A Review of Methods for Calculating Heat Transfer From a Wellbore

8/12/2019 A Review of Methods for Calculating Heat Transfer From a Wellbore

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WHOC12-240

A Review of Methods for Calculating Heat Transfer from a Wellbore tothe Surrounding Ground

P. SKOCZYLASC-FER Technologies

This paper has been selected for presentation and/or publication in the proceedings for the 2012 World Heavy Oil Congress[WHOC12]. The authors of this material have been cleared by all interested companies/employers/clients to authorize dmg events(Canada) inc., the congress producer, to make this material available to the attendees of WHOC12 and other relevant industry

personnel.

Abstract

Accurately estimating heat transfer from a wellbore tothe formation is important in heavy oil applications. In

primary or cold production, the viscosity of heavy oilchanges substantially as the oil cools, and an error inestimating the cooling in the wellbore can have a significantimpact on pumping systems. In thermal operations, energyefficiency is an important design consideration for whichheat transfer rates must be estimated. For 50 years,engineers calculating the rate of heat transfer from oil wellsto the ground have referred to the classical paper by

Ramey(1). Rameys formulation is simple and effective butonly at times longer than one week. Other authors have

presented improved formulations to work at shorter times.For shorter times, properly considering the effects of casingand cement layers becomes significantly more important. Attimes less than one day, this may be critical, such as whenexamining the stresses in the cement caused by thermal

gradients at the onset of steam injection. Many proposedmethods do not consider the effects of temperature on the

ground properties. This is an issue, particularly in thermalwells, as the thermal conductivity of rock is reduced at hightemperatures. It is also an issue in wells which pass through

permafrost, as some of the heat is absorbed by the ground aslatent heat. This paper compares results using acomprehensive model to those based on simple formulations,and discusses under what conditions the simple formulationsare adequate or not.

Introduction

There are many applications in which the rate of heattransfer to or from a well needs to be calculated. Theseinclude: Predicting the temperature of the fluid so that the

viscosity, and hence the flowing pressure losses, maybe estimated in a heavy oil well.

Estimating the amount of heat lost in a steaminjection well and/or the associated productionwells, in order to estimate quantities such as the totalthermal efficiency of a process.

Determining whether flow assurance issues may existin a proposed application due to formation/depositionof paraffin, asphaltenes, hydrates, or precipitates atlower temperatures.

Estimating the degree to which permafrostsurrounding a well in an Arctic region may thaw as aresult of injection or production through the well, ordue to the use of heat tracing on that well to assess theeffects of thaw on subsidence and wellboredeformation(2).

For these applications, or other similar ones, a quasi-steady state result may be adequate. For other applications,however, a transient solution may be required. An examplecould be in an examination of thermal stresses in cementsurrounding casing at the onset of steam injection(3). If thesestresses are too large, a wellbore integrity problem mayresult. In order to calculate the thermal stress, the thermalgradients must be known. The thermal gradients will changerapidly over the first few minutes or hours of steam injection,and will in fact be greatest early in the process, so no quasi-steady state solution will be adequate.

This paper reviews various methods of estimating heattransfer that have been presented in the literature, anddiscusses when these are appropriate for use, and when theymay not be. It also presents a numerical method which can

be used when simpler methods will not give accurate results.

Problem Formulation

For the scenarios presented here and to illustrate themethods described below, we will focus on the heat transferoccurring at a single depth in a well. To calculate the overallheat transfer between the wellbore and the formations that it

penetrates, one must repeat this at every relevant depth in thewell, taking into account the change in wellbore and groundtemperature with depth. This has been covered

before(1,4,5,6,7,8,9), and as such will not be repeated here. The


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assumption is also made here that heat is conducted evenly inall radial directions from the wellbore, allowing us to use anaxisymmetric simplification. For this type of problem, axialconduction of heat is much smaller than radial conduction, sowe will ignore the axial component, yielding a much simpler

problem of calculating a number of one dimensional heattransfer problems instead of a single two dimensional

problem.That is, for the purposes of this discussion, we will

calculate heat transfer from a particular depth in a wellbore tothe surrounding formation as an axisymmetric, one-dimensional, heat conduction problem in cylindricalcoordinates. The differential equation governing this is(10):

+ = ..................................................................... Equation 1This can be solved for wellbores using a finite inner

boundary and an infinite outer boundary. It is alsoconvenient to make the further simplification that the ground

properties are constant in space and time. (There may becases where this assumption is not valid; this will bediscussed later.) Some formulations, including in the classic

paper by Ramey(1), make a further simplification that theinner boundary can be considered infinitesimalthe well istreated as a line source.

Before the problem can be solved, however, boundaryconditions must be applied. At the outer (far field) boundary,a constant temperature boundary is generally applied. Thisconstant temperature boundary condition may not be validover the long term when wells are close together, but in mostcases it will be valid for a period of several years. At theinner boundary, several different boundary conditions may beconsidered. The three most commonly addressed boundaryconditions are constant temperature, constant heat flux, andconvection.

While the nomenclature conventions used by the varioussources referred to here were often quite different, for this

paper, all nomenclature has been converted to a consistentsystem, described in the Nomenclature section. In addition,some sources used different definitions for certainmathematical functions, such as the exponential integral.Where needed, conversions have been made to one consistentset of definitions in this paper.

Line Source Solutions

The line source problem has generally been addressed as aconstant flux problem. The solution for this problem, as

presented by Carslaw and Jaeger(10), is:

2

= Ei

..................................................... Equation 2

This is presented in a non-dimensional form, which willfacilitate certain comparisons later. The 2 is a usefuladdition, in part because it makes the term on the left handside consistent with a non-dimensionalisation presented byHasan & Kabir(4). Carslaw and Jaeger(10) noted that theexponential integral can be simplified for very small or verylarge input arguments. However, there is no valid reason inthis case to look at very short times (large input arguments tothe exponential integral). A line source does not suitably

represent a wellbore, which has finite size, at small times. Itis only as the heat front progresses further from the well thata line source approximation becomes a good representation.Ramey(1) made note of this in his paper, saying the linesource will often provide a useful result if times are greaterthan one week.

Carslaw and Jaeger(10)presented an approximation of theexponential integral which works for small input arguments(large times):

() + ln + ............................................... Equation 3If we take just the first two terms of this approximate

solution and substitute them into Equation 2, we get:

2 = + ln ..................................................... Equation 4Rameys(1)formulation is:

2

= ln

+ 0.29.............................................. Equation 5

Since /20.2886, Rameys solution is essentially identicalto Carslaw and Jaegers, which should not be surprising, asRamey directly referenced their result. This formulation hasnumerical problems at very small times, due to the smallargument (large time) approximation of the exponentialintegral. Furthermore, it gives negative results at Fouriernumbers less than approximately 0.45. These clearly cannot

be used, but even for Fourier numbers greater than 0.45 theaccuracy is very poor until the time gets sufficiently largethe error is less than 2% (relative to methods which doconsider the finite wellbore diameter) at Fourier numbersgreater than approximately 30. This translates to roughly aweek for typical wellbore calculations, which is consistentwith Rameys assertion.

