An improved two-phase model for saturated steam flow in multi-point injection horizontal wells under steady-state injection condition

Accepted Manuscript An improved two-phase model for saturated steam flow in multi-point injection horizontal wells under steady-state injection conditions Fengrui Sun, Yuedong Yao, Guozhen Li, Xiangfang Li, Tao Zhang, Chengang Lu, Wenyuan Liu PII:

S0920-4105(18)30362-0

DOI:

10.1016/j.petrol.2018.04.056

Reference:

PETROL 4908

To appear in:

Journal of Petroleum Science and Engineering

Received Date: 13 August 2017 Revised Date:

23 April 2018

Accepted Date: 25 April 2018

Please cite this article as: Sun, F., Yao, Y., Li, G., Li, X., Zhang, T., Lu, C., Liu, W., An improved twophase model for saturated steam flow in multi-point injection horizontal wells under steady-state injection conditions, Journal of Petroleum Science and Engineering (2018), doi: 10.1016/j.petrol.2018.04.056. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

An Improved Two-phase Model for Saturated Steam Flow in Multi-point Injection Horizontal Wells under Steady-state Injection Conditions

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Fengrui Suna,b,c**, Yuedong Yaoa,b,c*, Guozhen Lic,d, Xiangfang Lib,c, Tao Zhangb,c, Chengang Lub,c, Wenyuan Liuc,d a State Key Laboratory of Petroleum Resources and Prospecting, China University of Petroleum Beijing, 102249 Beijing, P. R. China b College of Petroleum Engineering, China University of Petroleum - Beijing, 102249 Beijing, P. R. China c China University of Petroleum - Beijing, 102249 Beijing, P. R. China d College of Mechanical and Transportation Engineering, China University of Petroleum - Beijing, 102249 Beijing, P. R. China Abstract

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Previous models neglected the effect of frictional heating on steam quality and enthalpy in wellbores when saturated steam is injected into multi-point injection horizontal wellbores. In this paper, new energy conservation equations were developed for increasing the calculation accuracy of steam quality in inner tubing (IT) and annuli. Then, coupled with the momentum balance equations, the distributions of steam quality and pressure along the IT and annuli were

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obtained by using the straight forward numerical method. The predicted results from the new model were compared against field data and previous models. It is found that: (a). The effect of heat loss to surrounding formation on the temperature profiles is weak. However, the heat loss has a significant influence on the profile of steam quality in wellbores. (b) When removing the item of

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frictional work from the energy balance equations, the predicted values of steam quality and enthalpy are lower than actual data. This error in energy conservation equation may be hidden

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when the predicted temperatures showed good agreement with field data. (c) With the help of multi-point injection technique, formation heating effect at both heel-point and toe-point can be improved.

Keywords

Heavy oil; multi-point injection; horizontal well; two-phase steam flow; steam quality; friction work 1. Introduction Thermal technique has been widely adopted in many aspects of engineering, and the mathematical modeling methods are widely used in describing and estimating the performance of fluid 1

ACCEPTED MANUSCRIPT behaviors in industry (Yang et al., 2016,2017a,b; Sheikholeslami et al. 2018a,b,c,d,e,f,g; Zhang et al., 2017a,b,c,d; Feng et al., 2018a,b; Hu et al., 2018). With the depletion of conventional oil and gas resources, heavy oil reserves are gaining more and more attention (Manuel Monge et al., 2017; Abimelech Paye Gbatu et al., 2017; Yang and Wu 2017; Pang et al., 2017; A.M. Elbaz et al., 2018;

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Martin Berntsen et al., 2018; Yuan et al., 2018; Leng et al., 2018; Liu and Bai et al., 2018; Antzela Fivga et al., 2018; Zheng et al., 2018). Steam has always been selected as the thermal carrier for heavy oil recovery (Fontanilla et al. 1982; Khansari et al. 2014; Dong et al., 2015; Sun et al. 2017a; Liu and Falcone et al., 2018; Duan et al., 2018). This is because the enthalpy of wet steam per unit

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mass is relatively high, and the cost is relatively low (Griston et al. 1987; Butler 1991; Bendiksen et al., 1991; Hight 1992; Liu 2009; Bratland 2010; Oosterkamp et al., 2015). In order to increase

steam at the well-bottom condition.

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the oil recovery efficiency, a mathematical model is in need to optimize the key parameters of wet

Based upon the assumption that the heat flow rate is steady-state in wellbores, and it is transient-state in formation, Ramey (1962) proposed a model for estimating steam temperature along the wellbores. However, their model is only applicable for incompressible liquid. Satter

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(1965) presented a model for estimating steam quality in wellbores with consideration of phase change. However, kinetic energy change during steam flow in wellbores was neglected in the energy balance equation, which brought some errors to the predicted result. Hason et al. (1991)

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focused on the physics of heat transmission in formation and presented a model for predicting the temperature distribution in formation. In their model, a formula was developed indicating that the

