hasan and Kabir (2002). Eq. (B-6) is a general expression. Not allelements, such as tubing insulation represented by the second term ofthe right side, may be present in a given wellbore. Note that alloverall-heat-transfer coefficients in this work are expressed in termsof the tubing-outside diameter.

Appendix C. Example calculations

In this section, we show step-wise calculations in two flowregimes using the KE1-22 well, reported by Garg et al. Slug flowoccurs in the lower section of the well, whereas the annular flow isdominant in the upper section. For two-phase flow calculations, weshow the use of a few selected models. The KE1-22 well producesgeothermal fluids from 2957 ft MDwith 8.819 in ID tubing. The well isdeviated by 2.87° from the vertical, thereby making the TVD of2953 ft.

The well delivers 0.198×106 lbm/h of hot water at the bottomholewith an enthalpy of 427.8 Btu/lbm. The following values of fluidproperties, obtained from interpolation of steam tables, and otherparameters are computed at the well bottom, where the pressure is477.4 psia.

C.1. Saturated steam/water property values:

hg = 1205:3Btu=lbm hL = 444Btu=lbmρL = 51:03lb=ft3 ρg = 1:02lb=ft3

μL = 0:144cp μg = 0:013cpσ = 28:8dynes=cm = 0:0635lbm=s2

d = 8:819in Ax = 0:424ft2

qms = 0:198 × 106 lbm=hx = h−hLð Þ= hg−hL

� �= 427:8−444ð Þ= 1205−444ð Þ= −0:0213:

Thus, we have single-phase flow of hot water at the bottomhole.The total pressure gradient, almost entirely owing to the static head(0.3543 psi/ft), is 0.3550 psi/ft. With this pressure gradient, wecompute a fluid pressure of 419.1 psia at 164 ft above the bottomhole.Heat loss over this section is minimal and fluid enthalpy is calculatedto be 427.3 Btu/lbm and the fluid is still single-phase water. Similarpressure gradient computation leads to a pressure of 371.9 psi at thenext location, at a depth of 2957 ft. At this point, the enthalpy iscalculated to be 426.6 Btu/lbm. With the saturated liquid enthalpy at371.9 psi being 415.9 Btu/lbm, we compute the following steamquality and velocity:

x = 426:6−405:7ð Þ= 1204:6−415:9ð Þ= 0:0136

qg = qmsx = ρg= 0:198 × 106 × 0:0136 = 0:79 = 3600

= 0:941ft3=s

vsg = qg = Ax = 2:218ft=s

vsL = qL = Ax = 2:455ft=s:

C.2. Slug flow regime

C.2.1. The homogeneous model

Gas� volumefraction = vsg = vm = 0:491

Mixturedensity; ρm = 0:491ρg� �

+ 0:509ð Þ ρLð Þ= 26:96lbm=ft3

Mixtureviscosity; μm = xμg + 1−xð ÞμL = 0:0136ð Þ 0:0166ð Þ+ 0:9864ð Þ 0:12ð Þ = 0:12cp

Rem = vmρmd = μm = 1;200;139:

Using the Chen (1979) correlation, the friction factor is estimatedto be

f =4

4 log ε=d3:7065− 5:0452

Re log� �h i2 = 0:01345:

A pipe-roughness factor of 0.00984 in was used resulting in a valueof 0.000112 for ε/d.

−(dp/dz)F= fmvm2 ρm /(2d)/144=0.0012 psi/ft

−(dp/dz)H=ρm g sinθ=2.445 psf/ft=0.1872 psi/ft−(dp/dz)A=ρm vm (dvm /dz)=0.00012 psi/ft−dp/dz=0.1885 psi/ft total pressure gradient at 2957 ft

C.2.2. The Ansari et al. (1994) modelThe vsg needed for transition from bubbly to slug flow is 1.30 ft/s

(Eq. (A-1)), whereas the vm needed for transition to churn flow(Eq. (A-1)) is 11.44 ft/s. The flow pattern is therefore, slug. The

