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Thermophysical properties estimation and performance analysis of 

superheated-steam injection in horizontal wells considering phase

change

Hao Gu a,⇑, Linsong Cheng a, Shijun Huang a, Bing Bo b, Yinguo Zhou a, Zhongyi Xu a

a Department of Petroleum Engineering, China University of Petroleum, Beijing, 18 Fuxue Road, Changping, Beijing 102249, Chinab Research Institute of Petroleum Exploration & Development, PetroChina, 20 Xueyuan Road, Haidian, Beijing 100083, China

a r t i c l e i n f o

 Article history:

Received 1 February 2015

Accepted 11 April 2015

Available online 25 April 2015

Keywords:

Superheated-steam injection

Thermophysical properties

Performance analysis

Phase change

Horizontal wells

a b s t r a c t

The objectives of this work are to establish a comprehensive mathematical model for estimating thermo-

physical properties and to analyze the performance of superheated-steam injection in horizontal wells. In

this paper, governing equations for mass flow rate and pressure drop are firstly established according to

mass and momentum balance principles. More importantly, phase change behavior of superheated steam

is taken into account. Then, implicit equations for both the degree of superheat and steam quality are fur-

ther derived based on energy balance in the wellbore. Next, the mathematical model is solved using an

iterative technique and a calculation flowchart is provided. Finally, after the proposed model is validated

by comparison with measured field data, the effects of some important factors on the profiles of thermo-

physical properties are analyzed in detail. The results indicate that for a given degree of superheat, the

mass flow rate drops faster after superheated steam is cooled to wet steam. Secondly, to ensure that

the toe section of horizontal well can also be heated effectively, the injection rate should not be too slow.

Thirdly, the mass flow rate and the degree of superheat in the same position of horizontal wellbore

decrease with injection pressure. Finally, it is found that when reservoir permeability is high or oil vis-

cosity is low, the mass flow rate and the degree of superheat decline rapidly.   2015 Elsevier Ltd. All rights reserved.

1. Introduction

Thermal recovery methods [1], such as CSS (cyclic steam stim-

ulation), steamflooding and SAGD (steam-assisted gravity drai-

nage)   [2], have already been proved effective and economic in

exploiting heavy oil reservoirs. Moreover, wet steam is usually

chosen as heat carrier when these methods are used, and one of 

the main reasons is that both the latent heat of vaporization and

the specific heat capacity of water are higher than those of any

other commonly-used liquid. In other words, injecting wet steaminto pay zones can release a large amount of latent heat and sensi-

ble heat to raise reservoir temperature and to lower oil viscosity.

However, superheated steam may also be a good choice for the

heat carrier. Compared with wet steam, superheated steam is char-

acterized by high steam quality, high temperature and low pres-

sure   [3], which guarantees that it has many advantages in

thermal recovery of heavy oils. For example, not only the specific

enthalpy of superheated steam is larger than that of wet steam

at the same pressure, but also superheated steam can further

improve flow environment in porous media   [4]   and promote

aquathermolysis of heavy oils  [5]. At present, cyclic superheated-

steam stimulation using vertical wells is widely applied in

Kenkiyak Oilfield, Aktyubinsk, northwest of Kazakhstan. But if an

oil layer is not thick enough, a horizontal well would be more pro-

ductive than a vertical well due to its larger reservoir contact area.

As superheated steam flows along a horizontal wellbore, its thermo-

physical properties, such as mass flow rate and the degree of super-

heat, always change with horizontal well length, more importantly,

superheated steam may undergo phase change and be cooled to wetsteam in a certain position of the wellbore, in this case, steam quality

is another key parameter that needs to be determined. Therefore,

one of the most important tasks in the design of superheated-steam

injection projects is to estimate these thermophysical properties

before the fluid inside the horizontal wellbore enters the formation.

The classic work in this area was firstly developed by Ramey [6],

who derived an important expression for fluid temperature as a

function of well depth and injection time by combining well-

bore/formation heat-transfer model with energy balance equation.

Hasan and Kabir   [7]   set up a detailed formation heat-transfer

model and proposed a new expression for transient heat-

conduction time function, which was further improved by

http://dx.doi.org/10.1016/j.enconman.2015.04.029

0196-8904/ 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +86 10 89733726.

E-mail address:   guhao110110@163.com (H. Gu).

Energy Conversion and Management 99 (2015) 119–131

Contents lists available at   ScienceDirect

Energy Conversion and Management

j o u r n a l h o m e p a g e :   w w w . e l s e v i e r . c o m / l o c a t e / e n c o n m a n

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Cheng et al. [8] who considered the effect of wellbore heat capacityon heat flow in cement/formation interface. Satter [9] presented a

method of predicting steam quality distribution by taking into

account the effect of condensation. Farouq Ali   [10]   proposed a

comprehensive mathematical model for calculating steam quality

according to energy balance in the injected fluid. Gu et al. [11] sug-

gested a simplified approach for estimating steam pressure and

derived a complete expression for steam quality in wellbores.

Although, the above classic researches are all about fluid injection

in vertical wells, they lay a solid foundation for estimation of ther-

mophysical properties of fluid in horizontal injection wells. Ni et al.