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formation temperature is a function of both the distance to wellbore and injection time. However, the wellbore heat capacity was neglected in their model, which was later improved by Cheng et al. (2011, 2012). With the development of technology, multi-point injection was proposed to relieve the phenomenon of uneven steam absorption when the horizontal section of the wellbores is extremely long and the reservoir is of serious heterogeneity. Caetano (1985), Hasan et al (1992), Antonio et al. (2000, 2002) and Yu et al. (2010) conducted a series of studies on the flow patterns of two-phase fluid in wellbores during the up-flow process, which has some differences from the downward flow process. This is because the difference in the flow direction has an obvious influence on bubble distribution and flow patterns (Hasan 1995). Griston et al. (1987) and Wu et al. (2011) presented simplified models for gas/water flow in annuli by regarding the annuli as a pipe, 2

ACCEPTED MANUSCRIPT which was proved useful in oil field (Yang et al. 1999; Ni et al. 2005; Kaya et al. 2001; Orkiszewski 1967; Beggs et al. 1973; Hasan et al. 2007). In recent years, the study on superheated steam is a topic of current interest. Based on the PVT properties of superheated steam, Zhou et al. (2010) presented a numerical model for modeling of

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superheated steam flow in wellbores, which laid a basic foundation for later studies. However, they failed to take the effect of frictional work on temperature of fluid and formation into consideration (Sun et al. 2017b). Based on Zhou et al.’s work, Xu et al. (2013a, 2013b) presented improved models for analyzing flow behaviors of superheated steam in vertical wellbores. In their

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studies, detailed oil displacement mechanisms were discussed compared against saturated steam. Gu (2016) presented a model for superheated steam flow in the horizontal section of the wellbores.

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Their model made a modification on the energy balance equation, which laid a solid foundation for following researchers. Fan et al. (2016) studied the profiles of superheated steam in the vertical and horizontal sections of the wellbores with consideration of phase change. However, their energy balance equations in both the vertical and horizontal wellbores cannot be used under high injection speed conditions. Pang et al. (2016) studied the superheated steam flow behaviors in the

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ground pipelines, which also cannot be used under the high injection speed condition. When the injection rate is higher than 150 t/d, the deviation of their predicted temperatures were obviously smaller than tested data (Liu et al., 2018a,b). Besides, the relationship between the injection rate

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and fluid temperature in their model was not properly modeled. They made the conclusion that the higher the injection rate, the smaller the fluid temperature in wellbores, which may not be in line

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with reality. Based on these previous studies, Sun et al. (2017b) proposed a modified energy balance equation for superheated steam flow in the vertical wellbores. The reason why the temperature of superheated steam drops when the injection rate is high enough was discussed in detail in their work. Then, Sun et al. (2017c,d,e,f,g,h,i) built a whole set of models for superheated steam/superheated multi-component thermal fluid flow in offshore wells, concentric dual-tubing wells, parallel dual-tubing wells and ground pipelines, etc. However, this previous models were mainly focused on the vertical section of the wellbores (equal mass flow process) and horizontal wellbores with conventional heel-point injection method. Dong et al. (2014a, 2016) presented numerical models for multi-component thermal fluid flow in the horizontal section of the wellbores with heel-point and toe-point injection technique. However, 3

ACCEPTED MANUSCRIPT their models cannot be used under high injection speed conditions (Dong et al. 2014b; 2017). Sun et al (2017j) improved Dong et al.’s model by modifying the energy balance equation. Therefore, the predicted temperatures from Sun et al.’s model had a good agreement with field data. Wu et al. (2012), Li et al. (2015) and Chen et al. (2017) presented numerical models for simulating

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gas/water flow in horizontal wells with multi-point injection technique. However, they also failed to take the effect of frictional work on temperature of fluid and formation into consideration. Sun et al. (2018a) presented a numerical model for simulating superheated multi-component thermal fluid flow in horizontal wells with multi-point injection technique, which laid a solid foundation

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for following researchers. However, it is single phase flow condition of superheated steam in wellbores, and the thermophysical properties of superheated steam/superheated multi-component

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thermal fluid are function of both pressure and temperature, which varies from that of saturated steam. The motivation of this work is to build a two-phase flow model for water/steam flow in wellbores, which can be used to predict steam quality along the wellbores. In this paper, a new model is proposed to estimate the pressure and steam quality of saturated steam in horizontal wells with multi-point injection technique. The new model has mainly three

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contributions to the existing body of literature: (a) New energy balance equations were developed for saturated steam flow in IT and annuli based on the law of energy conservation. (b) the prediction accuracy of steam quality was improved. (c) Type curves of saturated steam flow in IT

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and annuli were obtained with straight forward method. The practical usefulness of this work can be summarized as two aspects: (a) practical engineers

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are now able to predict the distributions of key parameters (e.g. pressure, temperature and steam quality) of wet steam in the horizontal wellbores with multi-point injection technique. (b) Economically, uniform absorption of steam along the horizontal wellbores can be obtained by selecting of reasonable injection parameters with the help of this model. 2. Model development

2.1 General assumptions As mentioned above, when the horizontal section of the wellbores is very long or the reservoir is of serious heterogeneity, conventional single-tubing wells may lead to serious non-uniform of steam absorption along the interface between annuli and oil layer (Gu 2016; Sun et al. 2018a). Therefore, multi-point injection steam injection technique was proposed to overcome these 4

ACCEPTED MANUSCRIPT difficulties. The schematic of saturated steam flow in IT and annuli is shown ideally in the Figure 1 (Sun et al. 2018a,e).

