PROCEEDINGS, Fortieth Workshop on Geothermal Reservoir Engineering

Stanford University, Stanford, California, January 26-28, 2015

SGP-TR-204

1

Analysis of Wellhead Production Temperature Derivatives

Kaan Kutun, Omer Inanc Tureyen, Abdurrahman Satman

Istanbul Technical University Maslak 34469 Istanbul Turkey

kutunk@itu.edu.tr, inanct@itu.edu.tr, mdsatman@itu.edu.tr

Keywords: Geothermal, wellhead temperatures, derivatives.

ABSTRACT

It is commonplace to take pressure and temperature measurements along the wells in geothermal reservoirs. Such profiles give good

insight into the characterization of the geothermal reservoir. Many useful information such as upper and lower boundaries of reservoirs,

deliverability of wells, the nature of the phase changes within the wells can be obtained as a result of these profiles. However in most

cases taking these profiles along the wells can be challenging. Furthermore these types of profiles cannot be considered at all times since

the wells would have to be shut in and there is always the economical factor.

In this study we look at in more detail the behavior of the wellhead temperature, the only place where temperature can be measured at

all times and is not an economical burden. We specifically study the behavior of the temperature derivative obtained at the wellhead.

Just as in a well testing diagnostic plot the temperature derivative at the wellhead displays clearly various flow regimes that take place in

the well. At early times when wellbore convection dominates, a log-log plot of dT/dlnt versus t gives a unit slope straight line similar to

early-time pressure transient behavior dominated by wellbore storage. The intermediate-time behavior is observed when heat transfer

between wellbore and the surrounding formations is felt and is recognized by -1/2 slope straight line on a log-log plot of dT/dlnt versus

t. At late times the wellbore heat transmission solution converges to the cylindrical-source solution transferring heat at constant

temperature.

1. INTRODUCTION

Generally speaking, methods and models describing the wellbore heat transmission is based on a wellbore heat balance with some

assumptions. The most critical assumptions refer to the condition between the wellbore and the surrounding formations. Either a

constant heat flux or a constant temperature assumption is used for this purpose. In the case of the general wellbore heat problem,

neither heat flux nor the temperature at the wellbore remains constant except in special cases. However, the solutions for the cases of

sources loosing heat at constant flux and constant temperature eventually converge at long times.

Most of the literature on wellbore heat transmission is based on the classical work by Ramey (1962). He derived the temperature

distribution in a well used for injecting hot fluid. Ramey (1964) expanded on this to estimate the rate of heat loss from the well to the

formation. Horne (1979) reexamined the problem to determine the wellbore heat loss in production and injection wells. Hagoort (2004)

assessed Ramey’s classical method for the calculation of temperatures in injection and production wells and showed that Ramey’s

method is an excellent approximation.

Carslaw and Jaeger (1959) present graphical and analytical solutions for the cases of internal cylindrical sources losing heat at constant

flux, constant temperature and the radiation boundary conditions. Solutions converge at long times. This is a sufficiently long time at

which temperature is controlled by formation conditions.

For small values of time heat flow in the wellbore is controlled by convection, rather than conduction. Ramey recommends using the

constant-temperature cylindrical-source solution if thermal resistance in the wellbore is negligible that is the case when the fluid flow

occurs thru casing only.

Our study here considers only single-phase fluids flowing in the well. The single-phase flow analysis is based on the determination of

the fluid temperature as a function depth and time. For simplicity, most of our work here assumes a geothermal well with single-phase

liquid flowing in a casing without tubing, however it is not difficult to derive the equations and give the expressions for fluid flow

within tubing case.

Ramey’s solutions for temperature are obtained in terms of depth. However the effects of varying formation temperature as a function of

depth in terms of geothermal temperature gradient as well as heat transfer to the surrounding formations by transient conduction are

considered in those solutions.

The equation for the evaluation of the temperature in a producing geothermal well is (Ramey, 1962):

𝑇 = (𝑇𝑏ℎ − 𝛼𝑦) + 𝛼𝐴 (1 − 𝑒−𝑦

𝐴) (1)

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where y is the distance upwards from the bottom of the well, Tbh is the downhole reservoir temperature, (Tbh-αy) is the temperature of

the earth (Te) assuming linear geothermal gradient, and α is the geothermal gradient (the increase of formation temperature with increase

in depth).