[12]   established a mathematical model for calculating mass flow

rate of wet-steam injection in horizontal wellbores, but they

ignored the energy change due to radial outflow when modeling

steam quality based on energy conservation principle, which wascorrected by Wang et al.  [13]. Dong et al.  [14] created a predictive

model aimed at thermophysical properties of multi-thermal fluidin perforated horizontal wellbores. Su and Gudmundsson   [15,16],

whose work was very crucial to determining the total pressure

drop in horizontal wellbores, carried out pressure drop experi-

ments in perforated pipes and suggested a governing equation

for friction factor of perforation roughness. Emami-Meybodi et al.

[17] developed a transient heat conduction model to estimate heat

transfer from horizontal wellbore to the formation.

The authors and their team have done a series of researches on

estimation of thermophysical properties in the cases of wet-steam

injection [18], unsteady-state steam injection conditions [19], con-

centric dual-tubing steam injection   [20]   and superheated-steam

injection in vertical wells   [21]. Based on previous studies, the

authors begin to focus on cyclic superheated-steam stimulation

using horizontal wells that is applied in KMK Oilfield,Aktyubinsk, Kazakhstan. However, superheated-steam injection

Nomenclature

 Ac   cross-sectional area of casing, m2

 Ad   drainage area, m2

B   volume factor, m3/m3

d p=dL   pressure drop gradient, Pa/mDci   inside diameter of casing, m

 f    friction factor, dimensionless f ci   forced-convection heat transfer coefficient on inside of 

casing, W/(m2 K) f ðt Þ   transient heat-conduction time function, dimensionless g    gravitational acceleration, m/s2

h   specific enthalpy, J/kgH    thickness of oil layer, mH L   liquid holdup, dimensionlessI    volumetric outflow rate of fluid injected into the forma-

tion, m3/sI r   injectivity ratio, dimensionless J 0   first kind Bessel functions of zero order J 1   first kind Bessel functions of first order J pi   productivity index, m3/(s Pa)K    permeability, lm2

K r   relative permeability, dimensionlessL   horizontal well length, mDL   length of perforation unit, mM r   volumetric heat capacity of pay zone, J/(m3 K)nperf    perforation density, m1

N    total number of perforations or perforation units p   pressure, Pa p   average pressure, PaD p   pressure drop, PaQ c   heat conduction rate, WQ in   energy carried by hot fluid at the inlet, WQ rad;i   energy transferred to the formation due to radial out-

flow, WQ out   energy carried by hot fluid at the outlet, Wr ci   inside radius of casing, mr co   outside radius of casing, mr h   heated radius, mr ph   radius of perforation hole, mr w   radius of horizontal wellbore, mRei   Reynolds number, dimensionlesss   skin factor, dimensionlessS w   average water saturation, dimensionlessS wi   initial water saturation, dimensionlesst    injection time, sT    temperature, KT deg   degree of superheat, KT ei   initial temperature of the formation, K

T interf    cement/formation interface temperature, KT    average fluid temperature, KDT    temperature drop, Ku   dummy variable for integration, dimensionlessU co   over-all heat transfer coefficient between fluid and

cement/formation interface, W/(m2 K)Du=u roughness functionm   velocity, m/smr   radial velocity, m/smsg   superficial gas velocity, m/sw   mass flow rate, kg/s x   steam quality, dimensionlessD x   steam quality drop, dimensionlessY 0   the second kind Bessel functions of zero orderY 1   the second kind Bessel functions of first order

Greek lettersa   thermal diffusivity of formation, m2/hb   unit conversion factor, dimensionless

e   roughness of casing wall, mh   well angle from horizontalkcas   thermal conductivity of casing wall, W/(m K)kcem   thermal conductivity of cement sheath, W/(m K)ke   thermal conductivity of formation, W/(m K)l   viscosity, mPa sq   density, kg/m3

sD   dimensionless timex   ratio of the formation heat capacity to the wellbore heat

capacity, dimensionless/   porosity of oil layer, dimensionless

Subscriptsacc acceleration

h horizontalm mixturens no-slipo oilperf perforation roughnesspot potential energyr reservoirs dry steamsuperh superheated steamv verticalw saturated wateri; j; k   index

120   H. Gu et al. / Energy Conversion and Management 99 (2015) 119–131

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in horizontal wells has not been widely reported in the literature.

In fact, it is a complex process, involving not only mass and heat

transfer, but also phase change. Firstly, after superheated steam

reaches the heel-position of a horizontal well, it would flow from

the heel to the toe under the effect of injection pressure, at the

same time, a part of flowing fluid outflows radially from the well-

bore to the oil layer because of pressure difference between the

fluid inside the wellbore and the formation. It should be noted that

the variable-mass flow in horizontal wellbores not only affects the

profiles of thermophysical properties of hot fluid, but also leads to

a completely different mathematical model. For instance, the

energy transferred to the formation on account of fluid outflow

should be included in energy balance equation; also, the governing

equations for fluid temperature and steam quality are implicit,

which will be presented later in detail. Secondly, the heat carried

by the fluid inside the horizontal wellbore is transferred to the

oil layer in two different ways. The first one is forced convection

or fluid outflow and the other one is heat conduction due to tem-

perature difference between the fluid inside the wellbore and the

surroundings. Finally, as superheated steam flows in the horizontal

wellbore, it may be cooled to wet steam in a certain position of the

wellbore, especially for a long horizontal well, a low injection rate,

and so on. This is because the temperature of superheated steam in

the horizontal wellbore usually drops much faster than fluid pres-

sure; therefore, it is also an important question to determine the

critical point where the steam quality begins to decrease.