Saturated steam flow

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θ

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Figure 1. Saturated steam flow in horizontal wells with multi-point injection technique.

Based on previous studies on superheated fluid flow in wellbores (Sun et al. 2018a,e), some

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basic assumptions are made below:

(a) the water and steam are moving together at the same velocity. (b) the flow regime is assumed to be steady-state.

(c) the flow parameters of saturated steam at heel-point of IT and annuli are constant. (d) Heat transfer in the horizontal direction is neglected.

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(e) Heat transfer rate from saturated steam in IT to oil layer is steady-state. (f) Heat transfer rate in oil layer is transient state. What is worth to mention is that this model is built only for saturated steam flow in wellbores.

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One may propose phase change can occur under the high velocity, long horizontal wellbore, or high geothermal temperature conditions. When the horizontal wellbore is long, wet steam may

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condense to hot water due to heat loss, or when the geothermal temperature is high, the wet steam may be heated to superheated state due to heat absorption. If these cases happen, previous models presented by Liu et al. (2018) and Sun et al. (2018a,e) are adopted to go on calculation. That is to say, this work only focuses on the two-phase flow of wet steam in the multi-point injection horizontal wellbores.

2.2 Modeling of saturated steam flow in IT The two-phase saturate steam flows from the heel-point to the toe-point in the inner tubing before it reaches the beginning of the annuli. There exists no mass exchange between the IT and annuli at this flow stage. Therefore, the mass flow rate in IT is kept as a constant during this flow process, as shown below: 5

ACCEPTED MANUSCRIPT dwIT 2 d ( ρ IT v IT ) = π rITi =0 dL dL

(1)

There are two independent variables in the wellbores (pressure and steam quality). Therefore, two independent equations are needed to obtain the distribution of pressure and steam quality along the wellbores. When saturated steam flows from the heel-point to toe-point in IT, there

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exists heat exchange between the IT and annuli. When the temperature of saturated steam in IT is higher than that in annuli, heat flows from IT to annuli, and when the temperature of saturated steam in IT is lower than that in annuli, heat flows from annuli to IT. The heat exchange rate per

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unit length of wellbores should be equal to the total energy change in IT. Based on the law of energy conservation, the energy balance equation can be expressed as:

(2)

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dQIT dhIT d  vIT2  = −wIT − wIT   + wIT g sin θ dL dL dL  2 

When saturated steam flows in IT, the fluid unit is subjected to pressure, gravity and friction. Based on the law of momentum conservation, the impulse of resultant force of these three forces should be equal to the change of momentum. Based on previous studies, the momentum balance equation of saturated steam flow in IT can be expressed as:

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d ( ρIT vIT2 ) fτ dpIT − ρIT g sin θ + 2 + =0 dL dL π rITi dL

(3)

2.3 Modeling of saturated steam flow in annuli

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Saturated steam flow in annuli can be divided into two types: one is injected from heel-point of annuli and the other is from toe-point of IT. For either of them, steam is constantly injected into oil

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layer due to pressure difference between annuli and oil layer (Sun et al. 2018a). Therefore, the mass flow rate of saturated steam decreases gradually in the course of flow. The mass balance equation of saturated flow in annuli can be given as:

wan = w0 − ∫ ρ an I an

(4)

When saturated steam flows in annuli, there exists heat exchange between IT and annuli. Besides, heat energy in annuli releases heat to the oil layer through heat conduction and mass transfer. The sum of heat losses in these three pathways should be equal to the total energy change of saturated steam in annuli. The energy balance equation can be given as:

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ACCEPTED MANUSCRIPT 2 2 d ( wan han ) d  wanvan  dQan dQIT I an ρan ( han + vr 2) − + =− −   + wan g sin θ dL dL dL dL dL  2 

(5)

When saturated steam flows in annuli, the fluid unit is subjected to pressure, gravity and friction (from wellbore and perforation). Based on the law of momentum conservation, the impulse of

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resultant force of these three forces should be equal to the change of momentum. The momentum balance equation of saturated steam flow in annuli can be expressed as:

d ( ρanvan2 ) dpan fτ ρanvan2 = ρan g sin θ − 2 − − f perf dL dL 4rwi π rwi dL

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3 Solving method of the mathematical model

(6)

Different from superheated steam, the temperature of saturated steam is the function of pressure.