A is a group of variables defined as

𝐴 =𝑤𝑐𝑓(𝑡)

2𝜋𝑘 (2)

where w is the mass flow rate, c is the thermal heat capacity (specific heat) of the fluid (assumed constant), k is the thermal conductivity

of the formation (earth), and f(t) is a dimensionless time function representing the transient heat transfer to the formation.

The function f(t) may be found from Ramey (1962 and 1964), or alternatively may be approximated from the line source solution for

flowing times greater than 30 days or so by the equation (Ramey’s large time solution):

𝑓(𝑡) ≈ −𝑙𝑛 (𝑟𝑤

2√𝜅𝑡) − 0.29 (3)

where κ is the thermal diffusivity of the formation and rw is the radius of the casing.

Heat transfer from the casing to the formation in terms of heat flow rate per unit of length is given by:

𝑞 =2𝜋𝑘

𝑓(𝑡)(𝑇 − 𝑇𝑒) (4)

where T is the temperature of the fluid in the casing and Te is the temperature of the formation. Thus the total heat-loss rate from a well

of total depth L can be given as (Horne, 1979)

𝑞𝑡 = 𝛼𝑤𝑐 [𝐿 + 𝐴 (𝑒−𝐿

𝐴 − 1)] (5)

Application of the Ramey equation for the wellbore fluid temperature requires as input f(t) function at a certain time. Various studies in

the literature propose expressions to estimate f(t). These expressions are given in the following discussion.

1.1 f(t) Correlations For A Cylindrical Source With Constant Temperature and With Constant Heat Flux

1.1.1 A circular cylinder with constant heat flux at the surface case

It is well-known that in this case the formulation has a solution in complex integral form (Van Everdingen and Hurst (1949); Carslaw

and Jaeger (1959)).

For large values of the time: For large values of the time, Carslaw and Jaeger (1959)’s solution yields the following expression for a

circular cylinder of radius rw, with constant heat flux at the surface (r=rw):

𝑇 − 𝑇𝑒 =𝑞

2𝜋𝑘

1

2𝑙𝑛 (

4𝑡𝐷

𝐶) (6)

where

𝑡𝐷 = 𝜅𝑡𝑟𝑤2 (7)

and C=eγ=e0.57722=1.781.

Ramey’s heat flux solution is given by Eq. 4. Comparison of Eqs. 6 and 4 gives:

𝑓(𝑡) =1

2𝑙𝑛 (

4𝑡𝐷

𝐶) = 𝑙𝑛(√2.246𝑡𝐷) = ln(1.4986√𝑡𝐷) (8)

For small values of the time: Kutasov (2003) indicates that for an infinite cylindrical source of heat the following expression

represents f(t) at small values of tD:

𝑓(𝑡) = 2√𝑡𝐷

𝜋 (9)

1.1.2 A circular cylinder with constant bore-face temperature case

Carslaw and Jaeger (1959) gave a solution for this case. However the solution is in a complex integral form and thus it is not repeated

here. Later on Ramey (1962, 1964, and 1981) presented f(t) results in graphical and tabulated forms. Kutasov (1987) and Kutasov and

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Eppelbaum (2005) used a semi-theoretical approach to obtain the following expression to represent f(t) for the case of a flow in a

wellbore with constant wellbore temperature at the bore-face:

𝑓(𝑡) = 𝑙𝑛 [1 + (1.571 −1

4.959+√𝑡𝐷)√𝑡𝐷] (10)

For large values of tD, Eq. 10 reduces to:

𝑓(𝑡) = ln(1.77√𝑡𝐷 + 1) (11)

In this study we propose the following simplified expression for f(t):

𝑓(𝑡) = 𝑙𝑛(1 + 1.7√𝑡𝐷) (12)

This equation is based on a best curve fit of Ramey’s f(t) data (Ramey; 1962, 1964, and 1981) in the pertaining time interval and is

accurate within 1% as will be discussed later. Furthermore for values of tD>20 the following expression within about 5% accuracy can

be used:

𝑓(𝑡) = 𝑙𝑛(1.7√𝑡𝐷) (13)

Satman et al. (1979, 1984) discussed the time-dependent overall heat-transfer coefficient concept to heat-transfer problems and

introduced the following expression to represent f(t):

𝑓(𝑡) = √𝜋𝑡𝐷 (14)

Notice that Eq. 14 is identical in form to Eq. 11 given by Kutasov and Eppelbaum (2005) after modification for small values of tD using

the simplification of ln(1+x)=x.