The main objectives of this work are to establish a comprehen-

sive mathematical model for estimating the above thermophysical

properties and to analyze the performance of superheated-steam

injection in horizontal wells. In this paper, governing equations

for mass flow rate and pressure drop in horizontal wellbores are

firstly established and phase change from superheated steam to

wet steam is taken into account. Moreover, implicit equations for

both the degree of superheat and steam quality are derived based

on energy balance principle. Then, the mathematical model is

solved using an iterative technique and a calculation flowchart is

provided. Finally, after the model is verified with measured fielddata, the effects of some important factors on the profiles of ther-

mophysical properties are analyzed in detail.

2. Mathematical model

Fig. 1 shows a simplified schematic of superheated-steam injec-

tion in a horizontal well. In this paper, to simplify the model calcu-

lation, several major assumptions are made:

(1) The injection conditions, including injection pressure, tem-

perature and mass flow rate of superheated steam at the

heel-position of the horizontal wellbore, do not change with

injection time.

(2) Heat conduction from the fluid inside the horizontal well-

bore to the cement sheath is steady-state, while heat con-

duction in the formation is transient.

(3) The physical and thermal properties of the formation are

independent of temperature.

(4) Perforation parameters (i.e. perforation density, diameter

and phasing) are the same along the horizontal wellbore.

 2.1. Mass flow rate in horizontal wellbores

Assuming that the length of the horizontal wellbore and the

perforation density are L and nperf , respectively, so the total number

of perforations is   N  ¼ Lnperf . Then, the horizontal wellbore is

divided into  N  segments and is numbered from 1 to  N , and each

segment contains only one perforation, as illustrated in   Fig. 2.

Moreover, take the  i-th perforation unit as an example, the mass

flow rate, fluid pressure and temperature at the inlet are assumed

tobe wi1; pi1 and T i1, respectively, and the corresponding valuesat the outlet are wi; pi  and T i , respectively, also, the volumetric out-

flow rate of hot fluid injected into the formation through the per-

foration is assumed to be I i.

The mass flow rate at the outlet of the i-th perforation unit, wi,

can be calculated by subtracting the sum of mass flow rates of hot

fluid that has entered the formation from the initial mass flow rate

at the horizontal well’s heel position, accordingly, the mass balance

equation can be written as

wi ¼ w0 Xi

 j¼1

ðq jI  jÞ;   1 6 i 6 N    ð1Þ

where w0  denotes the mass flow rate of superheated steam at the

heel-position of the horizontal well; q j  represents the average den-

sity of hot fluid in the  j-th perforation unit. If superheated steam

does not undergo phase change in this unit, its density can be

obtained from interpolation of superheated-steam tables [22], and

parts of the data used in this article are provided in  Table A.1 in

Appendix A. Alternatively, a more practical approach, namely

regression analysis, can be adopted. Here, according to the

Table A.1, the authors propose empirical correlations, which are

given by

(a) Schematic offluid flow in a horizontal injection well. (b) Structure of a horizontal wellbore.

cir cor 

wr 

Casing

Cement

Formation

Perforation

Oil layer

Horizontal well

Underburden

Overburden

Toe

Heel

Fig. 1.  Schematic of superheated-steam injection in a horizontal well.

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qsuperh ¼ qð p;T Þ ¼   25:822345 p

0:019266ðT  273:15Þ 0:203202 p þ 1:017353;

4 MPa 6 p 6 9 MPa   ð2Þ

qsuperh ¼ qð p;T Þ ¼   16:251218 p

0:017255ðT  273:15Þ 0:151339 p 0:904339;

9 MPa < p 6 14 MPa   ð3Þwhere p;T   and qsuperh are the pressure, temperature and density of 

superheated steam, respectively. Table 1 shows the maximum abso-

lute residual (Max AR), mean absolute residual (MAR) and the max-

imum relative residual (MRR) of the above empirical correlations.The acceptable residuals may support the reliability of the proposed

empirical correlations.

However, superheated steam may be cooled to wet steam as it

flows along the horizontal wellbore, and in this case, slippage

between gas and liquid phases exists due to density difference,

which should be considered in estimating the density of steam/wa-

ter mixture fluid. Usually, it is defined as

qm ¼ qðH L; T Þ ¼ qsð1 H LÞ þ qwH L   ð4Þwhere qm  represents the density of mixture fluid;  H L  is the liquid

holdup, which can be calculated by using the classic method pre-

sented by Beggs and Brill  [23]; qs   and qw   are the densities of dry

steam and saturated water, respectively, which can be obtained

from interpolation of saturated-steam tables  [24] or can be calcu-

lated by empirical correlations, and in the computer procedure,

they are given as follows [25]:

lnqs ¼ 93:7072 þ 0:833941T  0:00320809T 2

þ 6:57652 106T 3 6:93747 109T 4

þ 2:97203 1012

T 5273:15 K 6 T  6 645 K   ð5Þ

qw ¼ 3786:31 37:2487T þ 0:196246T 2

5:04708 104

T 3 þ 6:29368 107

T 4

3:08480 1010T 5; 273:15 K 6 T  6 640 K   ð6ÞBased on the above discussion, the expression of  q j   in Eq.  (1)

can be summarized as follows:

q j ¼qsuperh; j ¼ qð p j; T  jÞ;   superheated steam

qm; j ¼ qðH L; j; T  jÞ;  wet steam

(  ð7Þ

where p j   and T  j  are the average pressure and average temperature

of hot fluid in the   j-th perforation unit, respectively, p j ¼ ð p j1 þ p jÞ=2, T  j ¼ ðT  j1 þT  jÞ=2.

