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P TROL UM

Heat Losses During Flow of Steam Down a Wellbore

ABSTRACT

ABDUS SATTERJUNIOR MEM ER A/ME

Studies of wellbore heat transmission during the injec-tion of a hot fluid as either gas or liquid have appearedin the literature. The present investigation takes into ac-count the effect of condensation which is of practical sig-nificance when considering steam injection operations. Cal-culation procedures are given for superheated and for sat-urated and undersaturated steam.

Effects of injection mte time pressure temperature anddepth of a well on heat losses have been analyzed. Thebenefits of using a packer are discussed. Also presentedare correlations for estimating heat losses involved duringinjection of saturated steam.INTRODUCTION

Studies of wellbore heat losses during the injection ofa hot fluid have appeared in the literature. Ramey haspresented a useful solution for calculating temperature ofa heated gas as a function of depth and injection time.Ramey and others have also presented solutions for thehot liquid case.'-' No method is available to take into account the effect of condensation, which is of practical significance when considering steam injection operations. Thispaper presents a method of estimating the quality of acondensing fluid as a function of depth and time. Theapproach is basically the same as that used by Ramey.

The following injected fluids and conditions are considered in this study:1 Injection of superheated steamA. Decrease in temperature until cooled to the sat

urated stateB Condensation until condensed completely into hot(saturated) water

2. Injection of saturated or undersaturated steam; condensation until condensed completely into hot (saturated) water.

Although this investigation does not include cooling ofwater after steam is completely condensed, Ramey's solution for the hot liquid case can be used for this purpose.The effects of various process variables on heat lossesduring the injection of steam into a well are investigated.

Original manuscript received in Society of Petroleum Engineers of -fice Nov. 18. 1964. Revised manuscript received May 11. 1965. PaperSPE 1071) presented at SPE Production Research Symposium held inTulsa. May 3-4. 1965.lReferences given at end of paperDiscussion of this and all following technical papers is invited. Dis-cussion in writing three copies) may be sent to the office of the Jour-nal of Petroleum Technology. Any discussion offered after Dec. 31. 1965.should be in the form of a new paper. No discussion should exceed 10per cent of the manuscript being discussed

JU LY . 1965

PAN AMERICAN PETROLEUM CORP.TULSA, OKLA.

THEORYQUALITY OF CONDENSING STEAMPredicting the behavior of condensing steam flowingdown the injection well bore requires an estimate of thequality of steam, i.e., the mass fraction of vapor in themixture. Let us divide the total depth of the injection wellinto several segments which might be, but do not have tobe, of equal magnitude. Consider condensation within agiven depth interval 6.Z, the bottom of which is locatedat a depth Z from the surface Fig 1). The followingequation, derived in the Appendix, relates the quality ofsteam at the bottom of an interval to that at the top ofthe interval:

y [Z,t] = (6.Z),y [ (Z-6.Z), / ] + a -+ [A_ _ . : : + - a _ ~ Z _ - _ ~ ~ l T ~ L ~ ~ _A ( I )where

2)()B = ----- '---778g,L v 3)

See the nomenclature for symbol definitions.The major assumptions made in the solution are asfollows:1 Steam is injected at a constant rate, wellhead pressure, temperature and quality.2. A down-hole packer is used to prevent steam fromentering the tubing-casing annulus. The annulus is assumedto be filled with air at low pressure.3. The heat transfer in the wellbore is under steady-

Te =Tml------ - joIr STEAM

CASE OF CASE OFCONDENSING COOLINGI y[(Z-llZ).t] T[(Z-llZ),t].'''.'''''''-.''1_:_ ..I--+--+-+---i ,'+ i i I L

Z Y(Z,t) T(Z,t)FIG. I S C H D I A T I C DIAGRAM OF WELLBORE HEAT PROBU:;\f.

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heat transfer to the earth involvesconduction.

4. Kinetic energy changes are negligible.5. Any variation in pressure of the steam with depthto hydrostatic effects and frictional losses is negligible.6. There is negligible variation in thermal conductivitydiffusivity of the earth with depth.While friction and gravity could have opposing effects

the fluid pressure (the pressure at any point in thel bore would be decreased by friction, but increased bynot likely' to cancel each other. In aand/or in the case of a lower injection rate.assumption of negligible variation in pressure withis quite reasonable. In the case of a deep well and/or

higher injection rate, this assumption might introduceerror in the present method of calculation.

OF A COOLINGRamey s equation describing the temperature behaviorinjected gas as it moves down the well bore IS

T[Z.t] = aZ+T , -aA-AB

A

+ (1 , -- T , +aA +AB)e'z/'.