Figures 1 and 2 compare the various f(t) correlations discussed above. The constant heat flux line source model yields an f(t) being equal

to (1/2)Ei(1/4tD) where Ei(x) is the exponential integral of x. The numerical data representing the constant temperature at cylindrical

source model in Figures 1 and 2 are taken from Ramey (1981).

Figure 1: Comparison of f(t) correlations on a semilogarithmic graph.

A detailed analysis of the correlations given in Figures 1 and 2 indicates that Eq. 12 for all times, Eq. 9 for early times, and Eq. 13 for

late times proposed in this study are quite adequate for most engineering purposes. Equations 10 and 12 match the dimensionless time

function curve for cylindrical source with a constant temperature at wellbore presented by Ramey (1962, 1964, and 1981), Earlougher

(1977) and Hagoort (2004). The match is acceptable for practical engineering calculations and Eqs. 12 and 13 have much easier forms

to be used.

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Figure 2: Comparison of early- and late-time f(t) correlations on a semilogarithmic graph.

One particular observation about the heat transfer models compared in Fig.1 is that the line source with a constant heat flux yields

lower f(t)’s than the cylindrical source with a constant temperature, although they eventually converge at long times at about tD=105

(Hagoort, 2004). The difference in f(t)’s is considerable, about 50%, at tD=1 and becomes lower as tD grows, about 5% at tD=1000.

1.2 Temperature Solution At Early-Time Wellbore Convection Dominated Period

The parameter A in Eq. 1 expresses the ratio of the heat transported up the wellbore by convection to the heat loss by conduction in the

formation. For large values of A (high production rates), the heat transport is dominated by convection and little heat will be lost to the

formation.

Ramey’s solution (Eq. 1) containing the parameter A for the temperature distribution is based on the following two simplifying

assumptions in the wellbore heat balance: (i) the initial accumulation term up to the filling of the welllbore is negligible, and (ii) the heat

loss term can be replaced by the product of the f(t) function and temperature. The first assumption definetely does not hold during the

early stages when the hot fluid entering the wellbore at the bottom is displacing the initial wellbore fluid. In this period, the temperature

profile in the wellbore is coupled to the displacement front, the movement of which is controlled by the accumulation up to the filling of

the wellbore.

Using rigorous solution of the wellbore and formation heat-balance equations Hagoort (2004) proposed the following early-time

solution for the temperature ahead of the fluid front ( (y/L)>(ut/L));

𝑇 = 𝑇𝑏ℎ − 𝛼𝑦 + 𝛼𝑢𝑡 (15)

where u is the linear velocity in the wellbore, and L is the wellbore length. Early means a time long before the fluid front reaches the

wellbore. Equation 15 reflects the displacement of the initial fluid with the linear temperature profile from the wellbore, the profile

ahead of the front moves up linearly proportional to time.

Hagoort (2004) also presented the following early-time solution (ahead of the displacement front, (y/L)>(ut/L) ) based on heat balance

equations assuming constant heat-loss function:

𝑇 = 𝑇𝑏ℎ − 𝛼𝑦 + 𝛼𝐴{1 − 𝑒𝑥𝑝(−𝑢𝑡/𝐴)} (16)

Equation 16 represent the wellbore temperature distribution before the arrival of the front at the wellhead for t<L/u.

Comparing the rigorous solutions obtained from the heat balance model, Hagoort (2004) concluded that Ramey’s method is an

excellent approximation, except for an early transient period in which the calculated temperatures are significantly overestimated.

1.3 General Characteristics of Production Temperature Profiles

Figure 3 presents a comparison of temperatures obtained in a water-flowing well with temperatures computed for production a

conditions. The thermal properties used in modeling the temperature profiles and related discussions are given in Table 1.

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Table 1-Input parameters for the cases discussed

Earth temperature at surface, oC 20

Rock density, kg/m3 1952

Formation thermal conductivity, J/m.s.oC 2.92

Formation specific heat capacity, J/kg.oC 1000

Formation thermal diffusivity, m2/D 0.129

The water-production rate is 20 kg/s, the bottomhole reservoir temperature is 245 oC, the surface (earth) temperature is 20 oC, and the

wellbore radius is 0.15 m. Figure 3 illustrates the general features of the wellbore temperature distribution. The static temperature

profile (the geothermal gradient) represents the formation temperature as a function of depth for undisturbed wellbore conditions.