In addition, in Eq.  (1), the volumetric outflow rate of hot fluid

injected into the formation,  I  j, can be estimated by [12]

I  j ¼ J pi; jI r; jð p j  prÞ ð8Þwhere pr  is the average reservoir pressure, which can be calculated

with the method suggested by Chen [26]; J pi; j and I r; j  are the produc-

tivity index and the injectivity ratio for the   j-th perforation unit,

respectively, which can be calculated from [12,27]

 J pi; j ¼ b2p

 ffiffiffiffiK hK v

q   K vDL   K ro

Boloþ   K rw

Bwlw

ln

0:571

 ffiffiffiffiffi Ad; j

p r w

þs

0:75

ð9Þ

I r; j ¼2 ln

 Ad; j

r 2w 3:86

ln Ad; j

r 2w 2:71

ð10Þ

where b  is the unit conversion factor;  K h  and K v  are the horizontal

permeability and the vertical permeability, respectively;  DL   is the

length of each perforation unit,  DL ¼ L=N ;  Ad; j   is the drainage area

for the j-th perforation unit, which can be determined by referring

to the methods found in Ref.   [28];   r w   is the radius of horizontal

wellbore; s  is the skin factor; K r;B andl  are the relative permeabil-

ity, the volume factor and the viscosity, respectively, and the sub-

scripts   w   and  o  denote water phase and oil phase, respectively,

moreover, an overall mass balance on the drainage volume yields

the following equation for average water saturation that determinesK ro  and  K rw

Fig. 2.  Division of the horizontal wellbore into N  segments.

 Table 1

The maximum absolute residual, mean absolute residual and the maximum relative

residual of the proposed empirical correlations.

Empirical correlation Max AR MAR MRR (%)

Eq. (2)   1.224 0.301 4.204

Eq. (3)   3.532 0.778 4.228

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S w ¼ S wi þ   w0t 

qhw AdH /  ð11Þ

where  S w   and S wi   are the average water saturation and the initial

water saturation, respectively;  t   is the injection time;  qhw   is the

density of hot water in the reservoir;   H   and  /  are the thickness

and the porosity of oil layer, respectively.

 2.2. Pressure drop in horizontal wellbores

As fluid flows in a completely horizontal wellbore, the total

pressure drop is dominated by frictional effects   [15,16].

Consequently, if the mixing effects are ignored, then according to

momentum balance principle, the total pressure drop in an

inclined perforation unit can be expressed as

d pt ;i

dL ¼ d ppot;i

dL  d pwall;i

dL  d pperf ;i

dL  d pacc;i

dL  ð12Þ

where d pt;i=dL is the total pressure drop in the i-th perforation unit;

other terms in Eq.  (12) are introduced as follows:

The first term of the right side in Eq.  (12), d ppot;i=dL, represents

the pressure drop due to potential energy change, which can be

written as

d ppot;i

dL  ¼ qi g sin h   ð13Þ

where g  is the gravitational acceleration;  h  is the well angle from

horizontal. If the wellbore is completely horizontal, namely,

h ¼ 00

, then d ppot;i=dL  is equal to zero.

The second term of the right side in Eq.  (12), d pwall;i=dL, repre-

sents the pressure drop due to casing wall friction and a general

expression for it is

d pwall;i

dL  ¼  f wall;i

q0i

Dci

mi2

2  ð14Þ

where D ci  denotes the inside diameter of casing; m i   is the averagevelocity of fluid in the i-th perforation unit, which can be estimated

by

mi ¼ mi1 þ mi

2  ð15Þ

where mi1  and mi  are the velocities of fluid at the inlet and outlet of 

the i-th perforation unit, respectively,  mk ¼ wk=ðqk AcÞ   (k ¼ i  1 or

k ¼ i), Ac  is the cross-sectional area of casing,  Ac ¼ pD2ci=4.

In Eq. (14), the expression of  q0i  is given by

q0i ¼

  qsuperh;i ¼ qð pi; T iÞ   ; superheated steam

qns;i   ; wet steam

(  ð16Þ

whereqns;i is the no-slip density of steam/water mixture fluid in the

i-th perforation unit, which can also be determined by adopting the

method in Ref. [23].

In addition,  f wall;i  in Eq. (14) denotes the friction factor for pipe

flow. If superheated steam does not undergo phase change, it is

single-phase flow in the horizontal wellbore, and  f wall;i   depends

on flow patterns, Reynolds number (Rei) and the roughness of 

the casing wall (e). Yuan et al.  [29] summarized empirical correla-

tions that can be used to estimate sing-phase friction factor of pipe

flow, as displayed in Table 2. It should be stressed that the critical

region between laminar flow and turbulent flow is always

unsteady-state, and f wall;i  can be calculated according to the empir-

ical correlation for smooth pipe. However, if phase change occurs,

it is steam/water two-phase flow in the horizontal wellbore, and in

this case,   f wall;i  can be estimated by adopting the classic methodproposed by Beggs and Brill   [23]. In their study, the two-phase

friction factor is a function of no-slip friction factor, input liquid

content and liquid holdup, and the detailed calculation method

can be found in Ref.  [23].