LC [k ,+r,Uf(t)]271'r,Uk f

B = g .77Sg,C,

(4)

(5)

(6)basic assumptions made in this solution are as follows,

1. The fluid is a perfect gas.2. The fluid does not undergo any change of phase inwellbore.3. The rate of fluid injection is constant with a con

temperature at the wellhead.4. Any variation in thermal and physical properties ofearth and wellbore fluids is negligible.5. The heat transfer in the wellbore is under steady

heat flow to the earth is unsteadyconduction.

6. Kinetic energy changes and frictional losses are negIn the present investigation, changes in the over-all heatcoefficient with temperature (or depth) are con

on a depth-step basis (similar to thatthe condensing steam case) is, therefore, suggested.

the present approach, it is assumed that the thermalphysical properties of steam are constant within adepth interval. These properties could, however,from one depth interval to another.The differential equation from which Eq. 4 was derived

on a depth-step basis similar to the condensingcase, and the following equation relating the temperof the steam at the bottom of a given depth intervalthat at the top is obtained.

T[Z,t] = aZ+T ,-aA-AB+{T[(Z- t lZ) , tJ-T , -a (Z- t lZ)+aA+AB} e-AZ / . 7)

1ft)The time function f(t) describing the transient heat

from an infinitely long cylinder has been preby Ramey:

t t) = f - - ~ ),- 8)816

Ramey's graphical solution of the above equation forvarious boundary conditions can be used for estimatingf(t) at the end of a given injection time.WELLBORE THERMAL RESISTANCE

For the heat losses during injection of steam throughthe tubing, with air filling the annulus between the tubingand the casing, the following hea t-transfer coefficients havebeen considered:1. Convective heat-transfer coefficient between the injected fluid and the tubing surface;

2. Convective heat-transfer coefficient between the tubing and the casing through the medium of air; and3. Radiation heat-transfer coefficient between the tubing and the casing.In addition, the thermal resistance of the tubing wall

has been taken into account, while the resistance of thecasing wall has been neglected. The over-all heat-transfercoefficient using an arithmetic average is given by. ... = _ _ + (r, '-r,) + ___ .U V, k , U,+ Ur (9)

The heat-transfer coefficients can be found from heattransfer correlations presented in the literature. McAdams'has presented methods applicable for estimating the convective heat-transfer coefficient between the fluid and tubing for cooling and condensation of steam. The radiationheat-transfer coefficient between the tubing and the casing can be estimated by using a correlation also presentedby McAdams. The heat-transfer coefficient for natural convection across a layer of fluid (air) between two concentric cylinders has been reported by Fishenden." The overall heat-transfer coefficient given by Eq. 9 is dependentmainly upon the radiation coefficient.CALCULATION OF PER CENT HEAT LOSS

Considering the change in enthalpy and the change inpotential energy, the following equations can be used tocalculate the instantaneous rate of heat loss as a per centof the wellhead heat input.

Superheated Steam at the SurfaceLoss of Heat Through Cooling:

[H - H [Z,t] + 7gZ X 1000/ 7Sg . (10)7 Heat Loss = .H,-(HwhmLoss of Heat Through Condensation:

[H,-{Hw+Y[Z,t1L.}+ 7 ; ~ J X100% Heat Loss = .0..... = ~ : : : ~H, - Hwh mSaturated or Undersaturated Steam

Loss of Heat Through Condensation:[{Y,-Y[Z,t]}Lv+ gsZ ]X100. H L 77 g,% eat oss = H L _ )w+Y, v (Hw Tm

(11 )

(12)In these equations, the heat input is taken as the amount

of heat required to generate steam (at wellhead conditions) from water at the mean surface temperature of theearth.

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CALCULAn ON PROCEDUREA computer program was written for and executed onan IBM 704 computer. The program was composed of twomain parts. The first involved the calculation of temperature distribution in the case of cooling of a superheatedsteam, and the second was concerned with the quality distribution in the case of condensation of steam. The calculation procedure was on a depth-step basis at a given injection time. Several iterative calculations were required

for each depth step. For example, in the case of superheated steam being cooled, the temperature of the steamat the bottom of the depth interval and an average casingtemperature within tohe interval were assumed. The averagetubing temperature in the section was estimated by thearithmetic mean of the steam temperature at the top(known from the preceding depth step) and the bottom ofthe interval (assumed). The average air temperature wasalso estimated by the arithmetic mean of the tubing andcasing temperatures. Then, the over-all heat-transfer coefficient around the wellbore could be calculated. Finally,knowing the time function, the steam temperature at thebottom of the interval and the average casing temperature'Nithin the section were estimated. The calculated valueswere then compared with the assumed values. n case ofdiscrepancy, the calculated values were used as the nexttrial values. When the condition of steam was determinedthe instantaneous rate of heat loss as a per cent of the w l l ~head heat input was calculated. Several subroutines wereused for calculating the time function, over-all heat-transfercoefficient, and for finding the thermal and physical properties of steam (in the superheated and saturated region)and air from their tabulated values.