85A.R. Hasan, C.S. Kabir / Journal of Petroleum Science and Engineering 71 (2010) 77–86

computational approach for in-situ volume fraction and pressuregradient using the Ansari et al. model can be found elsewhere (Hasanand Kabir, 2007); we summarize the important results here:

Gas-volume fraction, fg=0.328Mixture density, ρm=35.35 lbm/ft3.−(dp/dz)F=0.00159 psi/ft−(dp/dz)H=ρm g sinθ=0.2455 psi/ft−(dp/dz)A=ρm vm (dvm/dz)=0.00012 psi/ft−dp/dz=0.2471 psi/ft total pressure gradient at 2957 ft.

C.2.3. The proposed (H–K) modelFor slug flow, we have

v∞T = 1:765ft=s; v∞ = 0:6952ft=s; �v∞ = 1:6235ft=sGas� volume fraction; fg = 2:218= 1:2�4:672 + 1:6235

� �� �= 0:3067

Mixture density; ρm = 36:43lbm=ft3

−(dp/dz)F= fmvm2 ρm/(2d)/144=0.0015 psi/ft

−(dp/dz)H=ρm g sinθ=2.445 psf/ft=0.2529 psi/ft−(dp/dz)A=ρm vm (dvm/dz)=0.00012 psi/ft−dp/dz=0.2545 psi/ft total pressure gradient at 2957 ft.

Aswithmany geothermal wells, our calculations show thatmost ofthis well (from wellhead down to 2100 ft) experiences annular two-phase flow. In the following, we show sample calculations for alocation 337 ft below the wellhead.

C.3. Annular flow regime

C.3.1. The homogeneous modelThe model computed pressure at 337 ft below wellhead=

182.7 psia

ρL = 54:83lbm=ft3 ρg = 0:400lbm=ft3

x = 915:6−789:2ð Þ= 2782:7−789:2ð Þ = 0:063

vsg = qg = Ax = qmsx= ρgAx = 20:044ft=s

vsL = qL = Ax = 2:222ft=s:

Gas� volumefraction = vsg = vm = 0:900

Mixture density; ρm = 0:9ρg� �

+ 0:1ð Þ ρLð Þ= 5:833lbm=ft3:

−(dp/dz)F= fmvm2 ρm/(2d)/144=0.00577 psi/ft

−(dp/dz)H=ρm g sinθ=2.445 psf/ft=0.0405 psi/ft−(dp/dz)A=ρm vm (dvm/dz)=0.0002 psi/ft−dp/dz=0.0465 psi/ft total pressure gradient.

C.3.2. The proposed (H–K) modelThe model computed pressure at 337 ft below wellhead=

142.8 psia

ρL = 55:57lbm=ft3 ρg = 0:315lbm=ft3

x = 920:95−738:6ð Þ= 2782:7−738:6ð Þ = 0:063

vsg = qg = Ax = qmsx = ρgAx = 20:044ft=s

vsL = qL = Ax = 2:222ft=s:

The minimum vsg needed for transition to annular flow is given byEq. (A-4)

vsg > 3:1 gσ ρL−ρg� �

=ρ2gn o1

=4

= 3:1 32:2 � 0:0936 55:57−0:315ð Þ=0:3152n o1

=4

= 19:83ft=s:

Therefore, we have annular two-phase flow. For fully developedannular flow, we use Eq. (2) with Co=1.0 and v∞=0.0. However,because actual vsg is close to the transition value, Co=1.018 (calculatedusing an exponential expression). Accordingly,

Gas� volumefraction; fg = 20:044= 20:044 + 2:222ð Þ½ � = 0:927Mixturedensity; ρm = 4:329lbm=ft3:

−(dp/dz)F= fmvm2 ρm/(2d)/144=0.0122 psi/ft

−(dp/dz)H=ρm g sinθ=0.0301 psi/ft−(dp/dz)A=ρm vm (dvm/dz)=0.0004 psi/ft−dp/dz=0.0427 psi/ft: total pressure gradient.