The third term of the right sidein Eq. (12), d pperf ;i=dL, represents

the pressure drop due to perforation roughness, which can be cal-

culated as

d pperf ;i

dL   ¼  f perf ;i

q0i

Dci

mi2

2   ð17Þwhere the friction factor of perforation roughness,   f perf ;i, is deter-

mined by the following implicit equation [15,16] ffiffiffiffiffiffiffiffiffiffiffi8

 f perf ;i

s   ¼ 2:5 ln

  Rei

2

 ffiffiffiffiffiffiffiffiffiffiffi f perf ;i

8

s 0@

1Aþ B Du

u  3:75   ð18Þ

where constant   B  and roughness function  Du=u   can be obtained

from

B ¼ ffiffiffiffiffiffiffiffiffiffiffi

8

 f wall;i

s   2:5 ln

  Rei

2

 ffiffiffiffiffiffiffiffiffiffiffi f wall;i

8

s 0@

1Aþ 3:75   ð19Þ

Du

u ¼ 7:0  2r ph

Dci

  n perf 

12

  ð20Þ

where r ph is the radius of perforation hole.

The fourth term of the right side in Eq.  (12), d pacc;i=dL, repre-

sents the pressure drop due to acceleration. For superheated-steam

single-phase flow, d pacc;i=dL  can be expressed as

d pacc;i

dL  ¼ qim

2i  qi1m

2i1

DL  ð21Þ

While for steam/water two-phase flow, d pacc;i=dL  can be calcu-

lated by

d pacc;i

dL   ¼ qm;imm;i

dmm;i

dL   ¼ qm;imm;imsg;i

 pt;i

d pt;i

dL   ð22Þ

where mm;i  and msg;i  are the mixture fluid velocity and the superficial

gas velocity in the  i-th perforation unit, respectively.

Substituting Eqs.   (13), (14), (16), (17) and (21)  into Eq.   (12)

yields the total pressure drop for superheated-steam single-phase

flow in the  i-th perforation unit,

d pt;i

dL ¼ qsuperh;i g sin h ð f wall;i þ f perf ;iÞ

qsuperh;i

Dci

mi2

2

qim2i  qi1m

2i1

DL  ð23Þ

However, for steam/water two-phase flow, d pt;i=dL   can be

obtained by substituting Eqs.  (13), (14), (16), (17) and (22)   intoEq. (12),

 Table 2

Empirical correlations for estimating sing-phase friction factor of pipe flow [29].

Flow pattern   Rei   f wall;i

Laminar

flow

Rei  6 2000   f wall;i ¼  64Rei

Critical

region

2000 < Rei  6 3000 –

Turbulent flow

Smooth pipe   3000 < Rei  6 59:7=e8

7   f wall;i ¼ 0:3164 ffiffiffiffiffi

Rei4p 

Transition

region59:7=e

87  <  Rei  6

665765lgee  f wall;i ¼ 1:8lg   6:9

Reiþ   e

3:7Dci

1:11 2

Rough pipe   Rei  > 665765lg e

e   f wall;i ¼   2lg3:7Dci

e

2

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d pt;i

dL ¼

qm;i g sin h ð f wall;i þ f perf ;iÞ qns;i

Dci

mi2

2

1 qm;imm;imsg;i= pt;i

ð24Þ

 2.3. Energy balance equation

As mentioned above, a part of heat carried by hot fluid inside

the horizontal wellbore is transferred to the formation because of 

radial outflow, while another part of heat is lost to the surround-ings due to heat conduction. Applying energy balance to the  i-th

perforation unit yields

Q rad;i þ Q c ;i ¼ Q in;i Q out;i   ð25Þ

where Q c;i  is the heat conduction rate in the  i-th perforation unit

and is discussed in detail in Appendix B; Q rad;i  represents the energy

transferred to the formation on account of radial outflow, including

enthalpy and kinetic energy,

Q rad;i ¼ I iqi   hi þ m2r ;i

2

!  ð26Þ

Fig. 3.  Calculation flowchart for the mathematical model.

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where hi  is the average enthalpy of fluid in the i-th perforation unit;

mr;i  is the velocity of radial outflow from the horizontal wellbore to

the formation, mr;i ¼ I i=ðpr 2ph

Þ.

In addition, Q in;i  and Q out;i  in Eq. (25) denote the energy carried

by hot fluid at the inlet and outlet of the   i-th perforation unit,

respectively,

Q in;i ¼ wi1   hi1 þm2

i12

  ð27Þ

Q out;i ¼ wi   hi þ m2i

2

  ð28Þ

where hi1 and hi are the enthalpies of fluid at the inlet and outlet of 

the i  -th perforation unit, respectively.

Before superheated steam undergoes phase change, the degree

of superheat is a key parameter that needs to be calculated, while

after phase change occurs, steam quality is another important ther-

mophysical parameter. In the following, how to obtain the govern-

ing equations for both the degree of superheat and steam quality

will be introduced based on the above Eq.  (25).