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There are two independent variables in the wellbores (pressure and steam quality). Given the fact that there are two flowing spaces (IT and annuli), there are four independent variables (

pan , qIT

and

pIT ,

qan ) in the horizontal wellbores with multi-point injection technique. In this

paper, straight forward numerical method is adopted to obtain the distributions of pressure and

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steam quality in IT and annuli. In this paper, straight forward numerical method means to straightly calculate the pressure and temperature of wet steam from the heel-point to the toe-point (from one end of the wellbore to the other end). Firstly, the governing equations (the energy and

as shown below:

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momentum balance equations in IT and annuli) are expressed in the form of difference equations,

∆QIT ∆ hIT ∆  v IT2  = − wIT − wIT   + wIT g sin θ ∆L ∆L ∆L  2 

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∆ ( ρ IT v IT2 ) fτ ∆ p IT = ρ IT g sin θ − − ∆L ∆L π rITi2 dL

2 ρan ( han + vr2 2) ∆ ( wanhan ) ∆  wanvan  ∆Qan ∆QIT − + Ian =− −   + wan g sinθ ∆L ∆L ∆L ∆L ∆L  2  2 ∆ ( ρ an van ) − f ρ an van2 ∆ p an fτ = ρ an g sin θ − 2 − perf ∆L ∆L 4 rwi π rwi dL

(7)

(8)

(9)

(10)

As shown in Figure 1, the flow direction of saturated steam in IT and annuli may be co-current or counter-current. Therefore, the solution of the model is discussed in two cases. Wu et al. (2012) studied saturated steam flow in horizontal wells with multi-point injection technique. While their 7

ACCEPTED MANUSCRIPT model cannot be used under high injection speed conditions, it showed basic method in model solving. Sun et al. (2018a,e) studied superheated multi-component thermal fluid flow in horizontal wells with multi-point injection technique. Detailed model solving process was presented in their work. Besides, their model was inherited to help validate the new model presented in this paper,

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which will be discussed in detail later. Based on previous works (Wu et al. 2012; Sun et al.

①

②

The flow direction of saturated steam in IT and annuli is co-current

The flow direction of saturated steam in IT and annuli is counter-current

qIT ,inlet pIT ,inlet

IT

qan ,inlet pan ,inlet

Annuli

mth

···

ith

1th

qan,outlet pan ,outlet

qan ,outlet pan ,outlet

Annuli

qan ,inlet pan,inlet

qIT ,outlet pIT ,outlet

qIT ,outlet pIT ,outlet

IT

qIT ,inlet pIT ,inlet

qan ,outlet pan ,outlet

qan ,outlet pan ,outlet

Annuli

qan,inlet pan,inlet

Unknown numbers

Unknown numbers

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Known numbers

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Annuli

mth

Interface B

Interface A

qan,inlet pan ,inlet

···

···

ith

Interface C

··· 1th

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Divide the IT and annuli into small segments

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2018a,e), the relationship between flow direction and model solving are presented in Figure 2.

Known numbers Solving direction

Solving direction

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Figure 2. Co-current and counter-current flow of saturated steam in IT and annuli: basics for model solving with coupling method (Sun et al. 2018a,e).

For clarity, it is stated that (a) the interface A is at 0m, B is at 100m, and C is at 200m; (b) while there exists only one wellbore in the reservoir, this wellbore is divided into two part for calculation purpose according to the flow direction of fluid in wellbores, as shown in the lower portion of Fig. 2. Therefore, if saturated steam flows from interface A to interface B, the pressure and steam quality at outlet of each segment can be obtained by finding the zero points of the equations below:

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ACCEPTED MANUSCRIPT f (TIT ,outlet ) QIT ,outlet − QIT ,inlet

=

∆L

+ wIT

hIT , outlet − hIT ,inlet ∆L

2 2 1  vIT ,outlet − vIT ,inlet + wIT  ∆L  2

  − wIT g sin θ 

(11)

=0

pIT ,outlet − pIT ,inlet

=

∆L

fτ

− ρIT g sin θ +

π rITi2 dL

(ρ +

2 IT ,outlet IT ,outlet

v

∆L

=0

+

QIT ,outlet − QIT ,inlet

− ∆L ∆L wan ,outlet han ,outlet − wan ,inlet han ,inlet ∆L

+

pan , outlet − pan ,inlet

+

∆L fτ + π rwi2 dL

−

∆L

(13)

2 2 1  wan ,outlet van ,outlet − wan ,inlet van ,inlet  ∆L  2

f ( pan , outlet ) =

+ I an

ρ an ( han + vr2 2 )

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=

Qan ,outlet − Qan ,inlet

ρ an ,outlet + ρ an ,inlet 2

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=0

  − wan g sin θ = 0 

g sin θ

( ρ an,outlet van2 ,outlet − ρ an ,inlet van2 ,inlet ) ∆L

(12)

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f (Tan ,outlet )

− ρIT ,inlet vIT2 ,inlet )

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f ( pIT ,outlet )

+ f perf

ρ

2 an , outlet an , outlet

v

+ρ

(14) 2 an ,inlet an ,inlet

v

8rwi

If saturated steam flows from interface C to interface B, the pressure and steam quality at outlet of each segment can be obtained by finding the zero points of the equations below:

f (TIT ,outlet ) QIT ,outlet − QIT ,inlet ∆L

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=0

− wIT

hIT ,outlet − hIT ,inlet

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=

∆L

− wIT

f ( pIT ,outlet )

=

pIT ,outlet − pIT ,inlet ∆L

+ ρIT g sin θ −

τf π rITi2 dL

2 2 1  vIT ,outlet − vIT ,inlet  ∆L  2

(ρ −

2 IT ,outlet IT ,outlet

v

  + wIT g sin θ 

− ρIT ,inlet vIT2 ,inlet )

∆L

(15)

