Fluid Temperature Distributation in Oil & Gas Wells - Engineering Essay


Fluid Temperature Distributation in Oil & Gas Wells - Engineering Essay
Predicting accurate temperature Profiles in flowing wells can improve the design of production facilities in petroleum engineering. Temperature Profiles in wellbore have application in cementing operations, accurate 2-phase

flow pressure drop prediction, Gas lift designs. Gas lift design can be enhanced by more accurate prediction of temperature at valve depth. In this way, the valves dome pressure can be set more accurately thereby improving the predictability of valve thorough put.

Existing temperature correlations are often inaccurate because they do not consider the effects of different fluids in the annullus and cooling and heating of the fluid resulting from phase change. Rigorous theoretical model are often complex and in convenient. They depend on many variables and require information about fluid composition.

This project work describes a method of predicting temperature distribution in a flowing wells. A model is derived from the steady-state energy equation that considers the heat transfer mechanisms found in the wellbore. An extensive data bank of temperatures from 3 wells was used in the model validation.

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TABLE OF CONTENTS

Page
Title page… . …. … … …. … … …. ….
Certification …. …. … ….. ….. …. …. …. ….
Dedication … …. …. …. … ….. …. …. ….
Acknowledgement … …. …. … … … ….
Abstract … … …. …. …. …. …. …. ….
CHAPTER ONE
Introduction … …. …. …. …. …. …. …..
CHAPTER TWO
Literature Review …. …. …. …. …. …. ….. ….
CHAPTER THREE
Theoretical Background and model development …. ….------------------------------------------------------------

CHAPTER FOUR
Result of Model Validation
CHAPTER FIVE
Analysis of Result

Conclusion
Recommendation
References
Nomenclature
Appendix A
Appendix B
Appendix C: Computer Programs and Output results

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CHAPTER ONE

INTRODUCTION

Heat loss from wellbore fluid depends on the formation temperature distribution. Fluid temperature distribution in wellbore is determined by rate of heat loss from the wellbore to the surrounding formation, which is a function of depth and production/injection time.

A model to predict fluid temperature during a steady-state 2-phase flow incorporates a thermal diffusivity solution. Convective and conductive effect is also incorporated in this solution. Ramey and Edwardson et al were the first to present theoretical model for estimating fluid Temperature as a function of depth and producing time. This model is only applicable to single phase fluids flow-because kinetic energy, friction and Joule-Thompson expansion were neglected.

Ramey’s work was improved by incoporating the effect of phase charge in fluid injection well. A presentation of excellent model on various resistances to heat transfer between the wellbore and the formation was also constructed by Ramey.

An expression for fluid temperature distribution in single-phase flow has the limitation of being applied when multiphase flow is encountered.

Interpretation of temperature logs for estimating water and gas injection profile was proposed. Noting, the usefulness, kirkpatrick presented a flowing gradient chart, though simple, lacked generality and accuracy.
This inaccuracy and thermal stress failure of casing in steam-injection wellbores, demands a proper understanding of wellbore heat transfer and accurate estimation of flowing fluid temperature.

Procedure for estimating wellbore fluid temperature has been suggested, model has been presented as the function of depth and producing time.
The problem, however remains that since Kinetic energy and friction was neglected, is therefore only applicable for single-phase flow.

Also, the assumption by Ramey of microscopic well radius in solving the temperature formation temperature distribution cannot be defended.

Resistances to heat transfer between wellbore fluid and formation, was ignored and consequently rendering the model inaccurate and stream lined.

Even qualitative estimation of flow rate from various producing wells depends on establishment of constant temperature different between the wellbore fluid and the formation. Ramey’s method is of limited use especially for estimating flow rate in multiple zones.
Fluid temperature distribution in wellbore is very crucial and can be used to estimated flow rate and Gas top Oil ratio in vertical and deviated wellbores.
In short a higher flow rate results in a lower temperature drop. A less complex algorithm which avoid complex calculation of overall thermal coefficient has been presented for estimating wellbore fluid temperature distribution which will in -coporate conductive, convective and radiative heat transfer and can be used predict temperature at various depths and production time.