 2.4. Implicit equation for the degree of superheat 

The enthalpy of superheated steam is related to fluid pressure

and temperature, which can also be obtained from interpolation

of superheated-steam tables [22], and parts of the data used in this

paper are listed in Table A.2 in Appendix A. Similarly, by regression

analysis, an empirical correlation is also recommended:

hsuperh ¼ hð p; T Þ¼ 2588:296398 þ 3:670906 ðT  273:15Þ

186852:258072

T  273:15  38:997194 p þ 3835:954647

 p

12:063850 ðT  273:15Þ p

  ð29Þ

where hsuperh  denotes the enthalpy of superheated steam. And the

Max AR, MAR and MRR for this empirical correlation are about

49.90, 9.82 and 1.87%, respectively.

Incorporating Eqs. (26)–(28) and (B-8) into Eq. (25) leads to an

implicit equation for fluid temperature inside the horizontal

wellbore,

Fig. 4.  Permeability distribution along the horizontal wellbore.

 Table 3

Basic parameters used for the field test at Well-453 in KMK Oilfield, Kazakhstan.

Parameter Unit Value Parameter Unit Value

Reservoir depth (Dr) m 287 Length of the horizontal wellbore (L) m 195.3

Thickness of oil layer (H ) m 15 Perforation density (nperf ) m1 12

Initial reservoir pressure ( pr;i) MPa 2.38 Radius of perforation hole (r ph) m 0.0075

Initial oil saturation (S oi) – 0.75 Inside radius of casing (r ci) m 0.0807

Porosity of oil layer (/) – 0.32 Outside radius of casing (r co) m 0.0889

Initial formation temperature (T ei) K 291.29 Radius of wellbore (r w) m 0.12

Oil viscosity at T ei  (lo) mPa s 1366 Roughness of casing (e) m 0.0000457

Volume factor of oil (Bo) m3/m3 1.05 Thermal conductivity of the cement (kcem) W /(m K) 0.933

Volume factor of water (Bw) m3/m3 1.01 Thermal conductivity of the formation (ke) W /(m K) 1.73

Drainage area for the horizontal

well ( Ad)

m2 29,700 Thermal diffusivity of the formation (a) m2/h 0.00037

570

580

590

600

610

620

630

0 50 100 150 200

Measured field data

Simulated results

   F   l  u   i   d   t  e  m  p  e  r  a   t  u  r  e   (   K   )

Horizontal well length (m)

0.2

0.4

0.6

0.8

1.0

1.2

0 50 100 150 200

Measured field data

Simulated results

   S   t  e  a  m   q

  u  a   l   i   t  y

Horizontal well length (m)

(a) (b)

Fig. 5.  Comparisons of simulated fluid temperature (a) and steam quality (b) with measured field data.

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I iqi   hi þm2

r;i

2

!þ   2pr coU cokeDL

r coU co f ðt Þ þ ke

ðT i T eiÞ

¼ wi1   hi1 þ m2i1

2

wi   hi þ m2

i

2

  ð30Þ

where   hi ¼ hð pi;T iÞ;   hk ¼ hð pk; T kÞ   (k ¼ i 1 or   k ¼ i) and

DT i ¼ T i  T i1. DT i  or T i  can be determined by solving Eq. (30) with

an iterative technique.

In addition, according to saturated-steam tables   [24], there is

one-to-one correspondence between saturated pressure and tem-

perature. By polynomial interpolation, Tortike et al. [25] proposed

an important empirical correlation to describe this relationship,

which is given by

T  ¼  f ð pÞ ¼ 280:034 þ 14:0856 ln  p

1000þ 1:38075   ln

  p

1000

2

0:101806 ln  p

1000

3

þ 0:019017 ln  p

1000

4

611 Pa

6  p 6 2:212 107

Pa   ð31ÞThus, the degree of superheat at the outlet of the  i-th perfora-

tion unit (T deg;i) is

T deg;i ¼ T i  f ð piÞ ð32Þ

 Table 4

Basic parameters used for the performance analysis of superheated-steam injection in

horizontal wells.

Parameter Unit Value

The degree of superheat K 40

Injection rate t/h 8

Injection pressure MPa 12

Horizontal and vertical permeabilities   lm2 1500 103

Oil viscosity mPa s 1500

Fig. 6.  Effects of the degree of superheat at the horizontal well’s heel position on the profiles of the degree of superheat (a), steam quality (b) and mass flow rate (c) in thehorizontal wellbore.

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 2.3.2. Implicit equation for steam quality

After superheated steam is cooled to wet steam, the fluid tem-

perature can be easily estimated by Eqs. (24) and (31). In this case,

the specific enthalpy of steam/water mixture fluid is a function of 

steam quality and temperature:

hm ¼ hð x; T Þ ¼  xhs þ ð1  xÞhw   ð33Þwhere x is the steam quality; hs and hw are the specific enthalpies of 

dry steam and saturated water, respectively, and in the computer

procedure, they are given by [25]

hs ¼ 22026:9 þ 365:317T  2:25837T 2 þ 0:00737420T 3

1:33437 105T 4 þ 1:26913 108T 5

4:96880 1012

T 6; 273:15 K 6 T  6 640 K   ð34Þ

hw ¼ 23665:2 366:232T þ 2:26952T 2 0:00730365T 3

þ 1:30241 105T 4 1:22103 108T 5

þ 4:70878 1012T 6; 273:15 K 6 T  6 645 K   ð35Þ

Combing Eqs.  (33) and (30) results in an implicit equation for

steam quality inside the horizontal wellbore,

I iqi   hm;i þm2

r;i

2

!þ   2pr coU cokeDL

r coU co f ðt Þ þ ke

ðT i T eiÞ

¼ wi1   hm;i1 þ m2i1

2

wi   hm;i þ m2

i

2

  ð36Þ

where   hm;i ¼ hð xi; T iÞ;hm;k ¼ hð xk; T kÞ   (k ¼ i 1 or   k ¼ i) and

D xi ¼ xi  xi1:   Similarly,  D xi   or   xi   can be obtained by solving Eq.