Use of a more complete evaluation of the exponentialintegral would remove the numerical difficulties at Fouriernumbers below 0.45. These methods, however, do notsignificantly improve the accuracy at Fourier numbers muchabove 0.45. While Rameys method achieves errors less than2% at Fourier numbers above 30, a more complete evaluationof the exponential integral only improves this to 2% accuracyat Fourier numbers above 27, which is not muchimprovement over the approximate solutions.

It is apparent, therefore, that to improve upon the resultsof these line source approximationsto estimate heat transferat earlier times the finite size of the wellbore must beconsidered.

Laplace Transform Methods andIntegral Results

For an infinite outer boundary and finite inner boundary,the heat conduction differential equation presented earlier ismost easily solved using Laplace transforms. For theconstant flux inner boundary, the result in the transformedspace is:


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(, )=

........................................ Equation 6

And for the constant temperature inner boundary, it is:

(, )= 2 ...................... Equation 7Note that for the constant flux condition, we have solved

for temperature; and for the constant temperature case, wehave solved for flux. Both of these cases were solved as afunction of radius and time. We are generally interested inthe flux or temperature at the outside of the wellbore, so wecan solve for the respective results at that radius. Note thatthe solutions presented above are those of Carslaw andJaeger(10), but their solutions were presented only astemperature. The flux result (in the constant temperaturecase) was obtained through an extra step done as part of the

present work.For the case with convection at the inner boundary, Jaeger

and Chamalaun(11)presented:

= 1 +

........................................................................................................................ Equation 8

To calculate flux from this, the following relationship isused after the inverse Laplace transform is calculated:

= 2 ........................................................ Equation 9The inverse Laplace transforms of these results are noteasily obtainable. Carslaw and Jaeger(10) discuss the

mathematics of doing this, but their results were given in theform of integrals which cannot be evaluated analytically.

Numerical methods can be used to obtain the inverse Laplacetransform in some cases, and these are fast and accurate forthe range over which they can be used. The method that wasused in this work(12) did not yield a solution at very small(1,000,000) Fourier numbers butworked well between these values.

The inversions using integrals give the following results(10)for the constant temperature case:

= (()() ............................. Equation 10and for the constant flux case:

2 =

()() .................. Equation 11The formulations in Equations 10 and 11 contain

dimensional values in the integrals, specifically time and thewellbore radius. This makes comparison to non-dimensionalresults difficult, and means that tables of results need to have

an extra dimension. Jaeger(13)also provides non-dimensionalversions of these results, which are, for the constanttemperature case:

= ()() ....................................... Equation 12and for the constant flux case:

= (()()) ...................................... Equation 13It can be shown that these are equivalent to the

dimensional versions. Jaeger did not state how thenondimensionalization was done, but simply showed that thetwo results were equal. Hasan and Kabir(4), however,

provided some insight into how it may have been achievedthey actually provide a result for the constant flux case,although to show that their equation is in fact equivalent tothe one above, one needs to use the following relationship:

()() ()()= ......................................... Equation 14The convection solution was presented by Jaeger and

Chamalaun(11)in a dimensionless form, as:

= ()()()() ................................................................................................................... Equation 15

where:

=

................................................................................. Equation 16

Note that the temperature differential in Equation 15 isdifferent from the one in the other versions (in that it refers tothe fluid temperature instead of the wellbore interfacetemperature). Jaeger and Chamalaun(11) also presented arevised version of this:

= + () (, ) ........ Equation 17where:

(, )= + arg ()() + ()()................... Equation 18

Note that in order to use this formulation, the arg(z)function, evaluated over the whole range of the integral,needs to be continuousthis means that its output is notalways in the to range as might be expected. Rather, itmonotonically increases as the z vector rotates around theorigin of the Argand diagram as ugoes from 0 to infinity.

Jaeger and Chamalaun(11)also presented simplified resultsfor very small or very large values of time which do not


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require integration. These are not presented here, but shouldbe consulted if needed.

The equations for all three boundary conditions containintegrals which cannot be evaluated analytically. Rather,numerical integration is required to evaluate them.Unfortunately the integrands tend towards infinity at verysmall times. A way to deal with this problem is the methodof asymptotic expansions(10). The Bessel functions andexponentials in the integrals can be replaced with simpleexpansions which are valid for either very small times or verylarge times. For example, the exponential function in theintegrals (where the argument is negative) tends toward 1 asthe variable of integration (u) gets very small, and towards 0as it gets very large. Similar simplifications (although notgenerally as simple) also exist for the Bessel functions

presented in those integrals. When these are applied, theintegrals can be solved analytically for regions near u=0 andfor very large values of u. Numerical integration is then onlynecessary for intermediate values of the variable ofintegration. Note that the revised version of the integral

presented by Jaeger and Chamalaun, as shown above inEquations 17 and 18, is more conducive to being evaluatednumerically, as its integrand does not tend towards infinity as

uapproaches 0.Several authors, including Van Everdingen and Hurst(14),

Willhite(5), Jaeger and Clarke(15), Hasan and Kabir(4), andJaeger and Chamalaun(11)have given tables of results for oneor more of these integrals, which can be consulted.

While tables can certainly be entered into a computerprogram, correlations are much simpler to use. Severalauthors have published correlations to these tabulated values.These include, for constant temperature boundary conditions,Chiu and Thakur(6) and Moini and Edmunds(7), and forconstant flux boundary conditions, Hasan and Kabir(4,8).These correlations are presented in Appendix A. Nocorrelations were found in the literature for the convection

boundary condition as part of this work.Carslaw and Jaeger(10) also presented formulations for

both the constant flux and constant temperature cases whichprovide solutions at very small or very large values of time.These are as simple to use as the correlations (i.e. nointegration or numerical inversions are needed), and so may

be quite convenient to use when short or long times are ofinterest.

Results Comparisons

In this section, the results for several of the solutionsdescribed above for the cases with constant temperature andconstant flux at the inner boundary are presented. Forcomparison, the values are presented as dimensionlesstemperatures, as used by Hasan and Kabir(4), and which isessentially the same as Rameys(1) dimensionless timefunction:

= 2 ........................................................................................ Equation 18The relative error presented in the plots shown below was

calculated using the following:

= || ............................................................................ Equation 19

where TD0is a reference value, which for the plots here wasthe value obtained by numerical integration as part of thiswork. All of the other values obtained from tabulated data,formulations, and correlations were compared against this. Inthe figures, markers represent tabulated data, while linesrepresent continuous sources of values obtained fromcorrelations, short/long time approximations, numericalinversions, and numerical integrations.

Figure 1 and Figure 2 show the dimensionless temperatureand relative error for the constant temperature inner boundarycase.

Figure 1 Constant temperature boundary models

Figure 2 Error in constant temperature boundary models

Examination of these figures shows that all the constanttemperature formulations yield similar and suitably accurateresults, with the exceptions of the Carslaw and Jaegerapproximations for small time results at longer times andlong time results at smaller times. These approximationsyield better than 1% accuracy if the Fourier number is less

than 0.32 for the small time version and greater thanapproximately 100,000 for the long time version.

