concentric dual-tubing steam injection: a new … dual-tubing steam injection: a new model for...

Concentric Dual-tubing Steam Injection: A New

Model for Predicting Steam Pressure in the

Annulus

Hao Gu, Linsong Cheng, Shijun Huang, Shuang Ai, and Shaolei Wei

Department of Petroleum Engineering, China University of Petroleum, Beijing, China

Email: [email protected], {guhao110110, fengyun7407, aishuang4415, leisurewin}@163.com

Abstract—It is not always easy to accurately predict

bottomhole steam pressure, temperature and quality when

we design concentric dual-tubing steam injection schemes

due to the complexity of two-phase flow in the annulus, also,

previous methods for estimating pressure gradient in annuli

are time-consuming. In this study, a new model is

established based on mass and energy balances to calculate

steam pressure in the annulus. A more rigorous

thermodynamic behavior of steam/water mixture is also

taken into account. More importantly, one-to-one

correspondence between pressure gradient and temperature

gradient of saturated steam is reasonably developed and

applied in further derivation. The results indicate that the

proposed model is more accurate and convenient in

predicting steam pressure than previous methods. Moreover,

the steam pressure in the inner tubing drops faster than that

of in the annulus even the wellhead mass flow rate of the

former is lower. The variation law of steam quality in the

inner tubing is different from that of in the annulus,

depending on the net heat losses of mixture fluid in each

tubing.

Index Terms—new model, steam pressure, concentric dual-

tubing, annulus, heat transfer

I. INTRODUCTION

Heavy oil reserves are widely distributed in the world

and they play an important role in today’s energy supplies

[1]. However, it is not always easy to efficiently develop

high-viscosity oil due to low mobility under initial

formation temperature. At present, thermal recovery

methods, especially for steam injection techniques, are

extensively used in the process of heavy oil production.

Injecting steam into wellbore would inevitably result in

heat exchange between the wellbore fluid and its

surrounding formation [2]. In addition, the steam pressure,

temperature and quality would change as fluid flows

down the wellbore. Therefore, predicting wellbore heat

losses and bottomhole steam properties are two major

tasks when we design steam injection projects. However,

there are always many difficulties in designing concentric

dual-tubing steam injection schemes because two-phase

flow in the annulus is much more complex than that of in

pipes, so is the estimation of pressure gradient.

Manuscript received December 4, 2013; revised April 15, 2014.

The calculation methods for pressure gradient in annuli

can be divided into two categories: empirical correlations

and mechanistic models [3]. Many researchers have

established different mechanistic models for two-phase

flow in annuli based on flow-patterns, such as Caetano [4]

(1985) model, Antonio et al. [5] (2002) model and Yu et

al. [3] model (2010). However, the definitions of flow

patterns, the transition criteria and the process of

calculating flow-parameters are very complicated and

time-consuming. Griston et al. [6] treated the annuli as

pipes on the bases of hydraulic diameter concept, while it

is just an approximation and will be discussed later.

For heat transfer between the wellbore fluid and its

surrounding formation, Ramey [7] proposed the

expression of fluid temperature in wellbores. His work

laid the foundation for later researchers though he only

considered single phase (ideal gas and incompressible

liquid) flow. Satter [8] took into account the effect of

condensation and presented a method for calculating

steam quality, but he ignored frictional losses and kinetic

energy effects. Holst and Flock [9] studied the effect of

friction on wellbore heat losses. Willhite [10] explicitly

analyzed three mechanisms (conduction, natural

convection and radiation) of heat transfer in the wellbores

and suggested a method for determining over-all heat

transfer coefficient.

Recently, more and more researchers combined heat

transfer model with two-phase flow model , such as

Fontanilla and Aziz [11] (1982), Alves et al.[12](1992),

Hasan . [13]-[16](1994,2007,2009),Bahonar et al.

[17](2010) and Mahdy et al.[18](2012).However, these

models are all about sing-tubing steam injection. Barua

[19] and Hight [20] mentioned concentric dual-tubing

steam injection, but neither set up concrete mathematical

models.

In this work, we propose a new model which is more

accurate and convenient to predict steam pressure in the

annulus. In addition, the characteristics of heat transfer in

concentric dual-tubing are also analyzed in detail.

II. MODELING FLUID PRESSURE

Fig. 1 shows the schematic diagram of concentric dual-

tubing steam injection well over the differential length of

zd .After steam reaches the wellhead along surface

245

Journal of Industrial and Intelligent Information Vol. 2, No. 4, December 2014

©2014 Engineering and Technology Publishingdoi: 10.12720/jiii.2.4.245-251

et al

pipelines, it would be simultaneously enter into the inner

tubing and the annulus between the inner and outer tubing.