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CHAPTER TWO
LITERATURE REVIEW

In recent years, considerable attention has been given to fluid temperature distribution in wellbores. There has been many studies on this subject ranging from heat transfer during two- phase flow in wellbores (Formation Temperature) to wellbore fluid Temperature. And later proposing a unified model for predicting flowing Temperature Distribution in Wellbores.

Various aspects of heat transfer between the wellbore fluid and the formation has been studied by many over the last few decades.

The usefulness of fluid Temperature measurement was realized as early as 1937 by Schlumberger et al ; Interpretation of temperature log for estimating gas and water injection profiles was proposed by Nowak2 and Bird3 in the early 1950s. Noting the importance, KirkPatrick4 Presented flowing Temperature gradient chart, though simple, lacked generality and accuracy . These deficiency and thermal stress failure of casing in steam injection wells emphasized the importance of proper understanding of wellbore heat transfer and accurate estimation of flowing fluid Temperature.

Lesem et al5 and Moss and White6 were the first to suggest procedures for estimating wellbore fluid temperature. However, Ramey7 and Edwardson et al8 were the first present a theoretical model for estimating fluid temperature as a function of producing time and depth. However, both works neglect the effect of friction and kinetic energy. Thereby making them only applicable for flow of single phase fluids. In addition, the assumption of in -finitessimal well radius by Ramey in solving the formation temperature distribution can be unreliable in some cases. The classical method for temperature prediction in wellbore proposed by Ramey coupled the heat transfer mechanisms in the wellbore and transient thermal behaviour of the formation.

Temperature equations for the case of injection of either single phase incompressible hot liquid or single phase ideal gas flow were derived.

In Ramey’s method, the transient thermal behaviour of the formation was determined by solutions of radial heat conduction problem in an infinite cylinder.

The resistances to heat flow in the wellbore due to presence of tubing wall and cement is incoporated in the overall heat transfer coefficient.

Griston and Willhite9 extended the application of Ramey’s approach by evaluating the usefulness of steam injection well and taking note of radiative heat transfer during steam injection.

Witterholt and Tixier10 employed the influence of fluid flow rate on temperature of the fluid in Ramey’s equation to measure the fluid temperature.

Witterholt and Curtis also employed effect of qualitative estimation of flow rate from various producing zones. The method depends on the effect of fluid flow on the inverse Relaxation distance, A, in Ramey’s model for fluid temperature Distribution. This method is of limited use , especially for estimating flow rates from multiple zones, because of its limited accuracy and its dependence on the establishment of constant temperature difference between wellbore fluid and formation.
An empirical calculation was developed by Shiu and Beggs for producing wells, to determine the relaxation distance defined by Ramey. This method is actually an attempt to avoid complex calculation of overall heat transfer coefficient and transient heat transfer behaviour of the formation .

Since application of Ramey’s equation are restricted to single phase flow in the wellbore, sagar et al extended Ramey’s method for wells with multiphase flow, accounting for kinetic energy effects and Joule-Thompson’s expansion.
This simplified method is based on field data., all these methods include severe assumption related to the thermodynamic behaviour of the flowing fluids and thus applicable only for limited operating conditions.
Alves et al12 presented a unified model for flowing Temperature prediction which can also be applied to producing and injection wells, under single-phase or two-phase gas liquid conditions over the entire inclination angle range from horizontal to vertical wellbores:
the diverse application of the aspects of heat transfer in both formation and wellbore necessitated the development of a Rigorous approach.

The model below allows computation of temperature at the formation/wellbore interface, when undisturbed formation temperature and wellbore heat flow rate are input. This represents the relationship between wellbore heat loses and wellbore temperature for steady-state, 2-phase flow:

where I -
wellbore fluid Temperature: Expression for variation of fluid temperature with depth has been obtained:

where
using appropriate boundary conditions on the above, Hassan and kabir13 modeled the expression as a function of well depth and also producing time.

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CHAPTER THREE
THEORETICAL BACKGROUND AND MODEL DEVELOPMENT

Heat loss experience by the fluid as if flows up the well results in lowering its temperature. As this fluids moves through the wellbore, there is transfer of heat between fluid and the earth due to the difference between fluid and geothermal temperatures.

This type of heat transmission is involved in drilling and production operations. This solution would assure that heat transfer to the earth will be unsteady radial conduction and that in the wellbore will be steady state.