Figure 3 and Figure 4 show the dimensionless temperatureand relative error for the constant flux case. In these figures,the data points labeled Hasan and Kabir Rigorous arefrom the table in their 1991 paper(4); they use the phraserigorous solution to describe this tabulated data and todifferentiate it from values obtained using theirapproximate correlation.

0.001

0.01

0.1

1

10

1.E-04 1.E-02 1.E+00 1.E+02 1.E+04 1.E+06 1.E+08

D

imensionlessTemperature

Fourier Number

Numerical Integration

Inverse Laplace

Carslaw & Jaeger

(Small Time)Carslaw & Jaeger

(Long Time)Jaeger & Clarke

Willhite

Chiu & Thakur

Moini & Edmunds

1.E-08

1.E-06

1.E-04

1.E-02

1.E+00

1.E-04 1.E-02 1.E+00 1.E+02 1.E+04 1.E+06 1.E+08

RelativeErrorin

Dimension

lessTemperature

Fourier Number

Inverse Laplace

Carslaw & Jaeger

(Small Time)

Carslaw & Jaeger(Long Time)Jaeger & Clarke

Willhite

Chiu & Thakur

Moini & Edmunds


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Figure 3 Constant flux boundary models

Figure 4 Error in constant flux boundary models

As with the constant temperature case the resultspresented in Figures 3 and 4 show that all of the constant flux

formulations yielded suitably accurate results, with theexception of the Carslaw and Jaeger approximations for smalltime results at longer times and long time results at smallertimes. These approximations yield better than 1% accuracy ifthe Fourier number is less than 0.05 for the small timeversion and greater than approximately 58 for the long timeversion.

The correlations which gave the best results, showing theleast disagreement with numerical results over the largestrange of Fourier numbers, were the revised Hasan andKabir(8)formulation for constant flux cases and the Chiu andThakur(6) formulation for the constant temperature cases.Getting into sufficiently small or large times, the Carslaw andJaeger(10)approximations perform better than the correlations,

but these have the disadvantage of only being accurate in anarrow range.

Notice that the plots in Figure 1 and Figure 3 look verymuch alike. In fact, as Ramey pointed out, these plotsconverge toward identical results at long times. However, atshort times, there remain differences. As time approacheszero, the two results approach a constant ratio, with thedimensionless temperature for the constant temperature case

being /2 times the value for the constant flux case at thesame Fourier number. At large times, there is less than 1%difference between the values in Figures 1 and 3 by the timea Fourier number of one million has been reached. After aFourier number of 30, Rameys one week, there is less than

10% difference between themgenerally good enough formany engineering applications, especially considering theother sources of error and the probability that the truewellbore boundary condition is not quite either a constanttemperature or a constant heat flux boundary condition.

Shortcomings of These MethodsWhile the results discussed above may be very valuable

and useful for many engineering applications in the oilfield,they do have their shortcomings. These include:

1. The formulations do not consider how the thermalmass of the wellbore itself affects the results. Severalauthors, including Ramey(1), Willhite(5) andHagoort(9), provided methods to consider theresistance to heat transfer of the wellbore itself, butthese methods ignored its thermal mass. This is not asignificant issue in long term cases, but if one needsto observe changes in the short term (less than a fewdays), neglecting this thermal mass may be a problem.

2. There is no consideration of boundary conditionswhich change over time. Even if one is injecting

constant temperature fluids (e.g. steam), or producingconstant temperature reservoir fluids, over time, the

boundary conditions downstream of the place whereflow enters the wellbore (i.e. the wellhead or theformation, depending on whether an injection well or

producing well is being considered) will in factchange over time. This is because the groundtemperature between the point where flow enters thewell and the point under consideration will changeover time. This in turn changes the amount of heatlost (or gained) by the wellbore fluids over time.Other instances of changing boundary conditionsinclude periods during which the well is shut in orrestarted.

3. There is no consideration of changing ground

properties. This is a concern in thermal wells inparticular, as the thermal conductivity of groundchanges substantially between 0C and 300C. Thisis illustrated for some soil types in Figure 5, as given

by Clauser and Huenges(16) (note their use of forthermal conductivity). It is also a concern in wells infrozen ground, as the thermal properties of frozen soilare different from those of unfrozen soils.

Figure 5 Effect of temperature on ground thermal

conductivity

0.001

0.01

0.1

1

10

1.E-05 1.E-02 1.E+01 1.E+04 1.E+07 1.E+10

DimensionlessTe

mperature

Fourier Number

Numerical Integration

Inverse Laplace

Carslaw & Jaeger (Small Time)

Carslaw & Jaeger (Long Time)

Hasan & Kabir "Rigorous"

Van Everdingen & Hurst

Hasan & Kabir

Hasan & Kabir (#2)

1.E-10

1.E-08

1.E-06

1.E-04

1.E-02

1.E+00

1.E-04 1.E-01 1.E+02 1.E+05 1.E+08

RelativeErrorin

DimensionlessTemperatu

re

Fourier Number

Inverse Laplace

Carslaw & Jaeger

(Small Time)Carslaw & Jaeger

(Long Time)Hasan & Kabir

"Rigorous"Van Everdingen & Hurst

Hasan & Kabir

Hasan & Kabir (#2)


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4. There is no consideration of latent heat effects. Thisis most often an issue in permafrost applications,where the ground around the wellbore is initiallyfrozen, but thaws over time, consuming significantthermal energy in the process. It may also be an issuein many thermal applications, as any ground watermay boil if the temperature reaches or exceeds thesaturation temperature at the local pore pressurevalue. (This is more likely to happen closer to surface,where the pore pressures tend to be lower.)

5. The results do not, in general, tell us anything aboutthe radial temperature gradients in the ground. This isrelevant for certain problems. For example, onemight be interested in knowing what the thermalgradients (and therefore the thermal stresses) in thewellbore cement sheath are following the onset ofsteam injection. One might also want to know wherethe thaw boundary is around a well in a permafrostregion.

Analytical solutions do not generally exist for thesesituations. Other methods, such as numerical solutionmethods, must be used to obtain results where they mayoccur.

Numerical Methods

Finite element analysis (FEA) can be used to obtainsolutions in such cases. Most commercial FEA packages cando heat transfer analyses. Finite difference (FD) methodsmay also be used. For simple geometries, FD methods areoften easier to develop and to couple with the temperaturecalculations in a wellbore simulator.

A detailed, one-dimensional, axisymmetric, finitedifference model was developed for this study. It allowed forvariable spacing of the nodes, varying properties of thevolumes around the nodes, and regions of different

properties. The variable nodal spacing is useful in that

greater accuracy can be obtained where the temperaturegradients are steepest near the wellbore while allowing morewidely spaced nodes further removed from the wellborewhere the temperature gradients are small. If one was forcedto have constant spacing, one would either lose accuracy (dueto not having enough nodes where they are most needed), orone would have to have a very large number of nodes (whichwould greatly increase computation time). The FD modelused in this study can handle all three boundary conditionsreferred to above.