Figure 1. Concentric dual-tubing steam injection well.

A. Pressure Gradient in the Inner Tubing

For two-phase flow in the inner tubing, the pressure

calculation model proposed by Beggs and Brill [21] is

adopted. The governing equations for mixture fluid over a

differential length of zd yield to the following equations:

Mass balance equation

0ininin

zz

W (1)

Momentum balance equation

zD

gz

p

d

d

2sin

d

d ininin

in

2

ininin

in (2)

where inW and in are the mass flow rate and the velocity

of mixture fluid in the inner tubing, respectively; in and

inp are the density and the pressure of mixture fluid,

respectively; is the two-phase friction factor; inD is the

diameter of inner tubing; g is the gravitational

acceleration; is the well angle from horizontal. The

third term of the right side in (2) can be given by

z

p

pz d

d

d

d in

in

sginininininin

(3)

where sgin is the superficial velocity of gas phase in the

inner tubing.

Substituting (3) into (2), an expression for the pressure

gradient in the inner tubing can be obtained,

insgininin

in

2

ininin

in

/1

2sin

d

d

p

Dg

z

p

(4)

B. Pressure Gradient in the Annulus

Griston et al. [6] treated the annuli as pipes based on

hydraulic diameter concept, however, hydraulic diameter

is not always a suitable representative characteristic

dimension for two-phase flow in the annulus [4]. In this

study, a new semi-analytical model for estimating

pressure gradient in the annulus is formulated.

The heat changes of mixture fluid in the annulus result

from two parts: heat losses to the formation, heat losses to

the inner tubing or heat absorption from the inner tubing.

A general energy balance on the fluid can be written as

sin2d

d

d

d

d

d

d

d12

outoutinout

out

gv

zz

h

z

Q

z

Q

W

(5)

where outW and out are the mass flow rate and the

velocity of mixture fluid in the outer tubing,

respectively; outh are the specific enthalpy of mixture

fluid; zQ d/d out is the rate of heat transfer from the

annulus to the formation; zQ d/d in is the rate of heat

transfer from the inner tubing to the annulus, but if the

value of zQ d/d in is negative, it indicates the annulus

absorbs heat from the inner tubing.

In (5), the kinetic energy change for per unit mass of

mixture fluid over the differential length of zd can be

similarly obtained from (3)

z

p

pz

v

z d

d

d

d

2d

d out

out

sgoutoutoutout

2

out

(6)

where sgout is the superficial velocity of gas phase in the

annulus.

Using basic thermodynamic principles, the enthalpy

gradient can be written in terms of the temperature and

pressure gradients [12]:

z

pCC

z

TC

z

p

p

h

z

T

T

h

z

h

Tp

d

d

d

d

d

d

d

d

d

d

outpmJm

outpm

out

out

outout

out

outout

(7)

where pmC is the heat capacity of mixture fluid at

constant pressure; JmC is the Joule-Thompson coefficient.

Substituting (6) and (7) into (5)yields,

sindz

d

d

d

d

d

d

d

d

d1

out

out

sgoutoutpm

outpmJm

inout

out

gp

pz

TC

z

pCC

z

Q

z

Q

W

(8)

The relationship between temperature gradient and

pressure gradient can be further created. Ejiogu and Fiori

[22], Tortlke and Farouq Ali [23] proposed different

correlations between temperature and pressure of

saturated steam based on steam table. In this study, the

Tortlke and Farouq Ali’s correlation is adopted due to its

high accuracy and validation throughout almost the entire

steam-saturation envelope, which is given by

432ln019017.0ln101806.0ln

38075.1ln085.14034.280

ppp

ppfT

(9)

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©2014 Engineering and Technology Publishing

The relationship between temperature gradient and

pressure gradient of saturated steam can be expressed as

z

p

p

pf

z

T

d

d

d

d

d

d (10)

Substituting (10) into (8) reduces to the pressure

gradient in the annulus

out

sgout

pm

out

outpmJm

inout

outout

d

d

sind

d

d

d1

d

d

pC

p

pfCC

gz

Q

z

Q

W

z

p

(11)

In (11), the heat capacity and the Joule-Thompson

coefficient of two-phase flow system can be estimated by

referring to the method suggested by Alves et al. [12].

Therefore, it is easy to calculate the pressure drop in the

annulus without using mechanistic models or empirical

correlations, which depend on complicated flow-patterns

or approximate treatment. But before (11) can be used, it

is necessary to estimate zQ d/d in and zQ d/d out

.