In other to derive a model which predict heat flow in a 2-phase system (e.g Oil well), it would be necessary to obtain, first, the wellbore fluid energy balance.

The temperature difference between the formation and wellbore fluid causes a transfer of heat from the fluid to the surrounding formation with decrease in fluid temperature and depth. At any depth formation temperature would vary with radial distance and production time. However, when steady-state flow has been attained, there is constant fluid temperature at any given depth due to turbulence.

Heat loss from fluid, therefore declines with time and is dependent on various resistances to heat flow between the hot fluid in the tubing and the surrounding earth.

To derive a model for temperature as a function of depth and time, we must as necessity establish the formation temperature distribution as a function of radial distance and time, given a constant heat flux from the well.

For a 2-phase system, first obtained the wellbore fluid energy balance, which would relate the fluid temperature with wellbore/ formation interface temperature and the heat flux, given the overall heat transfer coefficient in terms of well configuration shown in Appendix A.

3.1 FORMATION TEMPERATURE DISTRIBUTION
Assuming symmetry around the heat source or sink (Producing or injection well); In a short time-step, heat flux from wellbore may be assured to remain constant. Then an energy balance of the formation leads to the following partial differential equation derived in cylindrical coordinates for formation temperature variation with radial distance from well and production time:
(1)
where: Te – earth temperature; t - time, r –distance measured from the wellbore center is thermal diffusivity. Initial condition is that formation temperature at any depth is constant, leading to :
Lim Te = Tei (2)

At the infinite or outer boundary, formation is also constant with radial distance.
(3)
The third boundary condition is derived from the heat flow rate at the wellbore/formation interface which is governed by Fourier Law of heat conduction. Heat flow rate per unit mass of wellbore fluid per unit length of well, dq/dz is given by:
(4)
rwb- outer radius of wellbore and W is the wellbore fluid mass flow rate. To facilitate solution and have a more general applicability of solution, equations 1,3 and 4 are recast in dimensionless variables of rD (dimensionless radial distance)= r/rwb, thermal diffusivity = Ke/PeCe , and tD (dimensionless time =. Equations 1,3 and 4 becomes:
(5)
(6)
(7)
I can now introduce a dimensionless temperature, TD, analogous to dimensionless pressure in pressure transient analysis:
(8)

3.2 WELLBORE FLUID ENERGY BALANCE
An energy balance of fluid for a differential length, dz, for a 2-phase system would lead to the following equation:
(9)
where gc and J represent appropriate conversion factors (gc= 32.2 lbm.ft/lbf-Sec2, unity in S.I units, dimensionless.) H-fluid enthalpy fluid enthalpy, H, depends on its pressure and temperature, which allows us to write the following expression, for dH/dz:
= - (10)
Cpm is the heat capacity at constant pressure, Cpm where CJ is the Joule- Thompson coefficient.
Combining equation (9) and (10), yields:
(11)
the radial heat transfer between the fluid and the surrounding earth, expressed in overall heat transfer coefficient based on transport phenomena and transient heat transfer. Heat is transferred from the wellbore fluid to the earth overcoming the resistance offered by the tubing wall, tubing insulation, casing wall and cement as shown in fig1. The resistances are in series, and, except for the annulus, the only mechanism of transfer involved is conductive transfer. At steady state, the rate of heat flow through a wellbore per unit length of well, dq/dz, can be expressed:
(12)
uto¬- Overall heat transfer coefficient based on tubing –inside surface area, 2 to and temperature difference between the wellbore fluid and wellbore/ formation interface (Tf-Twb).
Therefore, the heat transfer rate per total mass flow rate W is:
(13)
the overall heat transfer coefficient based on tubing outside surface area, uto, depends on resistance. In general, the resistance to heat flow through tubing or casing metal may be neglected.