The full details of the method are not given here for thesake of brevity, but an example of the derivation method isshown in Appendix B. Additional examples were presented

by Skoczylas(17).

Consideration of Latent HeatLatent heat effects were considered using a method called

apparent heat capacity as referred to, for example, byPham(18), Osterkamp(19), and Mottaghy and Rath(20). Theoriginal implementation of this method was used in caseswhen the modeled substance would freeze at a singletemperature. As Pham(18)says, the latent heat is represented

by a peak of small but finite width in the c(T) curve, wherec(T) represents the specific heat of the material as a functionof temperature. The problem with this method, however, is itis possible for the calculation to ignore this peak, by

jumping of the latent heat peak(18). This can be preventedonly by making the peak wider or by making the time stepsvery short, so that the amount of heat entering a node whichis just below the peak is not enough to take the temperatureabove the peak in a single time step.

Ground, particularly when made up of fine grained soilssuch as silts and clays, does not freeze at a singletemperature, but rather it freezes over a temperature range.An example of this phenomenon is shown in Figure 6 below,as given by Williams(21). Therefore, one does not need toapply a very narrow, yet very tall, c(T) peak to implement anapparent heat capacity method. Rather, the peak is spreadout over the full range of temperature over which freezingoccurs. Pham(18) actually suggests that smoothing of the peakis a way to prevent jumping of the peak, but for the problemof freezing at a single temperature, he noted that this reducesaccuracy. In our case, this should not be an issue, becausethe c(T) relationship is naturally smoothed.

Figure 6 Ground percent unfrozen as function of

temperature

To use this concept in a finite difference model, no

changes need to be made to the model itself, but only to thevalue of the specific heat at every node at every point in timebased on its temperature at that time.

Note that there is also an analytical solution for a latentheat problem as presented by ziik and Uzzell(22). Thissolution was for a line sink, however, so it is only useful forlonger times. This approach may be useful, for example fordetermining the heat loss from a wellbore over its lifetime. Itwill not be considered further here, except to note that it wasused to validate the numerical methods developed during thisstudy.

Comparison of Numerical Methods withSimple Models

In some cases numerical solution methods (such as FEA

or FD models) may be necessary to get accurate results.These may include cases when we need to know the thermalgradients inside one or more layers of cement in a wellbore,or when the thermal mass of the casing and cement mayaffect the results. In other cases, the correlations noted abovemay be more than adequate for obtaining useful results. Sucha case may be when estimating the thermal efficiency of asteam injection well over its life. The question is: in whichcases can we use the simple models, and what, if any,modifications do we need to make to the inputs to use them?


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Case 1a: Consideration of the thermal mass of casing andcement (constant temperature boundary).

For example, let us consider 9.625 casing, concentricallycemented in a 13 diameter hole. A constant temperature

boundary condition (45C higher than the groundtemperature) was applied at the inner wall of the casing, andthis was compared to the result from the FD model of this

scenario to that obtained using the Chiu and Thakur(6)

correlation. In using the correlation, the Fourier number wascalculated using the diffusivity of the ground and thediameter of the wellbore/formation interface. This was alsocompared to a Fourier number calculated using the insidediameter of the casing. The former approach ignores thethermal mass of the casing and cement, while the latterassumes that the casing and cement have thermal massequivalent to that of the same volume of ground. Results are

presented from four versions of the finite difference model,labeled as in Figure 7:

1. full consideration of the casing, cement, and ground,Full Model;

2. consideration of only the ground outside the wellbore(with the boundary condition applied at the

cement/ground interface), Ground Only;3. consideration of the casing and cement with the samethermal properties as the ground, Ground to CasingID, and

4. consideration of the casing and cement with nothermal mass but with their actual resistance to heattransfer, Ground with Resistance.

The results are shown in Figure 7. Several insights can begained from examination of these results:

Ignoring the casing and cement and applying theboundary condition at the wellbore/formationinterface, as in the Ground Only case, and thecorresponding Chiu & Thakur C&T (WellboreInterface) case, is not valid. While these resultseventually approach the full model results, they arevisibly different (error greater than 10%) even afterone year of operation (the longest time shown in thefigure). Note that the Ground Only and C&T(Wellbore Interface) lines in Figure 7 are practicallyidentical such that only one line is distinctly visible inthe figure except at very short times (fractions of asecond).

Modeling the ground plus the thermal resistance ofthe wellbore (the Ground with Resistance case)works very well for times longer than approximatelyone day, but is very inaccurate at shorter times, inthat it shows much too low a value of heat transfer.If one is not interested in the results at very shorttimes, however, the results show that this can be avery effective method.

The Chiu and Thakur method, considering the casing

and cement to be equivalent to ground (the C&T(Inside Casing) case), is reasonably good, but with asmall error, for times longer than approximately oneday for the scenario considered here. (The actualmethod described by Chiu and Thakur is actuallymore like the Ground with Resistance method

plotted in the figure. However, their full method wasnot used hereonly their transient calculation for theground was used.)

Likewise, a finite difference method that considersthe casing and cement to be equivalent to ground (the

Ground to Casing ID case) yields results that arereasonably good but with some error as compared tothe full FD model for times longer than one day.

All of these methods diverge substantially from theresults of the full FD model for times shorter thanapproximately one day.

In the figure, the three finite difference resultswithout resistances (the Full Model, GroundOnly and Ground to Casing ID cases) appliedhave an upturned result as time approaches zero.This does not represent a physical phenomenon, butis a numerical artifact of the finite difference methodwhich disappears within approximately 10 time steps.The initial time step in this calculation was 0.1 s, andthe upturned result has disappeared by approximately1 second. This artifact doesnt appear in the case withresistance because the resistance serves to damp itout.

Figure 7 Model comparison: constant temperature boundary

Let us further compare the true result (the full FDmodel) to the closest simple result, that of the Chiu and

Thakur correlation, considering the wellbore to have the samethermal properties as ground (the C&T Inside Casing case).The following comments relate to the comparison of thesetwo cases

The full model result gives an order of magnitudemore heat transfer over the first ten seconds. This isthe time when heating of the casing steel is dominant.The steel has a high thermal conductivity, and easilyabsorbs a lot of heat for a short period of time.

From 10-100 seconds, the true heat transfer dropssubstantially. This can be regarded as the time whenthe full thickness of the casing has essentiallyreached the imposed casing ID temperature, and heatis being transferred to the cement.

From 100 seconds to one day (depending on the

desired accuracy), the true heat transfer is somewhatless than what is predicted by the simpler solution.During this time, heat is transferred from the casingto the cement layer, increasing its temperature. Thecement (as modeled here) has a lower thermalconductivity than the ground, so the heat transfer rateis less than it would be if it was replaced with amaterial with the same thermal conductivity as theground. (If a lower conductivity was used, such asthat of an insulating cement, one would expect thedifference to be greater.)