The dynamic temperature profiles for the production case are plotted at four successive times; 0.1, 1, 10, and 30 days. The potential

importance of time on wellbore heat transmission is shown by Fig. 3.

Figure 3: Temperature profiles at successive times for production case.

As seen in Figure 3, the dynamic production profile for the smallest time of 0.1 day is concave, reflecting the greater effect of heat

losses from wellbore to surrounding formations at shallower depths. Because of continuous heating of the formation by the hot water in

wellbore, heat losses become progressively smaller with time. Therefore, the temperature profile moves to the right, closer to the inlet

bottomhole temperature.

Figure 3 shows how the temperature profile inside the well evolves with time. It is important to note that, most of the change occurs

during the first 10 days. Then the changes in profile with time become very small. Two main mechanisms of heat transfer play a role in

the changing well temperature profile. The first is the convective heat transfer from the bottom of the well to the top via production of

the fluid. The second is the conductive heat loss to the surroundings of the well. In the final profile at 30 days, the wellhead temperature

gets closer to the initial reservoir temperature. This difference between the wellhead temperature and the bottomhole temperature is

because of the heat losses to the surroundings of the well.

The effect of time on temperature profiles are clearly demonstrated in Fig. 2. As production time increases, the temperature at a given

depth increases. The change in temperature at a given depth with time is very rapid at first, then decreases markedly. Heat transfer rate

at early times is controlled by convection in wellbore but at large times it is controlled by the rate at which heat is conducted within the

surrounding media.

2. WELLHEAD TEMPERATURE DERIVATIVE BEHAVIOR

The general theory on wellbore heat transmission is already discussed in previous sections. Figure 4 schematically illustrates the

general characteristics of the wellhead temperature (Ttop) as a function of time. The temperature behavior consists of four time periods;

an early-time period, a transition period, an intermediate-time period, and a late-time period. In Fig. 4, ΔT corresponds to the difference

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between the wellhead temperature Ttop and the surface earth temperature Tsurf. It is determined from Eq. 1 which can be written in terms

of the Ramey Number, NRa=2πkL/(wc):

∆𝑇 =(𝑇𝑏𝑜𝑡−𝑇𝑠𝑢𝑟𝑓)

𝑁𝑅𝑎𝑓(𝑡) [1 − 𝑒𝑥𝑝 (−

𝑁𝑅𝑎

𝑓(𝑡))] (17)

where Tbot is the bottom-hole temperature. In the early-time period the cylindrical source constant temperature solution (Ramey; 1962,

1964, 1981) represented by the dashed-curve overestimates the wellhead temperature. Equation 15 is used to determine the wellhead

temperature at early-time period and Hagoort (2004) indicates that the duration of this period depends on 𝜅𝐿/(𝑢𝑟2); the larger the

𝜅𝐿/(𝑢𝑟2), the longer the early-time period. A transition period exists after the early-time period and the wellhead temperature solution

converges with the cylindrical source constant temperature solution about one and half log cycle after the wellbore convection

dominated early-time period which is diagnosed by a unit slope straight line relationship on a log ΔT-log t graph to be discussed in the

following section. The wellhead temperature behavior in the intermediate-time period is represented by Eq. 1 with a dimensionless time

function f(t) described by an expression proportional to √𝑡𝐷 such as Eqs. 9 and 14. The duration of this period depends on 𝜅𝐿/(𝑢𝑟2);

the larger the 𝜅𝐿/(𝑢𝑟2), the shorter the intermediate-time period.

Throughout the time period in which the cylindrical source solution is valid Eq. 1 describes the wellhead temperature (or temperature at

any y) employing Eq. 12 for f(t). In the late-time period, Eq. 13 is used an approximation and simplification of f(t) for tD>20.

Figure 4: General characteristics of the wellhead temperature.

In the following sections the effect of time on the temperature behavior in terms of temperature derivatives is assessed.