Natural convection is the dominant heat transfer mechanism for fluid in the annulus. Resistance through cement layer could be important depending on its thickness. Using the dimensionless temperature, TD obtained in equation (8), we may write the expression for heat transfer from the wellbore/formation interface to the earth:
(14)

combining (13) and (14) to eliminate wellbore temperature, Twb, yields:
(15)
3.3 WELLBORE FLUID TEMPERATURE
The next is to obtain an expression for variation of fluid temperature with well depth by substituting the expression for dq/dz from equation (15) into (11):
(16)
letting, (17)
expressing (16) in terms of A, yields:
(18)
if the undisturbed formation temperature is assumed to vary directly with depth, thus,
Tei= Teibh - gTZ (19)
Where gT represents the geothermal gradient and Teibh is the undisturbed (Static) formation temperature at the bottom hole:

Equation (19) can also be applied when different geologic formation are encountered. At various depths with differing values of geothermal gradient. In this case, the computation may be divided into a number of zones with constant geothermal gradient being applied to each zone.
If it is assured that the test two terms in equation (18) does not vary with well depth, equation (18) becomes a linear differential equation: (20)
Where (21)
Equation (20) can be integrated for a constant, A, and boundary conditions:
(a) For a producing well at bottom hole, Z=Zbh
(b) Fluid and earth temperatures are generally known Tf =Tfbh and Tei =Teibh giving the expression for fluid temperature as a function of well depth and production time:
(22)
the value of the parameter, ,in equation(22) would depend on a number of variables, such as flow rates, gas/liquid ratio, wellhead pressure.
The calculation of is shown in Appendix A. The earth temperature at wellhead can be assumed to be equal to the surface fluid temperature (i.e. Tewh =Tfwh).
The geothermal gradient, gT can be determined by dividing the temperature drop of the formation and the fluid by the measured depth (i.e(Tformation –Tsurface )/depth acceleration due to gravity, g and the appropriate conversion factors, g both has the value of 32ft/sec2 and 32.17 lbf-ft/1bm-sec2 J and Cpm are respectively the mechanical equivalent of heat, 778ft-1bf/Btu and specific heat capacity at constant pressure.
=900 for vertical wells. It must be noted that the variable depths are measured in negative values which is a generally accepted convention.
Having set these constraints: equation (22) can further be simplified to yield:

simply:
(23)
j=1,2,3,…..
J, mechanical equivalent of heat, has a value of 778ft-1bf/Btu and A is the thermal Relaxation distance, in feet.
From equation (17) shown below:
Aj =
The overall thermal coefficient Uto can be computed using the model below ;for differential length and a unit mass flow rate; for tubing:
(24)
and for casing flow;
The equation is shown in appendix (B) The inverse relaxation distance, A, can be estimated using equation (17) but bearing in mind the existence of equality between dimensionless temperature, TD and transient heat conduction time function for the formation (earth) developed by Ramey:
(25)
is the thermal diffusivity of the earth, ke/CePe, ke, Ce and pe are the formation conductivity, heat capacity and earth (formation) density. And rto , the tubing –inside radius.
Using equation (23), the wellbore fluid temperature, Tf can be estimated. The working is shown in Appendix A.
Shown in Chapter four are data of 5355-ft deep flowing well used to validate my fluid Temperature calculation.
The assumption of f(t)=TD does not affect the solution at large times. Besides the equality of f(t) and TD is not surprising because both functions attempt to describe the temperature distribution in formation based on the same differential equation.

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CHAPTER FOUR
4.0 RESULTS OF MODEL VALIDATION

The following data were used to test the efficiency of the derived model in predicting temperature at any depth and production time. Temperature data from Amerada Hess of west Texas.

4.1 THE FLUID TEMPERATURE DISTRIBUTION
Fluid temperature distribution at any depth can be computed using the relations below:

where A=
the former equation can easily be converted to deviated well, for vertical wells ( =900 )Sin 90=1. and geothermal gradient, gT gT = (Tebh-Ts)/ZSin for deviated wells where = angle of deviation , g and gc are equivalent and acts as a conversion factor.

Variables Data report by Sagar etal Amerada Hess
cooperation Nafta Gas of Yugoslavia Chevron Escravos Nigeria
N 12 15 18
Tebh(F) 108 237.2 172
Ts(F) 76 50.1 89.5
?(ft2/hr) 0.04 0.04 0.04
Zbh(ft) 5355 6,792 8250
Ke(Btu/hr/ft/F) 1.4 1.4 1.4
dto(Inches) 3.375 2.875 3.875
GLR(scf/bbl) 68.2 - -
APIG (gravity) 34.3 35 30
Yg 1.05 0.75 0.79
dti(Inches) 2.875 2.375 3.375
pwh(Psia) 113 174 215
dwh(Inches) 9.0 7.5 12
dco(Inches) 7.5 5.8 8.304
dci(Inches) 7.0 5.5 8.0
Kan(Btu/hr/ft/F) 0.383 0.383 0.383
Kcem(Btu/hr/ft/F) 4.021 4.021 4.021
tp( hours) 158 165 160
qo (BBL) 59 6.8 1720
qw (BBL) 542 13.3 236
qg (SCF) 41000 3100 1899000
rw 101 1.o 1.0