RESULTS AND DISCUSSIONSFA C TOR S AFFEC TI N G H E A T LOSSESAssuming that the geothermal properties and casing andtubing sizes are constant, the major factors affecting heatlosses are (1) injection time, (2) injection rate, (3)depth of the injection well, and (4) injection pressure andtemperature in the case of superheated steam or injectionpressure in the case of saturated steam.

Figs. 2 through 7 demonstrate the effects of these factors. The selected values of geothermal properties and casing and tubing sizes used to obtain the results shown onthese figures are listed in Table 1. These values are representative of an average situation.Fig. 2 shows the variation of temperature and qualitywith depth and injection time while injecting 5,000 Ib/hrof superheated steam at 500 psia and 1,000F. The temperature and quality distribution varies most rapidly atearly times, but later in the life only small changes occurwith time. The point of condensation, Le., the depth atwhich condensation first occurs, moves downward rapidlywith time during the early life of injection. At greatertimes, the condensation point moves rather slowly downward. Steam could travel to about 1,100 ft withoutcondensation at the end of several hours of injection. Onthe other hand, even at the end of 10 years of injection,steam would be condensing beyond the depth of about3,100 ft. For the same injection conditions as used in Fig.

2, Fig. 3 shows the variation of heat losses with depth andinjection time, and Fig. 4 demonstrates the variation ofheat losses with time to a depth of 4,000 ft. As expected,the variation of heat losses with time is most rapid forearly times, but very slow thereafter.Fig. 5 shows heat loss as a function of injection tem

perature and depth at the end of one year (for an injectionJ U I . Y 1 9 6 5

TABLE 1-PROCESS PARAMETERS1 Mean earth surface temperature2. Geothermal gradient3 Thermal conductivity of earth4 Thermal diffusivity of earth5 Inside diameter of casing6 Outside diameter of casing7 nside diameter of tubing8 Outside diameter of tubing

75F0 011 F ft1.0 Btu /hr.ft F0.046 sq It hr5.989 in.6.625 in.2.441 in.2.875 in.

rate of 5,000 lb/hr of steam at 500 psia). The heat lossper p o u ~ of steam is higher with higher injection temperatures, which suggests an advantage in injecting saturatedsteam rather tohan superheated steam. I t should be pointedout, however, that even though heat losses are higher insuperheated steam, the amount of initial heat content isalso higher than that of the saturated steam.

One advantage which superheated steam has over saturated steam is that superheated steam might reach thebottom of a well without condensation. For example, forthe conditions illustrated by Fig. 5, the superheated steaminjected at 1,000F first starts to condense at a depth of2,512 ft, while saturated steam begins to condense as soonas it enters the well.Fig. 6 sh?ws how heat loss increases with injection pressure and With depth (at the end of one year of injection)for the case of injecting 5,000 lb/hr of superheated steam

at 1,000F. The heat loss is more pronounced in the condensing steam region for higher pressures. This is becausehigher pressures have correspondingly higher saturationtemperatures, and thus greater temperature gradientsaround the wellbore. Also, at a higher pressure the pointof condensation is encountered higher in the well than inthe case of a lower pressure because the degrees of superheat decrease with increasing pressure. This effect is morepronounced in the pressure range of 100 to 1,000 psia.

1000

II2000Wwll .. 3000-Ia..w0

IIi wzw1 - 0 : :z lo

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Fig. 6 shows that hot water (saturated) is encountered-at a depth of 5,240 ft in the case of 2,500-psia pressure.Beyond this point, the injected fluid would travel as liquiddecreasing in temperature.Fig. 7 indicates the variation of heat losses with injection rate and depth. I t is assumed that saturated steam at500 psia and 467F is injected for a period of one year.At a given injection rate, the heat loss is almost directlyproportional to the depth. This is to be expected, becausein the case of condensing steam, the temperature of steamis essentially constant and, hence, the thermal gradientaround the well bore varies only slightly with depth for agiven injection time. At a given depth, the heat loss as apercentage of the input at the wellhead is inversely proportional to the injection rate, provided other factors areconstant. In the sample graph, hot water (saturated) wasencountered before the maximum depth of 6,000 ft wasreached in the case of the injection rate of 1,000 Ib/hr.70.

