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