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CHAPTER FIVE
ANALYSIS OF RESULTS

While the figure 1 shows a very good agreement between the model prediction, measured temperature and Hassan & Kabir. The 2 data points nearest to the surface (0 and 500ft depth) shows higher fluid temperature than are predicted the model can accurately predict these temperature if a higher conductivity of 1.4 Btu/hroF is used for the formation near surface. The model allows conduction and convection as heat transfer mechanism for fluid in the tubing casing annulus .
The deviation between measured and predicted temperatures is more at the surface and near surface and reduces as depth increases and eventually meet at the bottom hole. This is because gas expansion occurs as multiphase fluid flows up to the surface and that enhances cooling due to Joule-Thompson and Kinetic energy effect and provides room for heat exchange. As depth increase, pressure and temperature increases.

The concordance among the proposed Hassan & Kabir multiphase prediction with measured temperature and the correlation developed is shown figure 1. and figure 3 for fluid temperature distribution in wellbores and represented in the coloured legend on the profile.
The three models in each of the three profiles :(Hassan & kabir), the developed model and measured temperature matches as depth of wellbore increases. It could not tally at the surface and 500ft depth with measured temperature due to Joule Thompson cooling as multiphase fluid flows up and pressure is reduced giving rise to gas expansion.
The percentage error between the model prediction and that of measured temperature (at 0-500ft.) is 4.1%.
Therefore, using correlation developed we must make a correction of predicted temperature at the surface by multiplying it by 4.1% and subtract from the predicted temperature.
Example 1: Fluid temperature @ 0ft = 91.64F
corrected measured temperature =91.640*(1-4.1/100) = 87.88F
There is a high agreement between predicted fluid temperature and that of Hassan & Kabir prediction. The profile matches as depth increases and almost coincides showing that there is high agreement amongst both models.
Figure 2. is a temperature profile for a flowing gaslift well, the predicted temperature of model and that of Hassan & Kabir for multiphase flow agrees with tend to twist. This is as a result of gas injection at 4200ft, this point of gas injection. The temperature is lowered due to presence of gas, gas expansion and Joule-Thompson cooling.

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WHY FLUID TEMPERATURE DISTRIBUTION ?

? Accurate temperature prediction is necessary in the effective design and execution of cementing programme choosing suitable cement slurry properties: placement and setting time during completion and worker operations
? It is important to know the wellbore fluid temperature before gas lifting operation at various depths to enable us know the temperature at various value depth.
? Also temperature profile of a well must be known before enhanced or secondary recovery: Steam injection. Gas and hot water injection among others.
? Packer design and selection.
? To enable in designing of Logging tool and for log interpretation
? Prediction of Wax and scale depositions in production tubing
? Determination of region in the tubing and casing liable to Corrosion.
• Wellhead and production equipment design

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CONCLUSION

An algorithm is presented for estimating wellbore fluid temperature. It allows wellbore heat transfer by conduction and convection and evaluate the formation by assuming that fluid temperature at the wellhead is equal to the earth temperature at wellhead.
Here, the importance of convective heat transfer is demonstrated. The need for radiation is eliminated due to unavailability of fluid emissivity which depend on surface finish and view factor among other variables.
The algorithm can easily be applied to deviated wells.
This algorithm was developed from basic thermodynamic principles to predict temperature profiles in two-phase and multiphase flows in wells. The simplified model represents an extension of the latest multiphase fluid temperature distribution correlation.
The developed model eliminates the need to estimate fluid temperature at wellhead. And to iterate for overall thermal co-efficient at various depths.

RECOMMENDATION

I recommend that when using the model. Accurate temperature at the near surface (0-500ft) would be determined by:
accurate temperature = Calculated temperature* (1-% error)
This would account for the Joule-Thompson cooling and expansion of gas and pressure reduction that occurs as a result of upward flow of multiphase fluid in wellbore.