III. MODELING WELLBORE HEAT TRANSFER

As fluid flows both in the inner tubing and in the

annulus, heat transmission can be separated into three

parts: (a).Heat exchange between the fluid in the inner

tubing and in the annulus. (b).Heat transfer from the fluid

in the annulus to the cement sheath. (c).Heat transfer in

the formation. If the wellhead injection conditions remain

constant, heat transfer inside the wellbore can be assumed

steady-state while heat transfer in the formation can be

assumed transient, Holst and Flock [9] discussed the

assumption in detail.

A. Heat Exchange between the Fluid in the Inner

Tubing and in the Annulus

There would be heat exchange between the fluid in the

inner tubing and in the annulus if the fluid temperature is

not equal. The rate of heat flow over the differential

length of zd can be given based on Fourier law

)(2d

doutintoto

in TTUrz

Q (12)

where tor is the outside radius of inner tubing; inT and

outT are the temperature of fluid in the inner tubing and in

the annulus, respectively; toU is the over-all heat transfer

coefficient for inner tubing, which can be written as

follows:

1

ftoti

to

tub

to

tifti

toto

1ln

hr

rr

rh

rU

(13)

where, ftih and ftoh are the forced-convection heat transfer

coefficient on inside and outside of inner tubing,

respectively; tub is the thermal conductivity of tubing

wall; tir is the inside radius of inner tubing.

Because the forced-convection heat transfer coefficient

( ftih and ftoh ) and the thermal conductivity of tubing wall

( tub ) are very high, the rate of heat exchange between

the fluid in the inner tubing and in the annulus would be

very fast even the temperature difference is small.

B. Heat Transfer from the Fluid in the Annulus to the

Cement Sheath

Similarly, for steady-state heat transfer from the fluid

in the annulus to the cement sheath, the rate of heat flow

can be expressed as

)(2d

dhoutdodo

out TTUrz

Q (14)

where, dor is the outside radius of outer tubing; hT is the

wellbore/formation interface temperature ; doU is the

over-all heat transfer coefficient between the outer tubing

and the cement sheath. Willhite [10] proposed the method

for calculating doU , which can be given by

1

co

h

cem

do

ci

co

cas

do

rcdi

do

tub

do

difdi

do

do

lnln

1ln

r

rr

r

rr

hhr

rr

rh

r

U

(15)

where, fdih is the forced-convection heat transfer

coefficient on inside of outer tubing; dir is the inside

radius of outer tubing; cir and cor are the inside radius and

outside radius of casing, respectively; hr is the outside

radius of the wellbore; cas and cem are the thermal

conductivity of casing and cement, respectively; ch and

rh are the convective heat transfer coefficient and the

radiative heat transfer coefficient, respectively, which can

also be estimated by referring to the correlations

suggested by Willhite [10].

Also, because fdih , tub and cas are large compared

with other terms in (15), the expression of doU can be

reduced to (16)

1

co

h

cem

do

rc

do ln1

r

rr

hhU

(16)

C. Heat Transfer in the Formation

For the transient heat transfer in the formation, the rate

of heat flow is the function of injection time and

temperature difference, the general expression is

tf

TT

z

Q eiheout 2

d

d

(17)

where, e is the thermal conductivity of the formation;

eiT is the initial temperature of the formation at any given

247



depth, sin0ei azTT ,0T is the surface temperature of

the formation, a is the geothermal gradient; )(tf is the

transient heat-conduction time function. Ramey [7],

[24] and Hasan et al. [25] proposed different

empirical expressions, however, Hasan et al. method is

adopted due to its accuracy, which is given by

5.1)6.0

1(ln5.04063.0)(

5.1)3.01(1281.1)(

D

D

D

DDD

tf

tf

，

， (18)

Combing (14), and (17) yields,

edodo

dodoouteieh

)(

)(

tfUr

tfUrTTT (19)

eiout

edodo

edodoout -)(

2

d

dTT

tfUr

Ur

z

Q

(20)

The method for predicting steam quality is presented in

Ref. [26]. Equations (4), (11), (12), (20) and steam

quality model need to be coupled and solved iteratively

for each segment.

IV. RESULTS AND DISCUSSIONS

A. Comparison with Field Data

The field test data obtained from Hight [20] is used to

verify the accuracy and reliability of the new model. In

the field test, the diameters of the inner and outer tubing

are 0.0483m (1.9in.) and 0.0730m (27/8in.), respectively,

the diameter of the casing is 0.1397m (51/2in.) Fig. 2

shows a comparison of calculated pressure with the

measured field data. As can be seen from Fig. 2, the

simulated results agree very well with the measured

values. Also, the maximum relative error is about 5.39%.

