A Comprehensive Wellbore Steam/Water Flow Model for Steam Injection and Geothermal Applications S.M. Farouq Ali, SPE, U. of Alberta

Abstract A comprehensive mathematical model was developed to simulate the downward or upward flow of a steam/water mixture in a well. Comparisons of model predictions with actual field data for both steam injection for oil recovery and geothermal production showed the validity of the model.

The proposed model is based on mass and momentum balances in the well bore and on heat balance in the wellbore and the surrounding media. Unlike the previous models, the pressure calculation accounts for slip and the prevailing flow regime, based on noted correlations. Furthermore, heat loss to the surrounding formations is treated rigorously. The overall heat transfer coefficient involved permits the consideration of a variety of well completions.

The model was employed for a series of tests to evaluate the 'effects of the injection pressure, in-jection rate, time, and well completion on the downhole steam pressure and quality. It was found that the slip concept and the flow regime are essential elements in wellbore steam/water flow calculations. Pressure drop was found to increase with a decrease in the injection pressure, as also with more obvious parameters. An increase in the injection pressure or tubing size and/or a decrease in the injection rate led to a decrease in steam quality at a given depth.

Introduction Most steam injection operations for heavy oil recovery involve injection of wet steam down the tubing and occasionally down the casing/tubing annulus. Computations of reservoir heating by the injected steam require a knowledge of steam pressure and quality at the formation face. At the same time, it is important to know the heat loss to the surroundings during flow in the well bore, as well as the casing temperature, for an appropriate well 01977520/81/00107966$00.25 Copyright 1981 Society of Petroleum Engineers of AIME

OCTOBER 1981

completion. Although several investigators 1-5 have presented well bore models for steam injection, none considered the flow regime concept in a unified approach. Furthermore, all models employed only approximate solutions of the one-dimensional radial heat conduction equation to investigate the heat loss.

Vertical two-phase flow of water and steam occurs in geothermal wells. Among the geothermal reser-voirs, only a few areas are classified as vapor-dominated systems, producing dry to superheated steam. All others are hot-water systems and generally produce a mixture of water and steam at the sur-face. 6 Here, again, it is necessary to consider two-phase flow in the wellbore, coupled with heat transfer, to predict steam pressure and quality at the surface. This problem was considered in detail by Gould. 7

The main purpose of this work was to develop an integrated and comprehensive well bore model to simulate vertical, nonisothermal, two-phase flow phenomena. The model is a combination of the previous model of Pacheco and Farouq Ali4 and the pressure/flow-regime correlations of Gould et al., 8 Chierici et al.,9 and Duns and Ros, IO as well as a number of refinements such as the rigorous treat-ment of heat flow, geothermal gradient, etc.

Mathematical Model The system to be modeled consists of three parts: (1) the fluid flow conduit (tubing or annulus), (2) tubing/casing annulus, casing wall, and cement, (3) the formation encircling the cement. Within the conduit, steady,. homogeneous, one-dimensional, two-phase flow is assumed. This is described mathematically by combining a two-phase mass balance with a momentum balance and is as follows (the vertical coordinate z is taken positive in the downward direction).

527

144 dp _ - ~ - dWj Pm Vm dVm _ d Pm +Pm + ---0. Z ge dz ge dz ................................. (1)

Approximating the acceleration gradient 'Ya in the manner of Orkiszewski, II where

wqs 'Ya = 144A 2 gel) ....................... (2)

and using the friction factor for two-phase flow (considered in the next section), Eq. 1 takes the following forms for downward and upward flow.