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NOMENCLATURE
A Inverse relaxation distance, ft
Cpm Heat capacity of wellbore fluid, Btu/lboF
Ce Heat capacity of the earth, Btu/lboF
Cj Joule-Thompson coefficient, dimensionless
d Pipe diameter, ft
f(t) Ramey’s solution for wellbore earth/interface temperature, dimensionless
g Acceleration due to gravity, ft/sec2
gc Conversion factor, 32.2lbmft/lbf sec2
gT Geothermal gradient oF/ft
H Fluid Enthalpy Btu/lb
K Conductivity, Btu/ft oF
Ka Conductivity of Annulus, Btu/ft oF
Kcem Cement Conductivity, Btu/ft oF
Ke Formation conductivity, Btu/ft oF
q Heat flow rate from or to the wellbore, Btu/hr
r Radial distance of the wellbore, ft
T Temperature, oF
TD Dimensionless temperature
Tei Formation temperature at any given depth and radial distance from well,oF
Teibh Formation temperature at Bottom hole, oF
TeWh Formation temperature at wellhead, oF
Tf Wellbore fluid temperature, oF
W Total mass flow rate, lbm/sec
Z Variable depth from surface, ft
Zbh Total measured depth from surface, ft
J Mechanical equivalent of heat, 778ft-lbf/Btu
Kan Thermal conductivity of annulus material, Btu/hr-ft-oF
PWh Wellhead pressure
q0 oil flow rate, STB/D
qg Gas flow rate, SCf/D
qw Water flow rate, STB/D
uc Heat transfer coefficient of casing
ut Tubing heat transfer coefficient of casing
U Overall heat transfer coefficient, Btu/hr.ft2 oF
Yg Gas specific gravity
Yo Oil specific gravity
Yw Water specific gravity
Dci Casing Internal diameter
Dti Tubing Internal diameter
Dto Tubing external diameter
Dco Casing outside diameter
Dwb Diameter of well bore

GREEK LETTERS
Heat diffusivity of earth ft2/hr
Specific gravity of produced gas
Oil specific gravity dimensionless
Water specific gravity dimensionless
Parameter which combines Joule-Thompson and Kinetic Energy effects.
Wellbore inclination with horizontal, degree fluid viscosity, Cp
Density, lbm/ft3
Earth density, lbm/ft3
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REFERENCES

Sagar, R.K., Dotty, D.R., and Schmidt, Z: “Predicting Temperature Profiles in a flowing well,” Paper SPE 19702 presented at 1989 SPE Annual Technical Conference and Exhibition, San Antonio, TX Oct.8 –11

Hassan, A.R. and Kabir, C.S.: “Heat transfer during 2 –phase flows in wellbores –part II- wellbore fluid Temperature”

Hassan, A.R. and Kabir, C.S.: “Heat transfer during 2 –phase flows in wellbores –part I-Formation Temperature”

Ramey, H.J. Jr.: “wellbore Heat transmission,” JPT (April 1962) 435 Trans AIME, No. 225.
Alves, I.N. Alhanati and Shiham, U.: “ A Unified Model for predicting flowing Temperature distribution in wellbores and pipelines “SPE 20632.

G.J Plisga, “Temperature in wells,” Sohio Alaska Petroleum company.
Farouq Ali, S.M.: “A comprehensive wellbore steam/water flow model for steam injection wells,” Paper 196337 presented at the SPE California Regional meeting, Ventura, CA, April 8 –10,1987

Shiu,K.C. and Beggs, H.D.: “ Predicting Temperatures in Flowing Oil wells,” J. Energy Resources Tech, (March 1989 1- 11)

Lesem, I.B. et al.: “A method of calculating the Distribution of temperature in flowing Gas wells,” Trans, AIME (1957) 210,pg 169.

Nowak,T.J.: “The Estimation of water injection profiles from Temperature surveys,” JPT (Aug. 1953) 203,Trans AIME,198.
Gany R. Wooley; “Computing Downhole temperature in circulating, injection and production wells”. SPE, Enertech Engineering and Research Co.
Willhite, G.P; “Overall Heat Transfer Coefficient in steam and Hot Water injection Wells”, JPT (May 1967) 607-615

Schlumberger, M., Doll, H. G., ‘Temperature Measurement in oil wells,” J. Inst. Pet. Technologist (Jan, 1937) 13, 159

Kick Patrick, C.V: “Advance in Gaslift Technology”, Drill. & Prod. Prac. (March 1959) 24 - 60.