100

1000

10000

100000

1000000

1.E-01 1.E+01 1.E+03 1.E+05 1.E+07

HeatTransfer(W/m

)

Time (s)

Full Model

Ground Only

C&T (Inside Casing)

Ground to Casing ID

C&T (Wellbore Interface)

Ground with Resistance


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For times longer than approximately one day (to oneweek, depending on the desired accuracy), the resultssuggest that the simple solution is adequate for manyengineering purposes. If the problem at handrequires an analysis at shorter times, however, thesimple method is not adequate. An example of acalculation which may require the more detailedsolution is a consideration of how the casing andcement react (with regard to thermal stress) whensteam is initially injected in a wellbore(3).

The results presented here were just for a very simplesystem of a single casing and cement in a formation. Clearly,the existence of other layers will complicate this. One wouldexpect that the greater the thermal mass and/or thermalresistance of the combined layers between the inside of thewellbore and the formation, the longer the time until theresults of the simple calculation approach those of the full FDmodel.

Case 1b: Consideration of the thermal mass of casing andcement (Convection Boundary).

Consider the same wellbore configuration, except that

instead of having a constant temperature boundary, aconvection boundary coefficient was applied in the wellbore.The problem geometry, thermal properties and far-field andfluid temperatures were the same as in the previous example,

but a convective heat transfer coefficient of 15 W/mK wasapplied at the inside of the casing. This would tend to besomewhat more realistic than the previous case, in that therate of heat transfer at the very early times is not forced to beextremely high as it is in a constant temperature boundarycase.

Figure 8 shows the results for this scenario. Somecomments about these results are as follows:

The Ground Only FD model, and thecorresponding case using the Jaeger andChamalaun(11) data, the Jaeger & Chamalaun(Wellbore Interface case), have greater error in thiscase than the corresponding results in the constanttemperature case. (Note that these two results largelyoverlie each other in the figure.)

The results for the Ground with Resistance FDmodel are somewhat better (closer to the full modelresults) in this scenario, but still show a reduced rateof heat transfer at early times.

Considering the casing and cement to be thermallyequivalent to ground (as in the Ground to CasingID FD model and the Jaeger & Chamalaun (InsideCasing) casesnote that these cases largely overlieeach other in the figure) seems to work reasonablywell at large times, although it slightly overpredictsthe rate of heat transfer.

Figure 8 Model comparison: convection boundary

Case 2: Variable ground properties

For this example, a scenario of a high temperature thermalrecovery well was considered with an injected steamtemperature of 300C. The thermal conductivities for thesamples shown in Figure 5 from 0C to 300C were roughlylinear from 3.2 W/mK at 0C to 1.5 W/mK at 300C. Wellconsider the same geometry as the previous problem, as wellas the same casing and cement properties. Well also assumethat the thermal conductivities of the casing and cementremain constant. The initial ground temperature will start at0C and steam will be injected at 300C. No phase change ofwater in the ground was considered. A constant temperature

boundary condition was used to simulate this case.Figure 9 (a and b) shows the heat transfer calculated using

a finite difference model considering the thermal conductivityof the formation varying with temperature. It also shows theresults using the Chiu and Thakur correlation with theconductivity set to what it is near the wellbore (HighTemp) and what it is far from the wellbore (Low Temp).

The difference is that Figure 9a shows the results with casingand cement being included along with the formation in themodel, while Figure 9b shows the results with the boundarycondition being applied at the formation interface.

Figure 9a Model comparison: effect of temperature on

thermal conductivity

100

200

300

400

500

600

700

1.E-01 1.E+01 1.E+03 1.E+05 1.E+07

HeatTransfer(W/m)

Time (s)

Full Model

Ground Only

Ground to Casing ID

Ground with Resistance

Jaeger & Chamalaun

(Inside Casing)

Jaeger & Chamalaun(Wellbore Interface)

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E-01 1.E+01 1.E+03 1.E+05 1.E+07

HeatTransfer(W/m)

Time (s)

Full FD Model

Chiu & Thakur (Low Temp)

Chiu & Thakur (High Temp)


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9

Figure 9b Model comparison: effect of temperature on

thermal conductivity

The results presented in Figure 9 show that, for thescenario considered, using both the high and low temperaturethermal conductivity values in the Chiu and Thakurcorrelation brackets the heat transfer for times longer than afew hours. When the casing and cement are included in themodel, the heat transfer is higher at very early times, whenthe casing is being heated, and lower just after that, when thecement (with a lower thermal conductivity than theformation) is being heated.

It does not make sense to apply the same adaptation to asimple model in the constant heat flux or convection case, asthe temperatures in the ground near the well will changesubstantially over time.

Case 3: Phase change

The figures below show an example for a well flowingwarm fluid in a permafrost interval. In this scenario, thegrounds latent heat is 140 MJ/m. The specific heat and

thermal conductivity were also considered to be functions oftemperature, in that one value was used for frozen ground andanother value was used for unfrozen ground. Partially frozenvalues were assigned a weighted average based on thetemperature and the percent of frozen ground. The densitywas considered to be constant. The wellbore includes a layerof casing and cement, with constant properties and no latentheat. The figures show results with and without phasechangethe only difference between the two cases is that theno phase change case had the ground latent heat set to zero.In the example, the ground temperature was set to -15C,while the wellbore temperature was +25C.

Figure 10 shows the rate of heat transfer for both casesand Figure 11 shows the ratio of heat transfer in the caseincluding the phase change to the one that does not.

Figure 10 Model comparison: effect of phase change

Figure 11 Model comparison: effect of phase change (ratio)

For the scenario considered, up to a time of approximately2000 s, the results for the two cases are nearly identical. This

is the time in which most of the heat is simply heating thecasing and cement (which were modeled the same in the twocases). After approximately 2000 s, however, the case withthe latent heat has a greater amount of heat transfer. Thedifference peaks at just over 20% after approximately40,000 s, after which it drops gradually to approximately 5%at 1.2 years. The temperature profile from the casing ID to aradius of 5 m into the permafrost, after 1 year, is shown inFigure 12.

Figure 12 Model comparison temperature profiles

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E-01 1.E+01 1.E+03 1.E+05 1.E+07

HeatTransfe

r(W/m)

Time (s)

FD Model



1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

1.E-01 1.E+01 1.E+03 1.E+05 1.E+07

HeatTra

nsfer(W/m)

Time (s)

No Phase Change

Phase Change

1

1.05

1.1

1.15

1.2

1.25

1.E-01 1.E+01 1.E+03 1.E+05 1.E+07

RatioofHeatTransfer

(PhaseChange/NoPhaseCha

nge)

Time, s

-15

-10

-5

0

5

10

15

20

25

0 1 2 3 4 5

Temperature(C)

Radius (m)

No Phase Change

Phase Change


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10

Because it takes more heat to increase the groundtemperature in the phase change case (due to the latent heat),the temperature stays lower longer. Because the temperaturein the ground is lower, there is a higher gradient, whichtranslates to a greater rate of heat transfer.

To determine how far from the well the ground had fullythawed (assuming that this is at 0C), Figure 12 shows that avery different answer would be reached if the phase change isneglected. The 0C boundary is at 1.12 m when the phasechange is considered, and at 1.28 m when it is not, an error of14%.