Downward Flow - g

dp = _1_ (pm g; -7j ). dz 144 ............ (3a)

1- 'Ya Upward Flow

dp = _1_ (pm t +7j ). dz 144 ............ (3b)

1- 'Ya Eq. 3b was given earlier by Gould. 7

The energy balance for the flow under con-sideration is given by

3 Q600+W~(hm+ v~ -~z)=o, ... (4) , ~ ~e~ ~~ where the enthalpy hm of a two-phase mixture can be represented by functions of pressure p and quality is:

The enthalpies of saturated liquid h wand dry steam hs were expressed by the approximate functions of pressure given by Farouq Ali 12 as follows (ap-proximately SOlo error).

hw = 91p o.2574 , ........................ (6)

and

hs = 1 ,119p o.01267) , ..................... (7)

where hs is the enthalpy of dry and saturated steam. Also, the change in velocity in Eq. 4 can be replaced by the change in the inverse of density of the mixture. As a result, Eq. 4 takes the following form.

dis C I dz +CVs +C3 =0, ................. (8) where C I' C2 , and C 3 are functions of pressure given by

C I = w(1, 119 pO.01267 - 91p o.2574), ......... (9)

C2 = w[(1, 119)(0.0l267)p -0.98733 - (91)

.(0.2574)p-O.7426] dp , ............ (10) dz

and

528

C3 = ~ + w[(91)(0.2574)P - 0.7426 dp 3,600 dz w

2 d ( 1) g] + I - A2 -d --- - - ...... (11)

ge ePm Z Pm ge1e

Within the wellbore, heat transfer may occur by radiation, convection, and conduction. This is given by the following equation for radial heat flow.

where the overall heat transfer coefficient, referred to the exterior tubing surface Ute' represents the net resistance to heat flow, offered by the flowing fluid, tubing, insulation, annulus fluid, casing wall, and the cement sheath. Based on a number of assumptions discussed by Willhite,13 Ute can be simplified as follows.

Ute = ~ [ln~ r te k hins

In reem

+ _r_ee_J-l, ................ (13) k heem

where the conduction-convection and radiation heat transfer coefficients (he and hp respectively) are nonlinear functions of temperature.

The flow of heat from the cement/formation interface to the earth is by conduction only. This is described by the following two-dimensional equation.

~ ~(?Ii) a2 Tj _ _ 1_?Ii a r a + 2 - . . .... (14) r r r az a at

Numerical solution of Eq. 14 permits the use of different values of thermal conductivities in the rand z directions, although these were taken to be equal in the present work.

The boundary and initial conditions for the solution of Eqs. 3 through 14 are as follows.

Initial Condition Tj = Tair + 'Yz at! = O. . ................. (15)

Boundary Conditions Tj=Tair atz=O . ..................... (16a)

Tj = Tres atz=zres. . ................. (16b) . _ aTj_ Q- -27rrkhj - atr-reem . ........... (16c) aT ar T, =0 at r= 00. ... (16d)

Two-Phase Flow in the Wellbore A review of the published two-phase pressure-drop calculation methods suggests that the determination

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TABLE 1 - FLOW CORRELATIONS

Flow Region bubble/plug slug/froth transition

mist

Correlation Gould et at. 8

Chierici et at. 9 Duns and Ros 10 Duns and Ros 10

of holdup (hence, the average mixture density) and the friction gradient should recognize the prevailing flow regime, since the existence of slippage as well as the nonhomogeneity of the mixture cannot be ignored. In the present work, the criteria proposed by Duns and Ros 10 were used to identify the flow regime. The correlations in Table 1 were used for each region. Details of the two-phase flow com-putations are given in Ref. 14.

Solution Procedure Eqs. 3 and 8 are first-order ordinary differential equations; they were discretized and solved by a procedure similar to that described in Ref. 4. Eq. 14 was expressed by finite differences, with a cylindrical mesh-centered grid (Fig. 1). The grid employed equal spacing in the vertical direction and geometric spacing in the radial direction. The spatial derivatives were implicit in time. The resulting equations were solved by use of a band algorithm, with procedures typical of reservoir simulation.

The entire fluid-flow/heat-flow computation employed an iterative procedure, updating the nonlinear coefficients and the overall heat transfer coefficient at each iteration.

Discussion of Results The model was used to conduct comparisons with previously proposed models for steam injection and geothermal production. A number of sensitivity studies were conducted for steam injection also. The results are discussed in this order.

Steam Injection Comparison With Field Data. Few measured field data on well bore steam temperatures have been published. The present model was used to simulate two such sets of data reported in the literature.

Fig. 2 shows the good agreement between measured and calculated steam temperature profiles after 116 hours of injection in a cyclic stimulation operation. 15 The maximum discrepancy over the depth is 3"F (1.7"C). The prevailing two-phase flow is predicted to be in the mist region.