0.0

0.5

1.0

1.5

2.0

2.5

0 50 100 150 200 250 300

Measured annulus

Measured inner tubing

Calculated annulus

Calculated inner tubing

Ste

am p

ress

ure

(M

Pa)

Well depth (m)

Figure 2. Comparison of calculated pressure with measured field data

B. Example and Analysis

The characteristics of heat transfer in the wellbore are

discussed in detail. Table I. is the basic parameters of the

wellbore and the formation and Table II is the wellhead

injection conditions.

TABLE I. BASIC PARAMETERS OF THE WELLBORE AND THE

FORMATION

Parameter value Parameter value

Inside radius of inner tubing(m)

0.01905 Surface temperature of the

formation(K) 303.15

Outside radius of inner tubing(m)

0.02415 Geothermal gradient

(K ·m-1) 0.031

Inside radius of outer tubing(m)

0.0380 Forced-convection heat transfer

coefficient

(W ·m-2·K-1)

3191.43

Outside radius of

outer tubing(m) 0.0445

Thermal conductivity of the

tubing and casing wall(W ·m-1·K-

1) 46.51

Inside radius of

casing(m) 0.0807

Thermal conductivity of the

cement(W ·m-1·K-1) 0.93

Outside radius of

casing(m) 0.0889

Thermal conductivity of the formation

(W ·m-1·K-1)

1.73

Outside radius of the wellbore(m)

0.1236 Thermal diffusivity of the

formation(m2/s) 1.03×10-

7

TABLE II. WELLHEAD INJECTION CONDITIONS

Parameters Inner tubing Annulus

Steam pressure(MPa) 5.50 5.30

Steam temperature (K) 543.35 541.00

Steam quality 0.60 0.60

Mass flow rate(kg/h) 3000 5000

Fig. 3 shows a comparison of the results from the

present model and that from Griston et al. [6] model. In

their study, the annulus was treated as a pipe and the

hydraulic diameter todie 2 rrD was used to replace

inD in (4) to estimate the pressure gradient in the annulus.

However, Fig. 3 indicates that the result of the new model

shows a good agreement with field data while the

calculated pressure from Griston model is much

smaller than test data. Therefore, the hydraulic diameter

is not always a suitable representative characteristic

dimension for two-phase flow in annuli.

4.7

4.9

5.1

5.3

5.5

0 50 100 150 200 250 300 350 400 450

Field data

Griston et al. model

New model

Ste

am

pre

ssu

re i

n t

he

ann

ulu

s (M

Pa)

Well depth (m)

Figure 3. Comparison of calculated pressure with Griston et al. model

Fig. 4 is the results of steam pressure in the concentric

dual-tubing. It is found that the bottomhole pressure in

the inner tubing is smaller than that of in the annulus

248



et al.

et al.

though the wellhead injection pressure is not the case.

Moreover, the wellhead mass flow rate in the inner tubing

is 3000kg/h, which is lower than that of in the annulus

(5000 kg/h), but the pressure in the inner tubing drops

faster than that of in the annulus. The reason may be that

the radius of inner tubing is much smaller and the fluid in

the inner tubing needs to overcome larger friction. The

variation law of temperature is the same with that of

pressure (Fig. 5). From the entire steam injection, the

average pressure gradient and temperature gradient in the

inner tubing are 0.0018MPa/m and 0.0221K/m,

respectively, while the corresponding values in the

annulus are 0.0010 MPa/m and 0.0125 K/m, respectively.

4.6

4.8

5.0

5.2

5.4

5.6

0 50 100 150 200 250 300 350 400 450

Annulus

Inner tubing

Ste

am p

ress

ure

(M

Pa)

Well depth (m)

Figure 4. Pressure in concentric dual-tubing steam injection well

534

536

538

540

542

544

0 50 100 150 200 250 300 350 400 450

Annulus

Inner tubing

Ste

am te

mp

erat

ure

(K

)

Well depth (m)

Figure 5. Temperature in concentric dual-tubing steam injection well

Fig. 6 is the variation law of heat transfer inside the

wellbore. It is observed that from the wellhead to about

320m, heat is transferred from the inner tubing to the

annulus and the rate of heat flow decreases with well

depth. This is because the temperature of fluid in the

inner tubing is higher than that of in the annulus and the

temperature difference also decreases with well depth.

But when the well depth exceeds 320m, the fluid in the

annulus begins to release heat to the inner tubing. In

addition, since the temperature of fluid in the annulus is

much higher than formation temperature, heat is always

transferred from the annulus to surrounding formation

and the net heat losses in the annulus are positive.