f- 60'

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tion, but decreases with the increasing injection time. Inthe case of a huff-and-puff steam operation where thesteam injection period lasts only for several days or weeks,a substantial amount of the injected heat can be conservedif a down-hole packer is used.Another benefit of using a packer is illustrated in Fig. 9,which shows how the casing temperature at the bottom ofthe well changes with the injection time when a packer isused. The casing temperature rises rapidly at early times.

u.:owI

CASING TEMPERATURE FOR CASE (A)

l3 300 - I _I

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lation of the hot water point. Requirements for using thiscorrelation are the same as those for using the heat-lossA sample calculation to illustrate the use of Figs. 10

and 11 is presented in the Appendix. It should be emphasized again that the correlation presented in Figs. 10 and11 can be used only for first approximations. For accurateresults, calculations should be made for a specific case,particularly where conditions are substantially differentfrom those on which the correlation is based.

Since heat-loss correlations in the case of injecting superheated steam are more complicated, and very little, ifany, gain in thermal efficiency can be obtained by injecting superheated steam compared to saturated steam, nocorrelation is presented for the latter case.Ramey achieved a good agreement between computedwellbore temperatures and those measured in gas and waterinjection wells. I t would also be desirable to verify computed wellbore qualities in actual field data from steaminjection wells. Unfortunately, no device is available atpresent to measure steam quality down a wellbore. Therefore, the results of this investigation are subject to verification in the field.

CONCLUSIONSAn approximate solution can be developed for estimating the quality of a condensing fluid while flowing down awellbore.Wellbore heat losses and casing temperatures are primarily affected by injection rate, time, pressure, temperature, quality, depth and type of injection well completion.Significant benefits can be obtained by using a packer.Correlations presented in this work might be used for estimating wellbore heat losses during the injection of saturated steam.

ACKNOWLEDGMENTThe author wishes to express his appreciation to J. D.Neil of the Pan American Petroleum Corp. for his valuable assistance in preparing the computer program.

NOMENCLATUREA = function defined by Eq. 5, ftA = function defined by Eq. 2, ft-OFa = geothermal gradient, OF ftB = constant defined by Eq. 6, F/ftB = constant defined by Eq. 3, ft- Co = specific heat of gas, Btu/lb-oF

f(t) = transient heat conduction time function forearth, dimensionless, see Ref. 1g = acceleration due to gravity, 4.17 X 10 ft/hI

g, = conversion factor, 4.17 X lOs Ibm-ft/lb,-hIH = enthalpy, Btu/lbH[Z,t] = enthalpy at a given depth and injection timeH w = specific enthalpy of saturated water at the injection pressure, Btu/lbHi = specific enthalpy of steam at the initial pressure and temperature, Btu/lbLv = latent heat of steam at the injection pressure,

Btu/lbH ,hm = specific enthalpy of water at T m Btu/lb

850

kh = thermal conductivity of earth, Btu/hr-ft-OFk , = thermal conductivity of tubing material,Btu/hr-ft-OFi = steam injection rate, lb/hr

Q = heat-transfer rate, Btu/hr1 , = inside radius of tubing, ftr. = outside radius of tubing, ftr/ = outside radius of casing, ftT, = temperature of casing, OFT, = temperature of earth, OFT = surface temperature of injected fluid, OFT , = mean surface temperature of earth, OFT = saturation temperature of steam at a givenpressure, OF

T[Z,t] = depth- and time-dependent temperature of theinjected fluid, OFt = time from the sta rt of injection, hrV = over-all heat-transfer coefficient, Btu/hr-ff-oFVI = convective heat-transfer coefficient betweenthe fluid inside the tubing and the tubing surface, Btu/hr-ft2-oFV, = radiation heat-transfer coefficient between thetubing and the casing, Btu/hr-ft2-oFV 2 = convective heat-transfer coefficient betweenthe tubing and the casing through the medium of air, Btu/hr-ff-oF

y = quality, i.e. mass fraction of vapor in themixture, dimensionlessy [Z t] = depth- and time-dependent quality of the in-jected fluid

y = quality of steam at the wellheadZ = depth below surface, ftt>Z = depth interval, fta = thermal diffusivity of earth, ff hr