Another set of measured dataS consists of the casing temperature history of a cyclic steam stimulation producer. After injecting 14,600 lbm/hr (160 Mg/d) of 751J!o quality steam into a 2,700-ft (825-m) well for 7 days, with pressure increasing from 1,430 to 1,800 psia (10 to 12 MPa), the casing temperature was measured as 460F (238C). The present model predicted a value of 455F (235C), showing good agreement. The prevailing flow regime was identified as slug/froth flow. Comparison With Previous Models. Fig. 3 shows a comparison of the results from Earlougher's3 model

OCTOBER 1981

z

(np 1) ~r-~--~------~

(i, k) ~j--""" __ ---">--____ ~ (n" k)

Fig. 1 - Grid system used for simulation of heat flow in the ground.

o.---------------------------~~--.

200 -- This Model --- Field Data

400

600 --..c: -a. 800

o

1000

2000 .::: .r::. -Q. 3000 Q) 0

4000

5000

100

0.2

\ \ \

0.4

\ I \ I X

Steam ----' \

0.6 08

Quality Fractional Heat Loss

10

200 300 400 500 600 Steam Temperature, 'F

Fig. 3 - Comparison of results from the present model with Earlougher's method after 7 days of injection of 6,000 Ibm/hr (2721 kg/h) of 80% quality steam at 500 pSia (3447 kPa)through 4-in. (10.16-cm) tubing.

and those from the present model for the well completion scheme of Case 3 in Earlougher's study, which is considered to be the most commonly used. Zero slip was assumed between the phases, and the density of the two-phase mixture was expressed as the inverse of the total specific volume. A simulation using the present model showed that the flow was in the slug/froth flow region, where the effect of slippage cannot be neglected since it influences the fractional volume occupied by each phase in the conduit. This is reflected in the considerable discrepancy between the two sets of results shown in Fig. 3. In fact, steam pressure increases with depth because the increase in the static pressure com-pensates for friction loss.

Pacheco and Farouq Ali4 compared their numerical model results with those from earlier models, with the observed discrepancies discussed in detail. Fig. 4 shows a comparison of the results with those obtained from the present model for one of the cases. Although steam quality and heat loss are in good agreement, steam and casing temperature profiles differ considerably. A possible explanation is that the authors used the same type of expressions for average density and friction loss as the ones used in Ref. 3 and that neither flow regime nor slip were taken into account. This may not lead to a large discrepancy if only mist flow prevails. The present model shows that, in fact, mist flow was present only in the shallow part, while flow transition occurred in the deeper part of the well. Thus, the inclusion of the flow regions and the effect of slip make the steam temperature profiles deviate as shown in Fig. 4. Casing temperature profiles must be considered in the light of formation temperature distribution. The

530

1.0

1000

2000 ---

Fractional .r::. Heat Loss C. Q) 3000 , 0 ,

I 4000 I I

I I

5000 I

100 200 300 400 500 600 Temperature, cF

-- This Model --- Pacheco and Farouq Ali

Fig. 4 - Comparison of results from the present model with those based on the Pacheco-Farouq Ali model after 100 days of injection of 10,000 Ibm/hr (4536 kg/h) of 80% quality steam at 500 psi a (3447 kPa) through 2.5-in. (6.35-cm) tubing.

discrepancy may be due to the more precise two-dimensional treatment of the latter and, hence, of the heat loss, whereas the authors used the function in Ref. 16.

Fig. 5 compares the casing temperature variation with time as reported by Leutwyler 1,17 with that predicted by the present model. The predictions are 5 to 10070 lower than the author's values, possibly because he took the tubing temperature to be con-stant at 600 F (316 C). The results from the present model are taken at the particular depth where the steam temperature was 600F (316C). [It was at-tained at 90-ft (30-m) depth at all times while in-jecting 20,000 lbm/hr (220 Mg/d) of 80% quality steam at 1,500 psia (10 MPa).] Since the present model considers two-dimensional heat flow, the overall formation temperature is lowered, leading to lower casing temperature. This is evident from Fig. 5, except at the outset when the heat front had not advanced far into the formation.