Case 4: Shut-ins

Consider a well which is shut down for three days afterevery 90 days of operation over a period of two years. Whenin operation, it was modeled as having a constant temperature

boundary condition. The effects of casing and cement wereignored for this scenario and constant ground properties wereassumed. Figure 13 shows the heat flux from the wellborefor this case. Results obtained using finite differencemethods with and without consideration of the shut-ins arecompared with results obtained using the Chiu and Thakur

correlation. The shutdowns clearly cause a disruption in theheat flux profile, but it is important to note that the resultsrapidly approach the case without shutdowns during theoperation period after each shutdown.

Figure 13 Model comparison: effect of shut-ins

The results for this case, as presented in Figure 13, showthat the simple correlations may be used to get a reasonablygood engineering assessment of thermal efficiency in caseslike this, provided that having some error during the periodsafter restarts does not have a significant impact on therequirements of the scenario being considered. Note that theChiu and Thakur result is not visible in the plot, despite beingon the legend, because it is overlain so closely by the NoShut Ins casethe difference between the two is 0.6% afterone hour, and declines to 0.3% at the end of the modeled

period.

Case 5: Changing temperature over time

Even when injection or production conditions are constantwith time, at depths further from the top of an injection well(or conversely, further from the bottom of a production well),the fluid temperature will change over time. This is becausethe amount of heat lost to the formation decreases over time,as the ground near the well heats up. In Figure 14, the heat

transfer results are shown for a case in which the temperaturein the well was increased linearly from 50C to 70C over a

period of two years, after which it remained constant at 70Cfor an additional three years. The cases of constanttemperatures of 50C and 70C, as calculated using the Chiuand Thakur correlation, are presented for comparison. Nocasing or cement is modeled in this case. The properties ofthe ground were assumed to be constant over the temperaturerange. The initial and far ground temperatures were set to20C.

Figure 14 Model comparison: effect of changing boundary

condition

As one would expect, the heat transfer profile for the lowtemperature case matches that of the full FD model very wellover the first days or weeks. Over time, it deviates, and theheat flux increases towards the high temperature case. Asone would also expect, when the temperature at the wellboreafter two years is held constant, the heat flux approaches thehigh temperature value.

Assumptions

In this study, we examined many methods of calculatingthe heat transfer from a short section of wellbore to thesurrounding formation. For each case presented, certainassumptions were made, depending upon the method. Someof the key assumptions, pervasive across most or all of themethods include the following:

There is no axial heat transfer (other than by masstransfer in the wellbore); all heat transfer is radial.

The section of wellbore being considered is shortenough that there is no significant change in thetemperature of either the wellbore fluid or the far groundover the length of the section. Sufficiently large changes

in the temperature over a segment length can causesignificant calculation errors and artifacts. In anywellbore model, one should check that the change intemperature along any one segment is much smaller thanthe temperature difference between the fluid and the farground. It is possible, however, to compensate for this;Ramey(1) did so, and Skoczylas(17) also did so in thecontext of a finite difference model.

The far (undisturbed) ground temperature is equivalentto the initial temperature, and is constant in time. Thisassumption can be overridden in a finite difference (or

1.E+02

1.E+03

1.E+04

1.E+05

0 200 400 600 800

HeatTransfer(W/m)

Time (d)

With Shut Ins

Chiu & Thakur

No Shut Ins

100

1000

0 500 1000 1500 2000

HeatTransfer(W/m)

Time (d)

Full Model




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11

FEA) calculation, where any initial temperature profilecan be considered.

The system is axisymmetric. Some key situations inwhich this assumption may be violated are:

o Non-vertical wells; in these, the fartemperature at some distance from thewellbore, perpendicular to the axis of the wellvaries with direction.

o Non-concentric tubularsfor example casingthat is not centralized prior to being cementedin place.

There is no mass transfer, other than in the wellboreitself. These models are not designed to handle casessuch as:

o In injection or production intervals; othermethods need to be considered, if necessary,in intervals open to the reservoir

o Movement of formation fluids adjacent to thewellbore. This can cause significant increasesin heat transfer, as illustrated by Liu et al (23).

The density of the ground is constant. While changes inthermal conductivity and specific heat are permitted inthe finite difference models, changes in density imply

movement of the ground or of fluid within the ground,and the consideration of this was deliberately avoided inthis study.

There is no time-dependent heat transfer other thanconduction. Most of the correlation-based modelsignore transient conduction anywhere other than theground, while the finite difference models allow fortransient conduction in casing and cement (and even intubing strings within the casing, under certainconditions). But none of the models described hereconsider (for example) the time dependency ofdeveloping natural conduction in a tubing-casingannulus.

ConclusionsUnder these assumptions (and others specific to each

method), several conclusions can be made, including thefollowing:

Numerical solutions, such as the finite differenceapproach are the most flexible and robust of themethods that can be used to determine wellbore heattransfer rates. They can handle complicated

problems in an accurate way that the otherapproaches simply cannot match. The only realdrawback to numerical methods is the requiredcomputational power and the associated time to reacha solution. (If overlapping effects from multiplewells are to be considered, FEA methods would tendto be required, rather than FD models.)

For many simple cases, correlations fit to the resultsof the exact solution methods can work very well.Such correlations, however, cannot generally be usedat shorter times (under one day in typical wellbores),regardless of the accuracy of the correlation, becausethey do not consider the thermal mass of the casingand cement (and any other tubulars/annuli in thewell).

Correlation results may not be accurate when theboundary conditions are changing. This is especially

true early in the life of a well or during other periodsof transition such as during shutdowns and restarts.Only when conditions have stabilized do the resultsobtained from such correlations approach an accurateresult.

Ignoring (such as by the use of correlations) theeffects of phase change in scenarios when it canoccur (e.g. in permafrost) can lead to significanterrors.

Once a changing boundary condition stabilizes, theprior history seems to have minimal importance, andthe results will, in a reasonable time frame, approachwhat they would have been had the boundaryconditions been held at that value from the start.Correlation methods can therefore be used after sometime from a startup (or a restart), or any other changein operating conditions.

In cases, where correlations are to be used forsimplicity or computational efficiency, the followingrecommendations are made:

o For constant temperature problems, theChiu and Thakur(6) method is recommendeddue to its simplicity and accuracy.

o For constant flux problems, the revisedHasan and Kabir(8) method isrecommended, also due to its simplicityand accuracy.

o For problems with a convective boundarycondition in the well, the Jaeger andChamalaun(11)method can work very well.Unfortunately, no accurate correlation isavailable over a full range of times. Thechoices for using the Jaeger andChamalaun method are currently:

Interpolate from a table of data.This is a reasonable approach inmany circumstances.

For very small or very large

times, use the approximationsprovided by Jaeger andChamalaun(11).