REFERENCES1. Ramey, H. J., Jr.: Wellbore Heat Transmission , Jour. Pel.T ech. (April, 1962) 427.2. Moss, 1. T. and White, P. D.: How to Calculate Temperature Profiles in a Water Injection Well , Oil and Gas Jour.(March 9, 1959) 57, No. II, 174.3. Squier, D. P., Smith, D. D. and Dougherty, E. L.: Calculated

Temperature Behavior of Hot Water Injection Wells , Jour.Pet. Tech. (April, 1962) 436.4. Fokeev, V. M. and Kapyrin, Yu. V.: Evaluation of HeatLosses Along the Wellbore and the Effect of Injecting LargeQuantities of Water on the Temperature Gradient in the Romashkino Field , Neft. Khoz. (1961) 39, No. 12,33.5. McAdams, W. H.: Heat Transmission, 2nd Ed., McGrawHillBook 'Co., Inc., New York (1942) 63, 168 and 268.6. Fishenden, M. and Saunders, O. A.: An Introduction to HeatTransfer, 1st Ed., Oxford U. Press, London (1950) 104.7. Keenan, J. H. and Keyes, F. G.: Thermodynamic Properties ofSteam, John Wiley and Son New York (1947).

PPENDIXDERIVATION OF AN EXPRESSIONFOR THE QUALITY OF CONDENSABLE

STEAM FLOWING THROUGH A WELLBORELet us consider condensation of steam within a depthinterval t>Z the bottom of the interval being located at adepth Z from the surface (see Fig. 1). By applying thelaw of conservation of energy to a differential element offluid height dx within the depth interval, the followingequation can be obtained:

i g .- i. ,dH+ ~ g c dx = dQ (A-I)The specific enthalpy of a mixture of vapor and liquidat a given saturation pressure is

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Neglecting the change in pressure down the hole due tofriction and gravity, the differential change in enthalpy ofa condensing fluid is thendB = Lvdy (A-3)

Substituting Eq. A-3 in Eq. A-I, the following equationis obtained:

- i ,Lvdy+ 7 ~ : g , dx=dQ (A-4)Ramey s approach is then followed to evaluate dQ as follows:

dQ = heat lost by the fluid= heat transferred to the casing= heat transferred to the surrounding formations.Heat transferred to casing is

dQ = 2-rrr,U (T , -T , ) dx (A-5)Heat transferred by conduction from the casing to thesurrounding formation is

(A-6)Combining Eqs. A-5 and A-6, the equation for casingtemperature is

T, = k fT,+T,r,Uf(t)r,Uf(t) +k f A-7)

Assuming that geothermal temperature changes linearlywith depth,T, = T .'+ax,where

Tm' = T ,+a(Z-.e:.Z) .(A-8)A-9)

Combining Eqs. A-4, A-5 and substituting Eqs. A-7and A-8, the following partial differential equation is obtained:

y T, A'B' T ,' axx 7 - A' = o.

whereA =

JUL Y 1 9 6 5

u, [k f+r,Uf (t)]2-rrr,Uk i

A-lO)

2)

B = -=-:=c::-I{---::c_7781{,L, 3)Since pressure drop is neglected, the temperature of thecondensing steam can be considered constant. The solutionof the above differential equation is then given by:

ax' (A B T ,' - T')y = 2A' A' x e(t) . A-ll)The function c(t) can be evaluated from the condition

that y = y [(Z - e:.Z), t] at x = 0, i.e., at the top of theintervalc(t) = y[(Z - e:.Z), t] A-I2)

The final expression relating to the quality of steamat the bottom of an interval e: z to that at the top of theinterval is then given bya(6.z)'y[Z,t] = y[(Z - 6Z),t] 2A'

[A'B' T,,, a(Z- 6Z) - T ] ~ ZA'

EXAMPLE CALCULATION1)

Calculate the approximate heat loss in the case of injecting steam through tubing (annulus is filled with air) at theend of one year of injection. The following data areavailable: injection rate, 3,000 lb/hr of saturated steam;injection pressure, 1,000 psia; and maximum depth ofwell, 3,000 ft.SOLUTION PROCEDUREAssume that the casing and tubing sizes and geothermal properties are essentially the same as those used forpreparing Figs. 10 and 11.

Step i -Check the depth of hot water point. From Fig.11, the hot water point is at a depth of 4,370 ft. Therefore, the maximum depth of 3,000 ft can be reached bythe condensing steam without its being completely condensed.Step 2 ~ F i n d the heat loss. From Fig. 10, find heat lossin per cent of input heat/IOO ft. Depth is 1.35 per cent.Therefore, heat loss at a depth of 3,000 ft is 40.5 per cent.

Note that the correlation of Fig. 10 is not valid beyondthe depth of the hot water point.

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