Effect of Injection Pressure. Steam injection pressure is determined primarily by depth and the formation face pressure to achieve the desired in-jection rate. Thus, well bore pressure drop is of concern. A number of parametric studies were carried out using the data given in Table 2. Fig. 6, for example, shows the relationship between the wellhead and the sand face pressures. Pressure loss increases as the injection pressure decreases. It is desirable to use a small-size tubing from the cost standpoint; however, the pressure drop may be prohibitive. For the conditions listed in Table 2, a 2Ys-in. (60-mm) tubing seems to be unsuitable for a 1,0OO-psia (7-MPa) injection pressure. A drop in

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lL. --This Model o 500 Q) - -- Leutwyler '-

::J

-ell '-Q) Q. E Q) ~ Cl c en

400

300

ell 200 ()

10 100 10000

Injection Time, hrs Fig. 5 - Comparison of results from this model with those

reported by Leutwyler for a tubing temperature of 600F (315C).

steam quality may lead to an increase in sandface pressure.

Fig. 7 shows the relationship between injection pressure and both the heat used and sandface steam quality. Heat used is defined as that fraction of the injected heat that reaches the sandface. It is evident that both of these parameters decrease as the in-jection pressure increases, An increase in steam pressure or temperature results in a higher heat loss -i.e., lower heat use leading to a decrease in steam quality. A larger tubing size is not desirable in view of the above. Effect of Injection Rate. Steam injection rate determines the fluid production rate and, thus, is an important variable. Injection rate strongly affects pressure changes during flow, as shown in Fig. 8. As one would expect, pressure at the sand face decreases

ell 'iii 2000 Q.

~ -0 0 0 (1)

-ell Q) 1500 '-

::J rJ) en Q) '-

a.. E ell Q) 1000 -(/)

1000 1500 2000 Injection Pressure, psia

Fig. 6 - Effect of injection pressure on steam pressure at 3,000 tt (914 m) for the conditions given in Table 2.

OCTOBER 1981

TABLE 2 - CONDITIONS USED IN PARAMETRIC STUDIES

Surface steam quality Injection rate, * Ibmlhr Injection pressure,. psia Injection time,. days Tubing 00, * in. Casing 00, in. Hole size, in. Emissivity of pipe surfaces Emissivity of aluminum paint Thermal conductivity of insulation (calcium silicate), Btu/tt-hr-oF

Thermal conductivity of cement, Btulft-hr- F

Thermal conductivity of the earth, Btu/tt-hr-OF

Thermal diffusivity of the earth, sq ftlO

Surface geothermal temperature, OF Geothermal gradient, FIft

0.8 20,000

1,500 10

2.875 7.000 9.625

0.9 0.4

0.04

0.55

1.4

0.96

.80.0 0.02

'Used if the value is not specified in the text. The annulus contains low-pressure air.

with an increase in rate and/or a decrease in the tubing size. A 2Ys-in, (60-mm) tubing is not ac-ceptable for most rates in Fig. 8.

Fig. 9 shows heat use and sand face quality as functions of injection rate. At a constant injection pressure, both heat use and steam quality increase as the injection rate increases, since heat loss decreases; in this instance, the smaller tubing size appears more desirable. Effect of Injection Time. Time enters into the present model by means of Eq. 14 and the depen-dence of the coefficients on the calculated tem-perature.

Fig. 10 shows the relationship between radial distance and temperature in the formation with time for a depth of 900 ft (275 m), a constant steam temperature of 600F (316C), and 2Ys-in. (73-mm)

>. 1.0

:!:: en ::J a 0.9 E ell Q) -

0.8 (/) c 0 ~ 0.7 ell N

..-

:::J ..- 0.6 ell Q) I

0.5

Heat Utilization 2 3/8"

2 7/8"

27/8"

1000 1500 2000 Injection Pressure, psia

Fig. 7 - Effect of injection pressure on fractional heat utilization and steam quality at 3,000 tt (914 m) for the conditions given in Table 2.