(16)

=0

f (Tan ,outlet ) = +

Qan ,outlet − Qan ,inlet

QIT ,outlet − QIT ,inlet

− ∆L ∆L wan ,outlet han ,outlet − wan ,inlet han ,inlet ∆L

+

+ I an

ρ an ( han + vr2 2 )

(17)

∆L

2 2 1  wan ,outlet van ,outlet − wan ,inlet van ,inlet  ∆L  2

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  − wan g sin θ = 0 

ACCEPTED MANUSCRIPT f ( pan,outlet ) = +

pan,outlet − pan,inlet ∆L

τf π rwi2 dL

(ρ +

−

ρan,outlet + ρan,inlet 2

2 an ,outlet an ,outlet

v

g sin θ

2 − ρan ,inlet van ,inlet )

∆L

+ f perf

ρan,outlet van2 ,outlet + ρan,inlet van2 ,inlet

(18)

8rwi

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=0 Based on previous study (Sun et al. 2018a,e), the detailed model solving process is shown as a

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flowchart in Figure 3.

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ACCEPTED MANUSCRIPT Start Divide the IT and annuli into small segments

Input initial conditions, such as pIT , qIT ,wIT , pan , qan ,wan the wellbore and formation parameters etc. Toe point

i=i+1

i=i+1

i=0

Guess the outlet saturated steam ' ' temperature qIT , outlet qan ,outlet

Guess the outlet saturated steam ' ' temperature qIT , outlet qan ,outlet

'

Calculate pIT , outlet , using Eq. (16)

'

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Calculate pIT ,outlet , using Eq. (12)

'

'

Calculate pan, outlet , using Eq. (18)

''

Calculate qIT , outlet , using Eq. (15)

Calculate pan ,outlet , using Eq. (14)

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''

Calculate qIT , outlet , using Eq. (11)

''

''

Calculate qan,outlet , using Eq. (13)

qIT' ,outlet − qIT'' ,outlet < ε 1 ' an ,outlet

q

−q

'' an ,outlet

< ε2

No

Calculate qan,outlet , using Eq. (17)

qIT' ,outlet − qIT'' ,outlet < ε 1

Yes

Yes

' '' qan ,outlet − qan ,outlet < ε 2

q

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qIT' ,outlet = qIT'' ,outlet

= qIT'' ,outlet

' an , outlet

'' = qan ,outlet

pIT ,outlet = pIT' ,outlet

pIT ,outlet = pIT' ,outlet

' pan ,outlet = pan ,outlet

' pan,outlet = pan ,outlet

qIT ,outlet = qIT'' ,outlet

qIT ,outlet = qIT'' ,outlet

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No

' IT ,outlet

q

' '' qan ,outlet = qan ,outlet

'' qan ,outlet = qan , outlet

'' qan ,outlet = qan ,outlet

Yes

i<=m-1

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Yes

i=0

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Heel point

i<=m-1 No

No

Output the pressure and steam quality distributions from interface A to interface B

Is the steam quality at interface B of the two calculations equal?

No

Yes Output the pressure and steam quality distributions from interface A to interface C Stop

Figure 3. Algorithm block diagram for solving saturated steam flow in IT and annuli (Sun et al. 2018a,e).

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ACCEPTED MANUSCRIPT 4 Results and discussions Based on the discussion above, in this section, the proposed model is validated by field data and previous models. The key thermodynamic parameters of the saturated steam were measured in the wellbore by using sensors. The authors obtained the data from the technical staff of CNPC. The

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injection pressure, temperature and steam quality at heel-point in IT are 3.8 MPa, 521.4 K and 0.85, respectively. The injection pressure, temperature and steam quality at heel-point in annuli are 4 MPa, 524.4 K and 0.8, respectively. The mass injection rate at heel-point in both IT and annuli is 3 kg/s. the basic parameters are shown in Table 1. The predicted results are shown in Figure 4.

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Table 1 Basic parameters used for calculation. Values

Parameters

Inner radius of IT/m

0.076

Thermal conductivity of IT/W/(m·K)

55.6

Outer radius of IT/m

0.0889

Thermal conductivity of casing/W/(m·K)

45.7

Inner radius of outer tubing/m

0.1016

Thermal conductivity of cement sheath/W/(m·K)

0.35

Outer radius of outer tubing/m

0.1143

Thermal conductivity of formation/W/(m·K)

1.73

Inner radius of casing/m

0.12426

Outer radius of casing/m

0.1397

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Parameters

2

Thermal diffusivity of formation/m /s

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Figure 4 shows the distributions of pressure and temperature of saturated steam in IT and annuli. It is observed from Figure 4(a) that: (a) Predicted results from both of the new model and previous models agrees with field data. However, this is the main reason why the errors in the old models

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are not discovered, which is discussed in detail later. (b) Saturated steam pressure decreases with distance to heel-point. This is because there exists shear stress in IT and annuli which leads to the