Perform a difficult numericalintegration. Unless theconditions fall outside the rangeof conditions shown in the Jaegerand Chamalaun table, thereshould be no real benefit to doingthis, while there is a significantcomputational cost.

o With correlation methods, the casing andcement (and other wellbore tubular/annuli,as appropriate) should be modeled asground, or an equivalent resistance should

be applied. The error from doing this issignificantly less than the error fromapplying the wellbore boundary conditionat the formation interface instead. Theresults will generally be valid for mostwellbore scenarios after approximately oneday of elapsed time from start-up.

o When the thermal conductivity of theformation varies with temperature and aconstant wellbore temperature exists, thethermal conductivity of the formation


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12

should be evaluated at the wellboretemperature.

o There is generally no reason to use linesource methods. They work fine at longtimes (so long as you are interested inconstant flux results), but they are no easierto use and provide no better results thancorrelation methods.

A Final Note

While this paper has not addressed the problem ofcalculating temperatures throughout the wellbore, a final noteon this topic is warranted. Many authors(1,4,5,6,7,8,9) have

presented approaches to predict wellbore temperatureprofiles, usually by using steady state or quasi-steady statesolution approaches. But in very short time cases (such asthe cement stress problem mentioned earlier), this is notadequate.

Consider a simple example of a small 3.5 tubingcemented inside a 6 diameter vertical hole, 300 m deep. Theground temperature near surface is 10C, and at 300 m it is

30C. At time 0, the well is filled with water at thermalequilibrium with its surroundings. Starting at time 0, hotwater at 70C is injected at a rate of 4 litres per second (thisgives a velocity of just over 0.88 m/s). What does thetemperature profile in the well look like after 170 seconds,when the fluid front has reached 150 m? (Note that this isassuming there is no mixing between the cold fluid and thewarm fluid; i.e. the fluid front is always at a single depthacross the cross section of the tubing.) This case wasexamined using separate finite difference models at severaldifferent depths along the well. Each element of fluid (thevolume contained within the tubing for each segment in thewell) was tracked as it moved from its initial location (orfrom surface for injected fluid) down the well, changingtemperature in proportion to the rate of heat transfer.

Figure 15 shows the results of this example for three differentboundary conditions: a constant temperature boundary(where the casing ID is assumed to be equal to the fluidtemperature), a convection boundary, and a perfectlyinsulated wellbore.

Figure 15 Temperature in wellbore soon after initiation of

injection

With examination of the three temperature profiles presentedin Figure 15, a key point is that the fluid in the wellbore

moves downward as fluid is injected. This means that warmcasing/cement that is deeper in the well is first exposed tocolder fluid from higher in the well before it is exposed to thewarmer injected fluid. The thermal stress experienced by thewellbore casing and cement would therefore be increasedrelative to what it would be if this effect was not considered,such as in a quasi-steady state solution.

Nomenclature

c= specific heat, J/kgKFo= Fourier numberh= convection coefficient, W/mK(), ()= Hankel functionsi = nodal index

J0,J1= Bessel functions of the first kindk= thermal conductivity, W/mK

K0,K1= modified Bessel functions of the second kindL{} = Laplace transform operatorq= heat flux per unit length, W/mr= radius, mrwb= wellbore radius, m

s= complex argument used in the Laplace transformed spacet= time, sT= temperature, CTD= dimensionless temperatureTD0= reference dimensionless temperatureTf = fluid temperature, CTwb= temperature at the wellbore/formation interface, CT= far ground temperature, CT = difference in temperature between thewellbore/formation interface and the far ground, Cu= variable of integrationV= nodal volume per unit length of wellbore, m/m

X,Y= variables used to collect termsY0, Y1= Bessel functions of the second kind

z= complex variable= thermal diffusivity, m/s= dimensionless convection coefficientr= nodal spacing, mt= time step, s = 0.5772156649, Eulers constant (also known as theEulerMascheroni constant)= density, kg/m= dimensionless convection function

References

1. RAMEY, H.J. JR., Wellbore Heat Transmission;Journal of Petroleum Technology, pp. 427-435, April1962.

2. XIE, J., and MATTHEWS, C.M., Methodology toAssess Thaw Subsidence Impacts on the Design andIntegrity of Oil and Gas Wells in Arctic Regions;2011, SPE 149740.

3. XIE, J., and ZAHACY, T.A., Understanding CementMechanical Behavior in SAGD Wells; WHOC11-557, 2011.

4. HASAN, A.R., and KABIR, C.S., Heat Transferduring Two-Phase Flow in Wellbores: Part I -Formation Temperature;1991, SPE 22866.

10

20

30

40

50

60

70

0 50 100 150 200 250 300

Temperature(C)

Depth (m)

Perfectly Insulated

Constant Temperature

Boundary

Convection Boundary


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5. WILLHITE, G. PAUL., Over-all Heat TransferCoefficients in Steam and Hot Water Injection Wells;

Journal of Petroleum Technology,May 1967.6. CHIU, K. and THAKUR, S.C., Modeling of Wellbore

Heat Losses in Directional Wells Under ChangingInjection Conditions;1991, SPE 22870.

7. MOINI, B., and EDMUNDS, N., Quantifying HeatRequirements for SAGD Start-up Phase: SteamInjection and Electrical Heating; WHOC11-513, 2011.

8. HASAN, A.R., and KABIR, C.S., Fluid Flow andHeat Transfer in Wellbores; Richardson, TX : SPE,2002.

9. HAGOORT, J., Ramey's Wellbore Heat TransmissionRevisited; SPE Journal. SPE 87305, 2004.

10. CARSLAW, H.S and JAEGER, J.C., HeatConduction in Solids;Oxford University Press, 1959.

11. JAEGER, J.C., and CHAMALAUN, T., Heat Flow inan Infinite Region Bounded Internally by a CircularCylinder with Forced Convection at the Surface;

Australian Journal of Physics, Vol. 19, pp. 475-488,1966.

12. http://www.cambridge.org/us/engineering/author/nellisandklein/downloads/invlap.m

13. JAEGER, J.C., Heat Flow in the Region BoundedInternally by a Circular Cylinder;Proceedings, RoyalSociety of Edinburgh, pp. 223-228, 1942.

14. VAN EVERDINGEN, A.F., and HURST, W., TheApplication of the Laplace Transform to FlowProblems in Reservoirs; Petroleum Transactions,

AIME, pp. 305-324, December 1949.15. JAEGER, J.C., and CLARKE, M., A Short Table of I

(0,1;x),Proceedings, Royal Society of Edinburgh, pp.229-230, 1942.

16. CLAUSER, C., and HUENGES, E., ThermalConductivity of Rocks and Minerals. Rock Physicsand Phase Relations - A Handbook of PhysicalConstants; AGU Reference Shelf. AmericanGeophysical Union., Vol. Vol. 3, pp. 105-126.

17. SKOCZYLAS, P., A Method for CalculatingTransient Temperature and Pressure Profiles forCrude Oil and Water Flowing in a Buried Pipeline;Univeristy of Alberta, 2001. M.Sc. Thesis.

18. PHAM, Q.T., A Fast, Unconditionally Stable FiniteDifference Scheme for Heat Conduction with PhaseChange; No. 11, 1985, Int. J. Heat Mass Transfer,Vol. Vol 28.

19. OSTERKAMP, T.E., Freezing and Thawing of Soilsand Permafrost Containing Unfrozen Water or Brine;

No. 12, Water Resources Research, Vol. Vol. 23,December 1987.