531

tubing. It is seen that the temperature at the cement/formation interface rises rapidly, but the radial propagation is slow. Fig. 11 shows tem-peratures at the casing exterior and at the cement/formation interface as functions of time at a depth of 3,000 ft (915 m). The increases in the two temperatures with time are noticeable during the early stages but become small at longer times. Effect of Tubing Insulation. Many methods have been proposed for reducing wellbore heat loss. Two of these - painting of the outside surface and in-sulation - are considered here. The beneficial effects

2000 C\l (/) a. ..,;-

-0 0 0 C'? eo Q) 1500 ....

:::J (/) (/) Q) ....

CL E C\l Q)

-(/) 1000

15000 20000 25000 Injection Rate, Ib/hr

Fig. 8 - Effect of injection rate on steam pressure at 3,000 ft (914 m).

10

Heat Utiltization 23/8" >--tU 0.9 27/8" :::J a E C\l Q) 0.8 -(/) c 23/8" 0 -C\l 0.7 N

-::::>

-C\l Q) 0.6 I

0.5~----~--------~--------~----~ 15000 20000 25000

Injection Rate, Ib/hr Fig. 9 - Effect of injection rate on fractional heat

utilization and steam quality at 3,000 ft (914 m). 532

of these schemes are evident from Fig. 12. A reduction in the emissivity of the tubing exterior

surface by painting (or other means) reduces the radiation heat transfer coefficient in the annulus, thus lowering the overall heat transfer coefficient. However, tubing insulation is more effective (and practical). Fig. 12 shows that aluminum paint reduces the value of Ute by a factor of two, reducing the heat loss by 380/0. When 0.5-in. (l3-mm) calcium silicate insulation is used, Ute is reduced by a factor of six and the heat loss is reduced by 75%. Ob-viously, a low heat loss helps to maintain a high

500.-------------------------------~

LL

ai 400 ....

:::J eo ....

Q) 300 a. E Q) I-c 200 a

:;:::; C\l E ....

a LL

100

t 2 5 10 20 50 100 Cement-Formation

Interface

Radial Distance from the Well bore Center, ft

Fig. 10 - Relationship between radial distance and for-mation temperature with time.

-l 500 Casing Surface

LL 0

ai 400 ....

:::J

-C\l ....

Q) a. 300 E Q) I-

200

10 100 1000 Injection Time, days

Fig. 11 - Effect of injection time on temperatures at casing exterior surface and cement/formation interface at 3,000 ft (914 m).

SOCIETY OF PETROLEUM ENGINEERS JOURNAL

steam quality and a lower casing temperature.

Geothermal Production Comparison With Previous Models. There are few published field data for comparing with the results from the present model for the case of geothermal production. Gould7 has reported limited data for validating the model developed. The pressure drop was calculated by use of the correlation proposed by Griffith and Wallis for bubble flow, Hagedorn and Brown for slug flow, and Turner and Ros for annular mist flow, based on field data. Fig. 13 provides a comparison of the field data with predictions from the present model. Flow was found to be in the slug flow region. Fig. 14 shows a comparison of steam quality for a number of Ute values as functions of depth calculated by Gould's model and the present

--.L: -Q. 0) 0

Or-----------------------------,

2000

3000

4000

5000

0

--Ordinary Tubing --- Painted Tubing --Insulated Tubing

0.2 0.4 0.6 Fractional Heat Loss

Steam Quality

0.8

Fig. 12 - Effect of insulation on fractional heat loss and steam quality.

LL

~ :;J

~ OJ c. E OJ r-E co OJ

Ci5

Fig.

500

450

400

0

350

o

----I --This Model I

,

0 Field Data i I

t 0

0

200 400 600 800 1000 1200 1400

Depth, ft

13 - Comparison of computed and field data (Wairakei 27 New Zealand) of production of 610,000 Ibm/hr (277 Mg/h) of 13.9% quality steam at 180 psia (1241 kPa) at wellhead through 8in. (20.32cm) inner casing.