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decrease of pressure along the flow path. It is observed from Figure 4(b) that: (a) Predicted results from both of the new model and previous models agrees with field data. This is because the temperature of saturated steam is the function of pressure. As a result, the field pressures are obtained directly from the temperature values and the predicted temperatures agree with field data. However, this is the main reason why the errors in the old models are not discovered, which will be discussed in detail later in the paper. What is worth to stress is that the heat conduction rate of dQIT dL and dQan dL are extremely important in matching the predicted results with field data. When these two parameters (the detailed calculation equations can be found in Appendix B and F) are measured accurately by 12

Values

0.0000007

ACCEPTED MANUSCRIPT oil companies, the model’s predicted results can be fitted with field data. 4.2

3.8 3.6 3.4 3.2 50 100 150 Distance to heel-point, m (a) IT (This model) Annuli (This model) Annuli (This model) Field data IT (Li et al. 2015 & Chen et al. 2017) Annuli (Li et al. 2015 & Chen et al. 2017)

200

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0

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Pressure, MPa

4.0

524

520

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Temperature, K

528

516

512

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EP

0

50

100 150 200 Distance to heel-point, m (b) IT (This model) Annuli (This model) Annuli (This model) Field data IT (Li et al. 2015 & Chen et al. 2017) Annuli (Li et al. 2015 & Chen et al. 2017)

Figure 4. Validation of the new model by field data and previous models

As mentioned above, Li et al. (2015) and Chen et al. (2017) proposed numerical models for simulating saturated steam flow in multi-point injection horizontal wells. However, their model cannot be used under high injection speed conditions (Satter et al., 1965; Willhite 1967; Pacheco et al. 1972; Farouq et al. 1981; Durrant et al., 1986; Alves 1992; Livescu et al., 2010; Bahonar et al. 2011; Gu 2016; Sun et al. 2017b, 2017d). If their modeling method of energy balance equations are adopted in our case, the energy balance equations can be given as:

13

ACCEPTED MANUSCRIPT 2 2 d ( wan han ) d  wanvan  dQan dQIT dW Ian ρan ( han + vr 2) − + + =− −   + wan g sin θ dL dL dL dL dL dL  2 

(19)

Where, dW/dL denotes the frictional work per unit length of the wellbores, J/(s·m). The main difference between the new model and previous models (Li et al. 2015; Chen et al.

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2017) are focused on the item of dW/dL. In fact, this item should not appear in this equation. What is to stress is that removing the item of dW/dL from the equation does not mean we do not take the frictional heating into consideration. On the contrary, removing the item of dW/dL from the equation is the right way to take the frictional heating into consideration. This is because the

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appearance of dW/dL in the equation causes a smaller value of the calculated fluid temperature. According to Eq. (B-1) and (F-1), the temperature difference between fluids or fluid and formation

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decreases with decreasing of fluid temperature. As a result, the formation temperature decreases with decreasing of fluid temperature due to the decrease of heat conduction rate. One can find that both of the fluid temperature and formation temperature decrease with increasing of frictional heating. Further, one can find the link of frictional work and the mass flow rate in wellbores: a high value of frictional work corresponds to a high flow rate. That is to say, the higher the

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injection rate in wellbores, the lower the fluid temperature in wellbores. However, when the item of frictional work is removed from the equation, one can find that a high value of mass flow rate corresponds to a high value of fluid temperature in wellbores as well as a high value of formation

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temperature. In other word, the effect of frictional heating on fluid/formation temperature is in fact taken into consideration by removing the item of dW/dL from the energy balance equation.

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Figure 5 shows the comparison of the predicted steam quality from the new model with field data and previous models. It is observed that there exists a good agreement between the predicted results and field data. However, the predicted steam quality from previous models are smaller than the new model, as well as field data. This is because the frictional work in previous models got lost and was covered by the good temperature results, as discussed above. In fact, steam quality represents the enthalpy level under a certain injection pressure condition. The smaller the steam quality, the lower the steam enthalpy. Furthermore, the predictions of superheated fluid (a mixture of superheated steam and non-condensing gases) were added for further understanding (Sun et al. 2018a,e). This is because the temperature of superheated fluid depends on its pressure. As a result, if there is a decrease in fluid enthalpy, the temperature of superheated fluid will decrease 14

ACCEPTED MANUSCRIPT accordingly. The temperature predictions of superheated fluid are shown in Figure 6. 0.860

0.830 0.815 0.800 0.785 0.770 50 100 150 Distance to heel-point, m IT (This model) Annuli (This model) Annuli (This model)

200

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0

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Steam quality, K

0.845

M AN U

IT (Li et al. 2015 & Chen et al. 2017)

Annuli (Li et al. 2015 & Chen et al. 2017) Annuli (Li et al. 2015 & Chen et al. 2017) Field data

Figure 5. Comparison of the predicted values in steam quality against field data and previous models.

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It is observed from Figure 6 that if dW dL appears in the energy balance equation (the red dashed line), the calculated temperatures of superheated fluid in both IT and annuli are smaller than field data. What is worth to mention is that Li et al. (2015) and Chen et al. (2017) did not

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study the superheated case. We just adopted their modeling method of the energy balance

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equations to extend their work to the superheated case.