C.3.3. The Ansari et al. modelThe model computed pressure at 337 ft below wellhead=

184.17 psia

ρL = 54:81lbm=ft3 ρg = 0:4027lbm=ft3

x = 933:1−809:7ð Þ= 2786:3−809:7ð Þ = 0:0624

vsg = qg = Ax = qmsx= ρgAx = 20:138ft=s

vsL = qL = Ax = 2:222ft=s:

The computational approach for in-situ volume fraction andpressure gradient in annular flow using the Ansari et al. model canbe found elsewhere (Hasan and Kabir, 2007); we summarize theimportant results here:

Dimensionless film� thickness;�δ = δ= d = 0:07348

fg = 1−4�δ = 0:706

Thefilm� frictionfactor; ff = fsm 1 + 300 δÞ = 0:264:ð

−(dp/dz)F= ffvc2ρc/{2d(1−2δ_)}/144=0.0276 psi/ft

−(dp/dz)H=ρm g sinθ=0.0054 psi/ft−(dp/dz)A=ρm vm (dvm/dz)=0.0001 psi/ft−dp/dz=0.0331 psi/ft total pressure gradient.

Fig. 4 in the text shows the pressure profiles calculated by variousmodels.

References

Acuña, J.A., Arcedera, B.A., 2005. Two-phase flow behavior and spinner data analysis ingeothermal wells. Paper SGP-TR-176, Proc., Thirtieth Workshop on GeothermalReservoir Engineering, Stanford University, Stanford, California, January 31–February 2.

Ambastha, A.K., Gudmundsson, J.S., 1986a. Geothermal two-phase wellbore flow:pressure drop correlations and flow pattern transitions. Paper SGP-TR-93, Proc.,Eleventh Workshop on Geothermal Reservoir Engineering, Stanford University,Stanford, California, January 21–23.

Ambastha, A.K., Gudmundsson, J.S., 1986b. Pressure profiles in two-phase geothermalwells: comparison of field data and model calculations. Proc., Eleventh Workshopon Geothermal Reservoir Engineering, Stanford University, Stanford, California,January 21–23.

Ansari, A.M., et al., 1994. A comprehensive mechanistic model for upward two-phaseflow in wellbores. SPEPF 9 (2), 143–151.

Barelli, A., Corsi, R., Del Pizzo, G., Scali, C., 1982. A two-phase flowmodel for geothermalwells in the presence of non-condensable gas. Geothermics 11 (3), 175–191.

Blalock Jr., H.M., 1972. Social Statistics, 2nd edition. McGraw-Hill Book Co., New York,NY, pp. 406–407.

Chadha, P.K., Malin, M.R., Palacio-Perez, A., 1993. Modelling of two-phase flow insidegeothermal wells. Appl. Math. Model. 17 (5), 236–245.

Chen, N.H., 1979. An explicit equation for friction factor in pipe. Ind. Eng. Chem.Fundam. 18 (3), 296.

Chierici, G.L., Giannone, G., Schlocchi, G., 1981. A wellbore model for two-phase flow ingeothermal reservoirs. Paper SPE 10315 presented at the SPE Annual TechnicalConference and Exhibition, San Antonio, TX, 4–7 October.

Duns Jr., H., Ros, N.C.J., 1963. Vertical flow of gas and liquid mixtures in wells. Proc., 6thWorld Pet. Congress, pp. 451–465.

Garg, S.K., Pritchett, J.W., Alexander, J.H., 2004a. A new liquid hold-up correlation forgeothermal wells. Geothermics 33 (6), 795–817.

86 A.R. Hasan, C.S. Kabir / Journal of Petroleum Science and Engineering 71 (2010) 77–86

Garg, S.K., Pritchett, J.W., Alexander, J.H., 2004b. Development of new geothermalholdup correlations using flowing well data. Report EXT-04-01760 prepared forIdaho National Engineering and Environmental Laboratory.

Gould, T.L., 1974. Vertical two-phase steam–water flow in geothermal wells. JPT 26 (8),833–842.

