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## Heat Transfer Mechanism in the Wellbore

The steady-state rate of heat flow, Q. between the outer surface of the tubing at temperature Ttbo and the outer surface of the cement sheath at temperature Trmo can be expressed as:

 (4.114)

Q = 2 ж rtbo U, ol (Tst — Tcm J Al

where:

 rtba Tst Tcm о Al Utot

heat flow through the wellbore. Btu/hr. outer radius of the tubing, ft.

temperature of the flowing fluid inside the tubing. °F. temperature at the outer surface of the cement sheath, °F. incremental length of casing or tubing, ft. overall heat transfer coefficient, Btu/hr sq ft °F

Subscripts:

tb = tubing.

tbo = outside of tubing.

tb, = inside of tubing,

с = casing.

Co = outside of casing.

c, = inside of casing.

cm = cement.

Utot is defined as the overall heat transfer coefficient and its value for any well completion can be found by considering the heat transfer mechanism of individual completion elements, i. e., the tubing, annular fluid, casing, and cement sheath. Heat flow through the tubing wall, casing wall and cement sheath occurs by conduction. Fourier (Willhite, 1967). discovered that the rate of heat flow through a body can be expressed as:

 (4.115)

dT

Q — — 2 7Г г к, — Д/ dr

Integrating Eq. 4.115 with Q constant, yields:

_ 2 7Г kj {Tt — T0) 4 ln(r0/r,-)

where:

kj = thermal conductivity of the ‘j’th completion element (tubing or casing or cement).

Tt — temperature at the internal surface.

T0 — temperature at the outer surface,

r, = internal radius of the completion element.

r0 = external radius of the completion element.

The casing annulus is generally filled with air or nitrogen gas. Heat flow through the annulus occurs by conduction, convection and radiation. Thus, the total heat flow in the annulus is the sum of the heat transferred by each one of these mechanisms. For convenience, the heat transfer through the annulus is expressed in terms of the heat transfer coefficient, Qcon (natural convection and conduction) and QTad (radiation). Hence:

Q = 2irrtbo (Qcon + QTlld) (Ttbo — TCt) А/ (4.117)

Inasmuch as the heat flow through the well completion elements is assumed to be a steady-state flow, the values of Q for each completion element remain unchanged at any particular time. Thus, solving for T and Q one obtains:

 = (Tat — Tt0 + (Trtl — 7)0 + (70 — Tc,) + (Tc, — TJ + (TCo — Tcmo)

 Tat ~ Tc,

 (4.118)

 1 ln(r(ijr(i, )

 Q

 rtb0 (Q con "Ь Qrad)

 2 7Г Д/

 kti

 rtb, H

 (4.119)

| In(rCo/rQ + ln(rcmo/rcJ

к

Comparing Eqs. 4.114 and 4.119, one obtains the general expression for the overall heat transfer coefficient:

 rtbo n{rcJrcJ

 rtb0

 rtbo

 Utot =

 +

 гл, Hs

 rtbQ n(rtbJrtK) +

 (Q con + Q rad )

 (4.120)
 +

rtbo bircmo/rCo

where:

Hst = film coefficient for heat transfer or condensation coefficient based on inside tubing or casing surface and temperature difference between flowing fluid and either of these surfaces.

Table 4.9: Thermal conductivity of different completion elements. (Af­ter Proyer, 1980.)

 Material Temperature (°F) Thermal conductivity (Btu/hr ft °F) Condition Calcium- 212 — 392 0.042 — 0.047 Dry silic. ate 392 — 572 0.047 — 0.054 Dry 572 — 686 0.054 — 0.06 Dry 32 — 212 0.36 — 0.46 Wet Cement — 0.20 — 0.40 Dry — 0.50 — 0.60 Wet Steel 212 — 392 26 572 24.85 752 23.12

In a similar manner, an expression for Utot can be derived for injection tubing insulated with commercial insulation of thickness Лг (= rlns — rtbo) and thermal conductivity kms:

rtb0 + rtbo In(rtbJrtb)) rtbo n{rlns/rtbo) + rtbo

T’tb, Hst ktb kms C)is (hcon “b hrad)

, rtbo HrcJrCx) rtbo ln(rcmo/rCo)]_1 .

+ к + к———————————————————— (

л-с /хстп

where:

hrad and hcon — are based on the surface area 2лтШ5А/

and the temperature difference T, ns ~ Tc{.

Subscripts:

cm — cement sheath.

; = internal surface.

о = external surface.

in = insulation material.

The overall heat transfer coefficient can be found once the values of klb, klns, kc, к cm i Qcon, Qrai and Hst are known. In Table 4.9, typical thermal conductivities of different completion elements are listed (Proyer, 1980).

The heat transfer coefficients. Qcon and Qrad. between the outer surface of the tubing and the internal surface of the casing can be determined by using the Stefan-Boltzman Law (McAdams, 1954) and the method proposed by Dropkin

 INJECTION RATE, ton/h Fig. 4.34: Wet steam pressure gradient as a function of steam pressure, injection rate, and steam quality, Emsland, Northern Germany. (After Goetzen, 1987; courtesy of ITE-TU Clausthal.)

et al. (1965), respectively. For detailed information, readers are referred to the original literature.

Using Eqs. 4.117 through 4.121, the following expression for the casing temper­ature can be derived:

гг, (HrcmJrCo) , 1п(гСо/гс,)^ _ „1ЛП1

4c, — 4cmo + I —- ——- h—- 7—— rtbo ut0t (lst — 4cm0) (4.122)

*cm Kc J

where:

Utot — overall heat transfer coefficient based on the outside tubing surface and the temperature difference between the fluid and cement-formation interface, Btu/hr sq ft°F.

To determine the casing temperature, the temperature at the cement-formation interface, Tcm<> and the temperature of the steam, Tst, must be known.