2.1 Early-Time Temperature Derivative Behavior

For the temperatures measured at the wellhead (y=L), Eq. 16 which represents the wellbore temperature profile before the arrival of the

displacement front at the wellhead is employed. Eq. 16 can be modified to:

∆𝑇 = 𝑇−𝑇𝑠𝑢𝑟𝑓 = 𝛼𝐴{1 − 𝑒𝑥𝑝(−𝑢𝑡/𝐴)} (18)

Using the relationship exp(-x)≈1-x since x is too small, Eq. 18 reduces to

∆𝑇 = 𝛼𝑢𝑡 (19)

Eq. 19 is identical to Eq. 15 given by Hagoort (2004).

Now we can take derivatives of ΔT with respect to time t in terms of dΔT/dt and dΔT/dlnt;

Unit Slope

t<L/ut<κL/(ur2)

Early-Time Wellbore Convection Dominated

Period

Cylindrical Source Constant TemperatureRamey (1962, 1964, 1981)

Late-Time Period

tD>20.0

Transition Period ~1.5 log cycle

Intermediate-Time Period,

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dΔT/dt=αu (20)

and

dΔT/dlnt=αut (21)

Equation 20 indicates that the time derivative of wellhead temperature change stays at a constant value, αu, and moreover Eq. 21 shows

that a log-log plot of dΔT/dlnt (or tdΔT/dt) versus t gives a unit slope straight line.

The early-time temperature behavior is similar to the wellbore storage phenomenon occuring in oil and gas production wells and

affecting short-time transient pressure behavior (Earlougher, 1977). The temperature behavior in geothermal wells and the pressure

behavior in oil and gas wells are both due to the liquid stored in the wellbore when the liquid level rises.

Equations 19 and 21 both have a characteristic that is diagnostic of wellbore convection dominated period: the slope of the ΔT and

dΔT/dlnt versus t graphs on log-log paper is 1.0 during wellbore convection domination, respectively. Temperature-time data falling on

the unit slope of the log-log plot reveal nothing about the surrounding formation properties since all production is from the wellbore

during that time.

Note that the location of the log-log unit slope can be used to estimate the value of αu from Eq. 19. ΔT and t are values read from a point

on the log-log unit slope straight line. αu is calculated from Eq. 19.

2.2 Intermediate-Time Temperature Derivative Behavior

Using exp(-x)≈1-x+x2, the Ramey’s wellbore heat transmission equation, Eq. 1, for the wellhead temperature reduces to the following

intermediate-time temperature solution valid after the displacement front reaches the wellhead:

∆𝑇 = 𝛼𝐿 −𝛼𝐿2

2𝜋𝑘

𝑤𝑐

2𝑓(𝑡) (22)

Our study indicates that the early time f(t) correlation, Eq. 9, given by Kutasov (2003) matches the Ramey’s f(t) values within the

reasonable accuracy for the time range after the convection dominated period. Thus assuming the f(t) correlation, Eq. 9, is valid the

derivatives of ΔT in terms of dΔT/dt and dΔT/dlnt are obtained from Eq. 22 as following:

𝑑∆𝑇

𝑑𝑡=

𝛼𝐿22𝜋𝑘

𝑤𝑐

4.5√𝜅

𝑟2

𝑡−3

2 (23)

𝑑∆𝑇

𝑑𝑙𝑛𝑡=

𝛼𝐿22𝜋𝑘

𝑤𝑐

4.5√𝜅

𝑟2

𝑡−1

2 (24)

Notice that the plot of log(dΔT/dt) vs. log(t) gives a straight line with a slope -3/2 whereas the plot of log(ΔT/dlnt) vs. log(t) gives a

straight line with a slope -1/2.

One further observation regarding the wellhead derivative solutions, equating the early-time and the intermediate-time wellhead

temperature derivative relationships, Eqs. 20 and 23, can give the intersection time of the early-time and intermediate-time straight lines

(tei) as

𝑡𝑒𝑖 = [1.4𝐿2𝑘

𝑤𝑐√𝜅

𝑟2𝑢]

2/3

= 2.68 [𝐿2𝑟3𝑘

𝑤2𝑐√𝜅]2/3

(25)

The intermediate-time behavior described by Eqs. 22-25 is observed when the effect of the heat transfer between the wellbore and the

surrounding formation becomes significant after the early-time convection dominated period. It could be of a short duration and even

may not exist depending on the scales of the heat transfer rate (q) and the wellbore convective flow. It can be observed if the the

wellbore convective flow is relativeley small and the early-time period is long and the conditions for f(t) valid for small values of time

(Eq. 9) is dominant.