Fig. 7 is the results of steam quality in the concentric

dual-tubing. The steam quality in the annulus decreases

sharply with well depth because the net heat losses are

large. However, the steam quality in the inner tubing

decreases slowly from the wellhead to about 190m, from

190m to the bottom, it increases with well depth. The

possible explanation for inflection point is 190m rather

than 320m is as follows: from 190m to 320m, though the

fluid in the inner tubing still loses heat to the annulus, the

temperature difference is lower and the heat losses are not

large enough to result in quality drop, on the contrary, the

increased energy transformed from potential energy

makes the specific enthalpy of mixture fluid increase

slowly from 190m to 320m(Fig. 8).When the well depth

exceeds 320m, the steam quality increase sharply since

the fluid in the inner tubing begins to absorb heat from

the annulus.

-600

-400

-200

0

200

400

600

800

1000

0 50 100 150 200 250 300 350 400 450

From inner tubing to annulus

From annulus to formation

Net heat losses in the annulus

Rat

e o

f h

eat

flo

w (W

/m)

Well depth (m)

Figure 6. Rate of heat flow in the wellbore

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0 50 100 150 200 250 300 350 400 450

Annulus

Inner tubing

Ste

am q

ual

ity

Well depth (m)

Figure 7. Steam quality in concentric dual-tubing steam injection well

2080

2120

2160

2200

2240

2280

2320

0 50 100 150 200 250 300 350 400 450

Sp

ecif

ic e

nth

alp

y o

f w

et s

team

(k

J/k

g)

Well depth (m)

Figure 8. Specific enthalpy of wet steam in the inner tubing

249



V. CONCLUSION

In this paper, a new model is proposed to predict steam

pressure in the annulus. The one-to-one correspondence

between pressure gradient and temperature gradient of

saturated steam is reasonably developed and applied in

further derivation. It is because of the improvement that

we can avoid using mechanistic models or empirical

correlations to separately calculate the pressure drop in

the annulus when simultaneously solving wellbore heat

transfer model. The characteristics of heat transfer in

concentric dual-tubing are also analyzed in detail. The

method suggested by this paper provides a reference for

the design of steam injection projects.

ACKNOWLEDGMENT

The authors wish to thank the Research Institute of

Petroleum Exploration & Development, Liaohe Oilfield

Company, Petro China. This work was supported in part

by a grant from National Science and Technology Major

Projects of China (2011ZX05012-004).

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Hao Gu was born in the city of Huanggang, Hubei, China in 1989.Mr.Gu received his

Bachelor degree in Petroleum Engineering at

Yangtze University in 2011.In 2013, he received his master degree from China

University of Petroleum, Beijing. At present, he is pursing Ph.D in Oil and Gas Development

Engineering from China University of

Petroleum, Beijing. His research interest includes thermal recovery

and numerical simulation.

Linsong Cheng was born in the city of Yingcheng, Hubei, China in 1965.Mr.Cheng

received his doctor degree from China

University of Petroleum, Beijing in 1994.Now he is the professor and vice president at the

Department of Petroleum Engineering, China University of Petroleum, Beijing. His research

direction includes thermal recovery, mechanics

of flow through porous media, the recovery method for low-permeability reservoir.

Shijun Huang was born in the city of Zhengzhou, Henan, China in 1974.Mr. Huang

received his doctor degree from China

University of Petroleum, Beijing in 2006. He has been a visiting scholar in A&M University.

Now he is a vice professor at Department of

Petroleum Engineering, China University of Petroleum, Beijing. His research direction

includes heavy oil production and numerical simulation.

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[1] W. S. Huang, H. P. Chen, and Z. Meng, “Evaluation techniques of reserves for heavy oil and oil sands,” in Proc. International

Petroleum Technology Conference, 26-28 March 2013, pp. 1-8.

Shuang Ai was born in Xian Ning city, Hu Bei, China in 1986. Mr. Ai holds a BS degree in

2009 a MS degree in 2012 in Petroleum

Engineering from China University of Petroleum (Beijing), China. He is PhD student

in Oil and Gas Development in China University of Petroleum (Beijing), China. He

has a strong inclination for transient flow and

numerical simulation of unconventional gas reservoirs.

Shaolei Wei, earned her Master Degree in Oil-Gas Field Development Engineering from

China University of Petroleum (East China) in

Qingdao, China, in 2012. At present, she is pursuing her ph.D in Oil-Gas Field

Development Engineering in China University of Petroleum – Beijing. Her areas of interest

include reservoir numerical simulation,

production optimization, and thermal recovery of heavy oil.

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