15

ACCEPTED MANUSCRIPT 655

605

580

555

530

505 0

50

100

150

Distance to heel-point, m IT (Sun et al. 2018) Annuli (Sun et al. 2018) Field data

200

SC

Annuli (Sun et al. 2018)

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Temperature, K

630

IT (Li et al., 2015 & Chen et al., 2017)

Annuli (Li et al., 2015 & Chen et al., 2017)

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Annuli (Li et al., 2015 & Chen et al., 2017)

Figure 6. Comparison of the predicted temperatures of superheated fluid from different models (Sun et al., 2018a,e; Li et al., 2015; Chen et al., 2017). Based on the discussion above, steam enthalpy values are discussed. The predicted results are shown in Figure 7. It is observed from Figure 7 that: (a) the predicted results from Li et al. and

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Chen et al.’s models are lower than the new model’s results. This is because their energy balance equation cannot be used under high injection speed conditions. The calculated total energy losses from their models are larger than the new model. Therefore, predicted steam enthalpy from their

EP

models are smaller than the new model. (b) When the saturated steam flows from interface A to B in IT, steam enthalpy decreases with distance to heel-point. However, the slope of the curve

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increases with distance to heel-point. This is because from interface A to B, the steam temperature in IT is lower than that in annuli, as shown in Figure 4(b). As a result, saturated steam absorbs heat from steam in annuli from interface A to B, which causes the increase in enthalpy gradient. However, when saturated steam flows from interface B to C in IT, it constantly releases heat to steam in annuli. This is because the saturated steam temperature in IT is higher than that in annuli, as shown in Figure 4(b).

16

ACCEPTED MANUSCRIPT 2.58E+06

Enthalpy, 106J/kg

2.55E+06 2.52E+06 2.49E+06 2.46E+06

2.40E+06 0

50

100 150 Distance to heel-point, m

200

SC

IT (This model) Annuli (This model) Annuli (This model) IT (Li et al. 2015 & Chen et al. 2017) Annuli (Li et al. 2015 & Chen et al. 2017) Annuli (Li et al. 2015 & Chen et al. 2017)

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2.43E+06

Figure 7. Comparison of the predicted steam enthalpy from different models.

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As mentioned above, when the horizontal wellbore is very long or the reservoir is of serious heterogeneity, strong uneven steam absorption phenomenon may happen. Therefore, steam injection at heel and toe-point simultaneously or alternately will relieve the uneven steam absorption phenomenon. Figure 8 shows the formation heat absorption rate along the horizontal wellbores. It is observed that with the help of multi-point injection technique, formation heating

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effect at both heel-point and toe-point are improved. Besides, the injection parameters in IT and annuli can be adjusted independently. As a result, the optimized formation steam absorption rate at heel-point or toe-point can be selected based on the actual physical properties of rocks.

1.24E+05 1.20E+05

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Heat absorption rate, 106J/s

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1.28E+05

1.16E+05 Annuli

1.12E+05

Annuli

1.08E+05 0

50

100 150 Distance to heel-point, m

200

Figure 8. Improvement of formation heating effect during steam injection process with the multi-point injection technique. Figure 9 shows the mass flow rate of saturated steam in IT and annuli. It is observed that: (a) the mass flow rate of saturated steam in IT is kept unchanged from interface A to C. This is 17

ACCEPTED MANUSCRIPT because there exists no mass exchange between IT and annuli (Sun et al. 2018a,e). (b) When saturated steam flows from interface A to B in annuli, the mass flow rate decreases with distance to heel-point. This is because the saturated steam is constantly injected into oil layer due to pressure drop from annuli to oil layer. That is to say, if conventional heel-point injection method is

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adopted, the mass flow rate near toe-point of wellbores will to the smallest. Therefore, the production of heavy oil near toe-point will be poor. However, with the help of multi-point injection technique, the mass flow rate at toe-point can be high only if a higher injection rate in IT is selected.

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2.8

2.5

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Mass flow rate, kg/s

3.1

IT

2.2

Annuli Annuli

1.9 50

100 150 Distance to heel-point, m

200

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0

Figure 9. Mass flow rate of saturate steam in IT and annuli.

5 Conclusions

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Previous studies showed perfect match in temperature values with field data. However, their models showed relatively poor accuracy in steam quality and enthalpy. This paper presented a new

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model for simulating saturated steam flow in horizontal wellbores with multi-point injection technique. New energy conservation equations were developed for saturated steam flow in IT and annuli. The calculation accuracy of steam quality and enthalpy of saturated steam were greatly improved. Some meaningful conclusions are summarized below: (a) The effect of heat loss on saturated steam temperature is weak. However, the heat loss has an obvious influence on steam quality, which represents the heat carrying capacity of saturated steam. (b) When removing the item of frictional work from the energy balance equation, the predicted values of steam quality and enthalpy are smaller than actual values. This error in energy conservation equation may be hidden when the predicted temperatures showed good agreement with field data. 18

ACCEPTED MANUSCRIPT (c) With the help of multi-point injection technique, formation heating effect at both heel-point and toe-point can be improved. Besides, the injection parameters in IT and annuli can be adjusted independently to control the steam absorption rate.