The validity of the temperature derivative solutions given above are checked using the wellbore temperature model described by Kutun

et al.(2014, 2015). The thermal properties used in modeling are given in Table 1. The well dimensions, the flow rate, and other relevant

data are given in Table 2. In all the synthetic applications, presented in this section, the parameters given in Table 1 and 2 are used

unless otherwise stated.

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Table 2- Input parameters for the cases discussed

Well depth, m 1050.0

Well radius, m 0.1

Bottom hole temperature, oC 144.09

Water production rate, kg/s 4.39

Geothermal temperature gradient, oC/m 0.118

Production time, days 50.0

Figures 5-8 give ΔT, dT/dt, and dT/dlnt versus time (t) plots obtained from the wellbore simulator by Kutun et al. (2014). Figure 5

shows the temperature derivative results for a 50 day production period whereas Fig. 6 illustrates the intermediate-time temperature

derivative with a -3/2 slope. Similarly Fig. 7 gives dT/dlnt versus time on a log-log plot whereas Fig. 8 exhibits the unit slope and -1/2

slope periods on the same plot. Figures 7 and 8 show ΔT versus t and dΔT/dlnt versus t on the same plot for comparison purposes.

Figure 5: dT/dt versus time plot.

Figure 6: dT/dt versus time plot and -3/2 slope.

-3/2 Slope

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Figure 7: ΔT versus time and dT/dlnt versus time plots.

Figure 8: ΔT versus time and dT/dlnt versus time plots and unit slope and -1/2 slope characteristics.

For this particular data the linear velocity in the wellbore u is 0.14 m/s and the fluid front reaches the wellbore at 0.087 day. As seen in

Figs. 5-8 the convection dominated period with a unit slope ends at about 0.003 day which indicates the validity of the condition t<L/u.

Thus the earlier temperature data represents the convection dominated period.

Figure 7 shows ΔT versus t on log-log plot. As indicated earlier it is possible to estimate αu from the unit-slope straight line. At

t=0.0001 day, ΔT=0.14 oC on the log-log unit-slope straight line so that ΔT/t is 1400 oC/day that is being equal to αu as Eq. 19 indicates.

The input value for αu is 1424 oC/day which matches the calculated value perfectly well.

Unit Slope

Half (-1/2) Slope

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The intermediate-time period for which the intermediate-time behavior and the cylindirical source solution as well are expected to begin

at approximately one and half log cyle after the end of the early-time period (0.003 day) that is 0.06 day. That is the case as observed in

Fig. 8.

Figure 8 shows that the early-time unit-slope straight line and the intermediate time -1/2 slope straight line intersect at about 0.03 day.

Using the input data in Eq. 25, the intersection time tei is determined to be 0.032 day that validates the accuracy of Eq. 25.

2.3. Late-Time Temperature and Heat-Transfer Behavior

Theoretically the heat transmission characteristics in a borehole heat exchanger and in a geothermal well are similar. The method of

estimating thermal conductivity from borehole heat exchanger temperature measurements is applied provided that the heat transfer rate

per unit length (q) is kept at a constant value.

For a geothermal well with flow in casing so that the thermal resistance term (Rb) is neglected, the total heat gained per length is

expressed by Eq. 4. Writing Eq. 4 in terms of the wellhead temperature and using Eq. 13 for f(t) yields:

𝑇 = 𝑞1

4𝜋𝑘𝑙𝑛(𝑡) + 𝑞

𝑙𝑛(1.7√𝜅

𝑟2)

2𝜋𝑘+ 𝑇𝑒 (26)

Assuming a constant heat transfer rate per length (q), then the plot of wellhead temperatures measured versus the natural logarithm of

time is expected to give a semilog straight line with a slope being equal to m and the thermal conductivity can be estimated from m.

However, practically speaking a constant q case (yielding a constant slope on a T-ln(t) plot) is not observed in the late-time temperature

behavior of a geothermal well (see Figs. 7 and 8).

As discussed and validated by computational analysis below, such a method of estimating thermal conductivity seems not possible for

the typical heat transmission in a geothermal well.