OCTOBER 1981

model. The cases for Ute = 0.5 Btu/hr-sq ft- of (3 W 1m2. K) and for adiabatic flow considered by Gould are not simulated, since in reality they are unlikely and would require thorough insulation. Good agreement is evident for Ute = 1 and 2 Btu/hr-sq ft- of (6 and 12 W 1m2. K), but a sizable discrerancy was found for Ute = 5 Btulsq ft- of (30 W 1m . K). In this instance, a large heat loss caused by the unfavorable well completion simulated serves to reduce two-phase flow to single-phase flow of water in the middle part of the well.

Toronyi and Farouq Ali 18 examined pressure and quality changes for geothermal production using a wellbore model coupled with a geothermal reservoir. Choosing as an example the case where the initial steam quality at the sandface was more than 10070, the steam pressure and quality at the wellhead was calculated as 525 psia (3.6 MPa) and 12.2070. The present model gave values of 474 psi a (3.3 MPa) and 14.7070, respectively. The discrepancy is explained by the assumptions of zero slip and a single flow regime.

On the whole, it can be said that the present model was successful in predicting available field ob-servations and compares favorably with the other two models for geothermal production.

Conclusions The following conclusions can be derived from the results of this work.

1. Both the slip concept and flow regime are essential in the calculation of pressure drop in the well bore , considering the wide range of flow con-ditions possible in steam injection and geothermal wells.

--.L: i5. 0) 0

Fig.

Or---~r----.-------_.

1000

2000

\ \ -- This Model \ --- Gould

\ \ \ \ \ \ \

3000~---~---~---~-~3---~ o 0.05 0.10 0.15 0.20 0.25

Steam Quality 14 - Comparison of results from the present model

with those from Gould's model after 10 days of production of 10,000 Ibm/hr (4536 kg/h) of 20% quality steam at 1,200 psia (8274 kPa) at reservoir through 6in. (15.24cm) inner casing.

533

2. In the case of steam injection, pressure loss increases as the injection pressure decreases and/or as the tubing size increases and the injection rate increases.

3. An increase in injection pressure or tubing size and/or a decrease in the injection rate leads to a decrease in steam quality and the heat reaching a given depth. Tubing insulation and painting can result in a significant reduction in heat loss.

4. Good agreement between the published field data for steam injection as well as geothermal wells and the predictions of the present model serves to establish the validity of the model. Comparisons with the previous models show the importance of the flow regime prevailing in the wellbore.

Nomenclature A area, sq ft (m2) is steam quality, fraction

g acceleration due to gravity, ft/s2 (m/s2) gc gravitational constant, 32.17 ft-Ibm/lbf-s2

(9.80665 m/s2) h specific enthalpy, Btu/Ibm (kJ/kg)

hc combined heat transfer coefficient for convection and conduction, Btu/hr-sq ft- of (kW /m2 . K)

h r heat transfer coefficient for radiation, Btu/hr-sq ft- of (kW /m2 . K)

Jc mechanical equivalent of heat, 788 ft-IbflBtu (1,000 gc)

kh thermal conductivity, Btu/hr-ft-of (kW/mK)

nr number of grid points in the r direction nz number of grid points in the z direction p pressure, psia (Pa)

qs volumetric flow rate of steam, cu ft/s (m3/s)

Q heat loss rate to surroundings, Btu/hr-ft (kW/m)

r radius, ft (m) t time, hours (seconds)

T = temperature, of (K) Tf formation temperature, of

Ute overall heat transfer coefficient, Btu/hr-sq ft-oF (kW/m2 K)

v velocity, ft/s (m/s) w mass flow rate, Ibm/s (kg/s)

Wf irreducible friction losses z vertical coordinate, positive downward,

ft (m) ex thermal diffusivity of the earth,

sq ft/hr (m2/s) 'Y geothermal gradient, F/ft (K/m)

'Ya acceleration gradient, dimensionless p density, Ibm/cu ft (kg/m3)

7f friction loss gradient, Ibflsq ft-ft (Pa/m) Subscripts

ce cern

ins

534

casing exterior cement insulation

m mixture (steam/water) res reservoir

s dry and saturated steam te tubing exterior w saturated water

References 1. Leutwyler, K. Jr.: "Casing Temperature Studies in Steam

Injection Wells," J. Pet. Tech. (Sept. 1966) 1157-1162; Trans., AIME,237.