20. MOTTAGHY, D., and RATH, V., Implementation ofPermafrost Development in a Finite Difference HeatTransport Code;[Online] RWTH-Aachen University.

http://www.eonerc.rwth-aachen.de.21. WILLIAMS, P.J., Unfrozen Water Content of FrozenSoils and Soil Moisture Suction;No. 3, Geotechnique,Vol. 14, September 1964.

22. OZISIK, M.N. and UZZELL, J.C. JR., Exact Solutionfor Freezing in Cylindrical Symmetry with ExtendedFreezing Temperature Range; Journal of HeatTransfer, Vol. 101, pp. 331-334,May 1979.

23. LIU, Z., STARK, S., and LUNN, S., Modeling ofWellbore Heat Loss for Thermal Operations at Cold

Lake A Convection Cell Approach; WHOC11-628,2011.

Appendix A Correlations

The correlations from the various sources presented in thispaper are listed here.

Ramey(1)(line source, constant flux):

= ln 0.29 ............................................... Equation A-1Hasan and Kabir(4)(constant flux):

= 1.128110.3 1.50.4063+0.5ln() 1 + . > 1.5 ................................................................................................................... Equation A-2

Hasan and Kabir(8)(constant flux):

= ln.+ (1.50.3719) ..... Equation A-3Carslaw and Jaeger(10)(constant flux, short times):

2 0.25..................................... Equation A-4Carslaw and Jaeger(10)(constant flux, long times):

(ln4 ) ................................................... Equation A-5Chiu and Thakur(6)(constant temperature):

= 0.982ln1+1.81.................................. Equation A-6Moini and Edmunds(7)(constant temperature):

log = 0.0024(log )+0.0446(log )0.3064(log ) 0.0126 .......................................... Equation A-7This correlation should only be used in the range of

Fourier numbers between 0.01 and 1000.

Carslaw and Jaeger(10)(constant temperature, short times):

() + + ................................ Equation A-8Carslaw and Jaeger(10)(constant temperature, long times):

2 () (()) ................................... Equation A-9


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Appendix B Derivation of a FiniteDifference Model

This Appendix contains an example derivation of a finitedifference heat transfer model.

Consider three adjacent radial nodes in an axisymmetricsystem. These nodes are an inner node (i-1), a middle node

(i), and an outer node (i+1). In the simplest case, which wewill consider here, the properties around the nodes areassumed to be constant and the nodes are equally spaced.

The heat flux from the inner node to the middle node canbe determined using the following radial heat transferequation:

= () ................................................................. Equation B-1Similarly, the heat flux from the outer node to the middle

node is:

= ()

.................................................... Equation B-2

Note that if the heat was flowing from the middle node toone of the other nodes, that flux value would be negative.

Before we can look at what happens during a time step,we need to know the volume around the centre node; thisvolume is considered to be uniformly at the temperature ofthe node. The volume per unit length or depth (since q ismeasured per unit length) is:

= 2 ......................................................................... Equation B-3During a time step, the total energy entering (or leaving)

the volume must be the same as the change in energy stored(or lost) in the volume:

(+ ) = ................................... Equation B-4In this notation, refers to the temperature at node i at

time t.Combining Equations A1-A4 yields:

() + () = 2 .............................................................................................................. Equation B-5

Some constants can be cancelled, we can substitutediffusivity for the combination of density, specific heat andthermal conductivity, and we can solve the resultingrelationship for the change in temperature of the centre nodeduring the time step, leaving:

= () + () ................. Equation B-6Alternatively, a more useful way of writing this might be:

= + ( ) + ( )..............

............................................................................................... Equation B-7

or by lumping variables as:

= + ( ) + ( )............... Equation B-8where:

= .................................................................. Equation B-9 = ................................................................ Equation B-10

It has not yet been specified whether most of thetemperature terms on the right hand side of the equation areevaluated at time tor t+t. It might seem obvious to use timet; this is called an explicit method. If we do this, thecalculation procedures are very simple, but there is a problemwith stability if the time steps are not kept sufficiently small,which may make our calculation take much longer. On theother hand, if we evaluate those temperatures at time t+t, wehave to solve a system of simultaneous linear equations, butwe do not have a stability problem with longer time steps.This is called an implicit method, and was chosen for thiswork. Equation A-8 is then rewritten as:

+ (1 + + ) = ....... Equation B-11This is now in a form which is easily adapted to matrix

methods for the simultaneous solution of a system of linearequations, such as by the method of Gaussian elimination.The methods for setting up the matrix and then solving it arenot discussed further here but were given by Skoczylas(17)fora more complicated two-dimensional heat transfer problemsolved with the FD method.

To derive the equations for different situations, such asvariable nodal spacing, or different properties around eachnode, or for the presence of a discontinuity at a boundary

between regions, the same basic process is used. That is,where energy flowing into the node during a time step isequated to the energy storage in the volume around the node.

Appendix C Inputs Used inScenarios Presented

Unless otherwise specified, the material thermal propertiesused in the problems described here are listed in Table C-1.

Material Thermal

Conductivity,W/mK

Density,

kg/m

Specific

Heat, J/kgK

Casing 45 7850 450Cement 1.2 2000 1000Formation 3.0 2200 1200

Table C-1 Thermal Properties

When used, the casing OD is 9.625, the ID is 8.835.The casing is cemented in a hole with a diameter of 13. Incases with no casing or cement, the hole diameter is also 13.


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Case 1aThe far-field and initial temperatures are 5C. The

imposed wall temperature at the inside of the wellbore is50C.

Case 1bThe far-field and initial temperatures are 5C. The fluid

temperature inside of the wellbore is 50C, and there is aconvection coefficient of 15 W/mK.

Case 2The ground thermal conductivity varies linearly from

3.2 W/mK at 0C to 1.5 W/mK at 30C.The far-field and initial temperatures are 0C. The

imposed wall temperature at the inside of the wellbore is300C. Note that in this case, there is no consideration of

phase change effects.

Case 3The far-field and initial temperatures are -15C. The

imposed wall temperature at the inside of the wellbore is25C.

The frozen ground thermal conductivity is 3.6 W/mK and

the unfrozen ground thermal conductivity is 2.5 W/mK. Thespecific heat of frozen ground is 1103 J/kgK and the specificheat of unfrozen ground is 1500 J/kgK. For partially frozenground, these properties are a weighted average, based on theunfrozen content. The unfrozen content is shown as afunction of temperature in Figure B-1. The grounds latentheat is 140 MJ/m

Figure B-1 Percent Unfrozen with Temperature

Case 4This case has no casing or cement. The far-field and

initial temperatures are 20C. The imposed wall temperatureat the inside of the wellbore is 75C.

Case 5This case has no casing or cement. The far-field and

initial temperatures are 20C. The imposed wall temperatureat the inside of the wellbore is 50-70C, as described in thetext.

0

10

20

30

40

50

60

70

80

90

100

-10 -8 -6 -4 -2 0

%

Unfrozen

Temperature, C

Documents

A Review of Methods for Calculating Heat Transfer From a Wellbore