(36) with an iterative technique.

3. Calculation flowchart for the mathematical model

As stated above, it is necessary to adopt iterative method to

solve the mathematical model. The main steps are given as

follows:

(1) Input given parameters and divide the horizontal wellbore is

into N  segments.

(2) Judge the state of the fluid based on Eq.  (31).

Fig. 7.  Effects of injection rate on the profiles of mass flow rate (a), the degree of superheat (b) and steam quality (c) in the horizontal wellbore.

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(3) If superheated steam does not undergo phase change, itera-

tive method is used to calculate both the temperature drop

and pressure drop in each perforation unit, or iterative

method is used to calculate both the steam quality drop

and pressure drop in each perforation unit.

(4) Output  wi; pi; T i   and T deg;i   for superheated steam, while for

wet steam, output wi; pi; T i   and xi.

(5) Repeat steps (2), (3) and (4) until the toe-position of the hor-izontal wellbore is reached.

The calculation flowchart for the mathematical model is shown

in Fig. 3.

4. Results and discussion

4.1. Validation of the model with measured field data

In this section, to validate the proposed model, simulated

results are compared with measured field data. The field test was

performed at Well-453 in KMK Oilfield, Aktyubinsk, northwest of 

Kazakhstan. In the field test, the fluid pressure and temperature

at the heel position of the horizontal well, namely,  p0   and T 0, are

9.86 MPa and 621.2 K, respectively, based on Eq.   (31), the degree

of superheat is about 38.1 K. In addition, the mass flow rate (w0)

and the injection time (t ) are 6.23 t/h and 12 days, respectively.

Fig. 4  shows permeability distribution along the horizontal well-

bore. It is clearly observed that the oil layer is heterogeneous,

and the permeability in each segment can be estimated with arith-

metic average method. Other basic parameters used for the field

test are listed in Table 3, mainly including the formation and fluid

properties, the horizontal wellbore dimensions, and so on.

Fig. 5(a) and (b) shows comparisons of simulated fluid temper-

ature and steam quality from the mathematical model and those

from the test field data, respectively. Firstly, it is observed that

superheated steam undergoes phase change at a horizontal well

length of about 149.39 m, and from the heel position to the phase

change point, the simulated fluid temperature declines with hori-

zontal well length while the simulated steam quality is always

equal to 1, which agrees fairly well with the measure field data.

Secondly, from the phase change point to horizontal well’s toe

position, the calculated fluid temperature nearly keeps constant

and the calculated steam quality drops gradually, which also show

good agreement with the test values. More importantly, error anal-

ysis is further conducted. The results indicate that the maximum

absolute error in the prediction of fluid temperature is about

6.8 K, and because the measured values are relatively high, the

maximum relative error is only about 1.15%. Also, the maximum

absolute and relative errors in the prediction of steam quality are

about 0.073 and 8.30%, respectively, which are also acceptable in

engineering calculation and should support the reliability of the

proposed model.

Fig. 8.  Effects of injection pressure on the profiles of mass flow rate (a), the degree of superheat (b) and steam quality (c) in the horizontal wellbore.

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4.2. Performance analysis of superheated-steam injection in horizontal

wells

In this section, the performance of superheated-steam injection

in horizontal wells is analyzed based on the above validated model.

The basic parameters used for the following calculation are pro-

vided in   Table 4   and other related parameters are displayed in

Table 3.

4.2.1. Effect of the degree of superheat 

Fig. 6 shows the effects of the degree of superheat at the hori-

zontal well’s heel position (T deg;0) on the profiles of thermophysical

properties in the horizontal wellbore. From Fig. 6(a) and (b), it is

easily found that for a given   T deg;0, the degree of superheat

decreases with horizontal well length and the steam quality is

equal to 1 before phase change occurs. Moreover, the lower the

T deg;0   is, the shorter the distance between the phase change point

and the heel-position of the horizontal well is. Therefore, to ensure

that it is superheated steam rather than wet steam that is injected

into the oil layer in cyclic superheated-steam stimulation, it is

necessary to enhance   T deg;0, especially in long horizontal wells.

As can be seen from  Fig. 6(c), the mass flow rate in the same

position of the horizontal wellbore increases with   T deg;0. The

main reason can be explained as follows: although, according

to Eqs.  (8) and (9), high temperature helps to reduce the oil vis-

cosity and inject larger volume of superheated steam into the oil

layer per unit of time, the difference in the volumetric outflowrate is very little, this is because when the fluid temperature is

high enough, the difference in the oil viscosity caused by little

temperature difference can be negligible. More importantly,

when the fluid pressure is the same, a little higher degree of 

superheat leads to much lower density of superheated steam,

as presented in   Table A.1 in Appendix A, in other words, the

mass outflow rate of superheated steam injected into the oil

layer is relatively slow in the case of a little higher   T deg;0. From

Fig. 6(c), it is observed that after phase change occurs, the mass

flow rate in the horizontal wellbore drops faster due to the

increase in fluid density.