2. Holst, P.H. and Flock, D.L.: "Wellbore. Behavior During Saturated Steam Injection," J. Cdn. Pet. Tech. (Oct.-Dec. 1966) 184-193.

3. Earlougher, R.C. Jr.: "Some Practical Considerations in the Design of Steam Injection Wells," J. Pet. Tech. (Jan. 1969) 79-86; Trans., AIME, 246.

4. Pacheco, E.F. and Farouq Ali, S.M.: "Wellbore Heat Losses and Pressure Drop in Steam Injection," J. Pet. Tech. (Feb. 1972) 139-144.

5. Herrera, J.O., Birdwell, B.F., and Hanzlik, E.J.: "Wellbore Heat Losses in Deep Steam Injection Wells SI-B Zone, Cat Canyon Field," paper SPE 7117 presented at the SPE 1978 California Regional Meeting, San Francisco, April 12-14.

6. White, D.E., Muffler, L.J.P., and Truesdell, A.H.: "Vapor-Dominated Hydrothermal Systems Compared with Hot-Water Systems," Econ. Oeol. (Jan.-Feb. 1971) 75-97.

7. Gould, T.L.: "Vertical Two-Phase Steam-Water Flow in Geothermal Wells," J. Pet. Tech. (Aug. 1974) 833-842.

8. Gould, T.L., Tek, M. Rasin, and Katz, D.L.: "Two-Phase Flow Through Vertical, Inclined, or Curved Pipe," J. Pet. Tech. (Aug. 1974) 915-926; Trans., AIME, 257.

9. Chierici, G.L., Ciucci, G.M., and Sclocchi, G.: "Two-Phase Vertical Flow in Oil Wells - Prediction of Pressure Drop," J. Pet. Tech. (Aug. 1974) 927-938; Trans., AIME, 257.

10. Duns, H. Jr. and Ros, N.C.J.: "Vertical Flow of Gas and Liquid Mixtures in Wells," Proc., Sixth World Pet. Cong., Frankfurt (1961) Sec. II, Paper 22-PD6, 451-465.

11. Orkiszewski, J.: "Predicting Two-Phase Pressure Drops in Vertical Pipe," J. Pet. Tech. (June 1967) 829-838; Trans., AIME,240.

12. Farouq Ali, S.M.: Oil Recovery by Steam Injection, Producers Publishing Co. Inc., Bradford, PA (1970).

13. Willhite, G.P.: "Over-all Heat Transfer Coefficients in Steam and Hot Water Injection Wells," J. Pet. Tech. (May 1967) 607-615.

14. Sugiura, T.: "An Advanced Well bore Model for Vertical Two-Phase Flow of Steam and Water," MS thesis, Penn-sylvania State U., University Park, PA (1978).

IS. Bleakley, W.B.: "Here Are Case Histories of Two Thermal Projects," Oil and Oas J. (Oct. 26,1964) 123-130.

16. Ramey, H.J. Jr.: "Wellbore Heat Transmission," J. Pet. Tech. (April 1962) 427-435; Trans., AI ME, 225.

17. Carslaw, H.S. and Jaeger, J.c.: Conduction of Heat in Solids, Oxford U., London (1948).

18. Toronyi, R.M. and Farouq Ali, S.M.: "Two-Phase, Two-Dimensional Simulation of a Geothermal Reservoir and the Wellbore System," Soc. Pet. Eng. J. (June 1977) 171-183.

SI Metric Conversion Factors Btu x 1.055 056 E+OO kJ

OF CF - 32)/1.8 C ft x 3.048* E-Ol m

lll. x 2.54* E+OO cm Ibm x 4.535 924 E-Ol kg psia x 6.894 757 E+OO kPa sq ft x 9.290 304* E-02 m2

*Conversion factor is exact. SPEJ Original manuscript received in Society of Petroleum Engineers office Feb.

21.1980. Paper accepted for publication March 14. 1980. Revised manuscript received July 22, 1981. Paper (SPE 7966) first presented at the SPE 1979 California Regional Meeling, held in Ventura, April 1820.

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