Goyal, K.P., Miller, C.W., Lippmann, M.J., 1980. Effect of measured wellhead parametersand well scaling on the computed downhole conditions in cerro prieto wells. Proc.,Sixth Workshop on Geothermal Reservoir Engineering, Stanford University,Stanford, California, December.

Hagedorn, A.R., Brown, K.E., 1965. Experimental study of pressure gradients occurringduring continuous two-phase flow in small-diameter vertical conduits. JPT 17 (4),475–484.

Harmathy, T.Z., 1960. Velocity of large drops and bubbles in media of infinite andrestricted extent. AIChE J. 6 (2), 281–288.

Hasan, A.R., Kabir, C.S., 2002. Fluid flow and heat transfer in wellbores. Textbook Series,SPE, Richardson, Texas.

Hasan, A.R., Kabir, C.S., 2007. A simple model for annular two-phase flow in wellbores.SPEPO 22 (2), 168–175.

Hasan, A.R., Kabir, C.S., Sayarpour, M., 2007. A basic approach to wellbore two-phaseflow modeling. Paper SPE 109868 presented at the SPE Annual TechnicalConference and Exhibition, Anaheim, California, 11–14 November.

Hasan, A.R., Kabir, C.S., Wang, X., 2009. A robust steady-state model for flowing-fluidtemperature model in complex wells. SPEPO 24 (2), 269–276.

Hughmark, G.A., 1963. Pressure drop in horizontal and vertical gas liquid flow. Ind. Eng.Chem. Fundam. 2, 315–320.

Kabir, C.S., Hasan, A.R., 2006. Simplified wellbore flow modeling in gas-condensatesystems. SPEPO 21 (1), 89–97.

Orkiszewski, J., 1967. Prediction of two-phase pressure drops in vertical pipe. JPT 19 (6),829–838 Trans., AIME, 240.

Ortiz-Ramirez, J., 1983. Two-phase flow in geothermal wells, development and uses of acomputer code. Stanford Geothermal Program Report SGP-TR-66. StanfordUniversity, Stanford, California.

Ramey Jr., H.J., 1962.Wellbore heat transmission. JPT 14 (4), 427–435 Trans., AIME, 225.Satman, A., Alkan, H., 1989. Modelling of wellbore flow and calcite deposition for

geothermal wells in the presence of co2. Unsolicited manuscript SPE 19350.Shi, H., Holmes, J.A., Durlofsky, L.J., Aziz, K., Diaz, L.R., Alkaya, B., Oddie, G., 2005a. Drift-

flux modeling of two-phase flow in wellbores. SPEJ 10 (1), 24–33.Shi, H., Holmes, J.A., Diaz, L.R., Durlofsky, L.J., Aziz, K., 2005b. Drift-flux parameters for

three-phase steady-state flow in wellbores. SPEJ 10 (2), 130–137.Shoham, O., 1982. Flow pattern transition and characterization in gas–liquid two-phase

flow in inclined pipes. PhD Disertation, U. of Tel Aviv, Tel Aviv, Israel.Taitel, Y., Barnea, D., Dukler, A.E., 1980. Modeling flow pattern transition for steady

upward gas–liquid flow in vertical tubes. AIChE J. 33 (2), 345–354.Tanaka, S., Nishi, K., 1988. Computer code of two-phase plow in geothermal wells

producing water and/or water–carbon dioxide mixtures. Paper SGP-TR-113. Proc.,Thirteenth Workshop on Geothermal Reservoir Engineering, Stanford University,Stanford, California, January 19–21.

Upadhyay, R.N., Hartz, J.D., Gulati, M.S., 1977. Comparison of calculated and observedpressure drops in geothermal wells producing steam–water mixtures. Paper SPE6766 presented at the SPE Annual Technical Conference and Exhibition, Denver,Colorado, 9–12 October.

Zuber, N., Findlay, J., 1965. Average volumetric concentration in two-phase flowsystems. Trans. ASME J. Heat Transfer 87, 453–468.

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