Fig. 9.  Effects of reservoir permeability on the profiles of mass flow rate (a), the degree of superheat (b) and steam quality (c) in horizontal wellbores.

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4.2.2. Effect of injection rate

Fig. 7 shows the effects of injection rate (w0) on the profiles of 

thermophysical properties in the horizontal wellbore. As can be

seen from Fig. 7(a), when w0 ¼ 5 t/h and w0 ¼ 6 t/h, the mass flow

rate drops to zero at a certain position of the horizontal well, there-

fore, to ensure that the toe section of the horizontal well can also

be heated effectively, the injection rate should not be too slow.

From Fig. 7(b), it is clearly found that the slower the injection rate

is, the faster the degree of superheat drops and the shorter the dis-

tance between the phase change point and the heel-position of the

horizontal well is. For instance, when w0   is equal to 5 t/h, 6 t/h and

7 t/h, respectively, superheated steam undergoes phase change at

horizontal well lengths of about 131.14m, 154.81m and178.05 m, respectively, but it is still superheated steam at the

toe-position of the horizontal well when the injection rate is not

less than 8 t/h. Consequently, to ensure that the fluid temperature

is still very high before the fluid enters the formation, the injection

rate should also not be too slow. Fig. 7(c) indicates that after phase

change occurs, steam quality declines rapidly, and the slower the

mass flow rate is, the faster the steam quality drops. For example,

when  w0   is equal to 5 t/h, the average steam quality gradient is

about 0.5949/100 m, while the corresponding values for w0 ¼ 6 t/h

and w0 ¼ 7 t/h are 0.5614/100 m and 0.2391/100 m, respectively.

4.2.3. Effect of injection pressure

Fig. 8  shows the effects of injection pressure on the profiles of 

thermophysical properties in the horizontal wellbore. FromFig. 8(a) and (b), it is observed that in the same position of the hor-

izontal wellbore, both mass flow rate and the degree of superheat

decrease with injection pressure. The reason can be given as fol-

lows: according to Eq.  (8), high injection pressure helps to inject

more fluid into the oil layer per unit of time, resulting in a slower

mass flow rate in the wellbore, which further leads to a faster

decrease in the fluid temperature. It should be noted that when

the injection pressure is equal to 13 MPa, the degree of superheat

drops to zero at a horizontal well length of about 183.22 m, where

the steam quality begins to decline, as illustrated in  Fig. 8(c).

4.2.4. Effect of reservoir permeability

Fig. 9 shows the effects of reservoir permeability on the profiles

of thermophysical properties in the horizontal wellbore. It isobviously found that the higher the reservoir permeability is, the

faster the mass flow rate declines, as illustrated in   Fig. 9(a).

This is because hot fluid can be more easily injected into

high-permeability heavy oil reservoirs, according to Eqs.   (8) and

(9). Thus, it can be concluded that in a heterogeneous reservoir,

low-permeability zone may not be fully heated due to the difficulty

of hot fluid injection. More importantly, fast decline in the mass

flow rate caused by high reservoir permeability can lead to not

only a fast decrease in the degree of superheat before superheated

steam undergoes phase change, but also a fast drop in the steam

quality after phase change occurs, as shown in  Fig. 9(b) and (c).

Consequently, it is highly possible that it is wet steam rather than

superheated steam at the toe-position of horizontal wellbores,

especially in high-permeability heavy oil reservoirs. In this case,based on the above analysis, enhancing the degree of superheat

and injection rate may be two effective methods to solve this

problem.

4.2.5. Effect of oil viscosity

Fig. 10 shows the effects of oil viscosity on the profiles of ther-

mophysical properties in the horizontal wellbore. According to Eqs.

(8) and (9), hot fluid can be more easily injected into low-viscosity

heavy oil reservoirs, so when the oil viscosity is equal to 500 mPa s,

both the mass flow rate and the degree of superheat drop very fast

and the phase change point is also close to the toe-position of the

horizontal well.

5. Conclusions

The following conclusions can be derived from the results of 

this work.

 The proposed comprehensive mathematical model is proved to

be reliable in engineering calculation and can be used to esti-

mate the thermophysical properties of superheated steam in

horizontal injection wells, and phase change behavior of super-

heated steam is taken into consideration.

 In the same position of horizontal wellbore, the mass flow rate

increases with  T deg;0, but for a given  T deg;0, the mass flow rate

drops faster after superheated steam is cooled to wet steam.

  To ensure that the toe section of horizontal well can also be

heated effectively, the injection rate should not be too slow,more importantly, the slower the injection rate is, the shorter

Fig. 10.  Effects of oil viscosity on the profiles of mass flow rate (a) and the degree of superheat (b) in the horizontal wellbore.

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the distance between phase change point and the heel-position

of horizontal well is, and after phase change occurs, the slower

the injection rate is, the faster the steam quality drops.

 Both the mass flow rate and the degree of superheat in the same

position of horizontal wellbore decrease with injection

pressure.

 When reservoir permeability is high or oil viscosity is low, the

mass flow rate and the degree of superheat decline rapidly.

 Acknowledgements

This work was supported by the Research Institute of Petroleum

Exploration and Development, Petro China and the National

Science and Technology Major Projects of China (2011ZX05024-

005-006 and 2011ZX05012-004).

 Appendix A. Supplementary material

Supplementary data associated with this article can be found, in

the online version, at   http://dx.doi.org/10.1016/j.enconman.2015.

04.029.

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