Figure 9 illustrates the typical late-time wellhead temperature behaviors of two cases for a geothermal well obtained from the data given

in Tables 1 and 2 for w=4.39 and 43.9 kg/s, respectively. The wellhead temperature in Figure 9 changes according to Eq. 1. The

temperature-log(time) behaviors of both cases do not exhibit any semilog straight line relationship. The wellhead temperature for

w=4.39 kg/s case increases sharply at early times and the increase becomes lower as the time increases. For particular cases, especially

when the flow rate is high and NRa (= 2πkL/(wc)) becomes small, the wellhead temperature seems to reach an apparently stabilized

value and the temperature change becomes negligibly small. Such a behavior is observed for w=43.9 kg/s case in Fig. 9.

Figure 9: Wellhead temperature versus time plot.

For the time region when the temperature change becomes negligible or assumed to be reached to an apparently stabilized wellhead

temperature, (1/q) versus log(time) plot yields a semilog straight line relationship. This can be theoretically shown when Eqs. 4 and 13

are combined to give:

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𝑞 =2𝜋𝑘

𝑙𝑛(1.7√𝜅

𝑟𝑤2 𝑡)

(𝑇 − 𝑇𝑒) (27)

and

1

𝑞=

1

2𝜋𝑘(𝑇−𝑇𝑒)𝑙𝑛 (1.7√

𝜅

𝑟𝑤2) +

1

4𝜋𝑘(𝑇−𝑇𝑒)ln(𝑡) (28)

Equation 28 indicates that (1/q) versus ln(t) plot should give a semilog straight line with a slope being equal to

𝑚 = 𝑆𝑙𝑜𝑝𝑒 =1

4𝜋𝑘(𝑇−𝑇𝑒) (29)

Figure 10 presents (1/q) versus log(t) relationship for the case with w=43.9 kg/s case. The wellhead temperature data given in Fig. 9

were used to construct the plot. The slope of the semilog straight line is determined to be 4.92x10-4 s.m/(J-log cycle) from the (1/q)

versus log(t) plot which corresponds to 2.136x10-4 s.m/(J-ln cycle) from the (1/q) versus ln(t) plot. Assuming an average (T-Te) value of

122 oC valid for the time region in which the semilog straight line exists (t=5-100 days as observed in Fig. 10), the thermal conductivity

is calculated from the slope to be 3.05 J/m.s.oC which matches well with the input value of 2.92 J/m.s.oC.

Figure 10: (1/q) versus time plot.

The discussion above demonstrates that the wellbore heat transmission solutions at sufficiently long times converge to the cylindrical-

source solution transfering heat at constant temperature rather than the constant heat flux, and the heat transfer is controlled by

surrounding formation conditions.

One another observation deduced from the analysis of this case is that the beginning of the semilog straight line behavior is related to

the stabilization time characteristic of the wellhead temperature to be discussed in the following section of this study. Notice that the

semilog straight line behavior begins at about 5 day as seen in Fig. 10. This is about the beginning of the period in which the wellhead

temperature reaches to a stabilized and almost a constant value and the heat transmission model becomes a cylindrical source with a

constant temperature.

4. CONCLUSIONS

The geothermal wellbore heat transmission is studied here. The dimensionless time function representing the transient heat transfer

between the wellbore and the surrounding formation is assessed in detail and the adequate approximations valid for practical

engineering purposes are presented.

Kutun et al.

12

The effect of time on the wellhead temperature behavior in terms of temperature derivatives is assessed. The following conclusions are

obtained:

1) At early times, the time derivative of wellhead temperature stays at a constant value and a log-log plot of dΔT/dlnt versus t gives a

unit slope straight line. The early-time temperature behavior is similar to the wellbore storage phenomenon occuring in oil and gas

production wells and affecting short-time transient presure behavior.

2) The intermediate-time behavior is observed when the effect of heat transfer between the wellbore and the surrounding formation

becomes significant after the early-time convection dominated period. It is recognized when the log-log plot of dΔT/dlnt versus t

gives a -1/2 slope straight line and/or dΔT/dt versus t gives a -3/2 slope straight line.

3) The wellbore heat transmission solutions at sufficiently long times converge to the cylindrical-source solution transferring heat at

constant temperature rather than the constant heat flux, and the heat transfer is controlled by surrounding formation conditions. At

late-times when the temperature change becomes negligible or assumed to be reached to an apparently stabilized wellhead

temperature, (1/q) versus log(time) plot yields a semilog straight line relationship.

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