Moss J.T and White, P.D; How to calculate Temperature Profile in a Water Injection Well,” Oil & Gas Journal. (March 9, 1959) 57, NoII, 174.

Willterholt, E. J. andd Tixier, M.P.: “Temperature Logging in Injetion Wells,” Paper SPE 4022 presented at the 1972 SPE Annual fall Meeting San Antonio, TX Oct. 8-11.
Edwardson, M.J et al: “Calculation of formation Temperature Disturbances caused by mud circulation, “JPT (April 1962) 416-26; Trans., AIME, 225.
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APPENDIX A

There are certain parameters we must compute before finally calculating the wellbore fluid temperatures, Tf , These parameters are itemized:
a. Geothermal gradient, gT
b. , dimensionless correction parameter which depends on Joule-Thompson expansion and GLR
c. dimensionless transient heat conduction time function of the formation, f(t) = TD
d. The overall thermal transfer coefficient, Uto
e. The inverse Relaxation distance, A , ft.
f. Using the model (23) to estimate the wellbore fluid temperature.

a. GEOTHERMAL GRADIENT, gT
The undisturbed temperature of the formation, Tei, is generally assumed to vary linearly with depth.Thus, Tei=Teibh-gTZ………………………………………………….. A1
where gT represents the geothermal gradient and Teibh is the undisturbed formation temperature at the bottom hole, Z
gT = (Teibh – Tei)/Z…………………….A2
Numerically, using the data in chapter 4
gT = (108oF – 760F)/5355ft
Geothermal, gT = 0.0059757oF/ft
b. DIMENSIONLESS CORRECTION PARAMETER,
This parameter depends on Joule-Thompson expansion and cooling effects and can be estimated by empirical correlation for a unit mass flow rate W=1Lb/sec:
=-0.002978+1.006X10-6Pwh+1.906X10-4W-1.047X10-6GLR-0.3551gT+3.229X10-5API +0.004009Yg…………………………..A3
Gas –to-liquid Ratio, GLR =
GLR = qg /(qo+qL)=41X103scf/(542+59)STB
GLR =69.374 scf/STB
Thus, upon substitution into A2, can be estimated:
c. THE SPECIFIC HEAT CAPACITY OF TUBING FLUID, Cpm
In other to compute the fluid temperatures at various depths, we need to compute first the specific heat of tubing fluid which is determined by :
…………………..A4
where typical values for oil, Cpo and water, Cpw are respectively 0.485 and 1.0 Btu/1bm.oF

d. THE MASS FLOW RATE, W
The mass fluid flow rate is calculated by:
…………A5
where Oil, water and gas flow rates are:
qo, qW, qg respectively .”APIG” is used to represent the API Gravity of oil.
e. THE OVERALL HEAT TRANSFER COEFFICIENT, U
During exchange of heat in the wellbore, heat is transfer from earth/formation to the fluid is in two parts:

(i) Overall heat transfer for fluid through tubing is given by:

(ii) For casing flow, we have:

therefore, the overall heat transfer coefficient is thus:
or explicitly:

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f. THE THOMPSON COEFFICIENT,
To account for cooling and expansion as a result of gas in the well bore a dimensionless constant, Joule-Thompson correction factor is introduced, computed by Sagar et al :
=1.006X10-6Pwh+1.906X10-4W-2.978X10-3-1.047X10-6 GLR+3.229X10-5APIG+4.009X10-3Yg-0.35511gT
where PWh, W and gT are wellhead pressure , mass flow rate and geothermal gradient of earth.

g. TRANSIENT HEAT CONDUCTION TIME FUNCTION, f(t)
from equation (25) f(t)=
using the data provided:
f(t)=T
f(t)= 2.306

h. WELLBORE FLUID TEMPERATURE, Tf
Model (23) permits the estimation of the temperature of fluids as a function of depth and producing times. This model is suitable for slide-rule calculation.

Where j=1,2,3………N
Temperature profiles can therefore be obtained at various depths using the thermal conductivity of the earth to be 1.4 Btu/hr-ft-0F

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