Bo

the volume factor of oil, m3/m3

Bw

the volume factor of water, m3/m3

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Nomenclature

the shear force of saturated steam flow in IT, N

f perf

the friction factor of perforation roughness, dimensionless

f wi

the convective heat transfer coefficient of casing, W/(m2·K)

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fτ

the gravitational acceleration, g = 9.81 m/s2

hIT

the enthalpy of saturated steam in IT, J/kg

h fITi

the forced convection heat transfer coefficient on inside wall of the IT, W/(m2·K)

h fITo

the forced convection heat transfer coefficient on outside wall of the IT, W/(m2·K)

hIT ,an

the enthalpy of the wet steam including the steam phase and the liquid phase, kJ/kg

hw

the enthalpy of the liquid phase, kJ/kg

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g

the volume flow velocity of saturated steam from annuli to oil layer, m3/s

Ir

the ratio between injection rate and production rate, dimensionless

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EP

I an

J an

the injection index, m3/(s·Pa)

J0

the first kind Bessel functions of zero orders

J1

the first kind Bessel functions of the first orders

Kh

the permeability in the horizontal direction of the oil layer, m2

Kv

the permeability in the vertical direction of the oil layer, m2

Kro

the relative permeability of oil phase, dimensionless

19

ACCEPTED MANUSCRIPT

the length of the horizontal wellbore, m

Lv

the latent heat of vaporization, kJ/kg

n perf

the perforating density, m-1

pIT

the saturated steam pressure in IT, Pa

pan

the saturated steam pressure in annuli, Pa

pr

the pressure in the oil layer, Pa

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L

the relative permeability of water phase, dimensionless

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Krw

the heat exchange rate between IT and annuli, J/s

Qan

the heat loss rate from annuli to oil layer, J/s

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QIT

the inside radius of the IT, m

rITi

the inside radius of the IT, m

rwi

the inside radius of casing, m

rITo rw o

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rITi

the outer radius of IT, m

the outer radius of the casing, m

the outer radius of the cement sheath, m

rph

the radius of the perforation hole, m

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EP

rcemo

T

the temperature under a certain pressure condition, K

TIT

the temperature of the wet steam in IT, K

Tan

the temperature of the wet steam in annuli, K

Th

the temperature at the interface between cement sheath and formation, K

Tei

the formation temperature, K

U ITo

the heat transfer coefficient, W/(m2·K)

U wo

the heat transfer coefficient, W/(m2·K) 20

vIT

the flow velocity of saturated steam in IT, m/s

vr

the radial steam injection rate, m/s

van

the flow velocity of saturated steam in annuli, m/s

wIT

the mass flow rate of saturated steam in IT, kg/s

wan

the mass flow rate of saturated steam in annuli, kg/s

w0

the initial mass flow rate in annuli, kg/s the steam quality, dimensionless

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x

the second kind Bessel functions of zero order

Y1

the second kind Bessel functions of the first order

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Y0

Greek alphabet

ρIT

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ACCEPTED MANUSCRIPT

the density of saturated steam, kg/m3 the density of saturated steam, kg/m3

ρ IT ,an

the density of the two-phase steam, kg/m3

ρw

the density of the liquid phase, kg/m3

ρg

the density of steam phase, kg/m3

θ

the well angle from horizontal, rad

λIT

the thermal conductivity of IT, W/(m·K)

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ρIT

λcas

the thermal conductivity of casing, W/(m·K)

λcem

the thermal conductivity of the cement sheath, W/(m·K)

λe

the thermal conductivity of oil layer, W/(m·K)

µws

the viscosity of the wet steam, mPa·s

µs

the viscosity of the steam phase, mPa·s

µw

the viscosity of the liquid phase, mPa·s

21

ACCEPTED MANUSCRIPT ω

the ratio between heat capacity between formation and wellbore, dimensionless

τD

the dimensionless time ( τ D = α r t rcem ) 2

Acknowledgements The authors wish to thank the State Key Laboratory of efficient development of offshore oil

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(2015-YXKJ-001). This work was also supported in part by a grant from National Science and Technology Major Projects of China (2016ZX05039 and 2016ZX05042) and the National Natural fund of China (51490654). The authors recognize the support of the China University of Petroleum (Beijing) for the permission to publish this paper.

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Appendix A. Density of saturated steam.

During the solving process of the above model, equations for steam density calculation are needed.

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In this paper, accurate empirical formulas are adopted for the iterative calculation. The expressions for steam density are expressed below (Gu 2016; Sun et al., 2018b,c):

ρIT ,an

 1− x x  = +   ρw ρs 

−1

ρw = 3786.31 − 37.2487 × T + 0.196246 × T 2 − 5.04708 × 0.0001× T 3

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+6.29368 × 0.0000001× T 4 − 3.0848*0.0000000001× T 5 ρ g = e −93.7072 + 0.833941×T − 0.00320809×T

2

+ 6.57652*0.000001×T 3 − 6.93747 ×10 −9 ×T 4 + 2.97203×10 −12 ×T 5

(A-1)

(A-2) (A-3)

Appendix B. Heat exchange between IT and annuli.

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When steam flows in the inner tubing or annuli, there exists heat exchange due to temperature drop from IT to annuli. This physic is modeled by the equations below (Fontanilla et al. 1982; Gu

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2016; Sun et al. 2018a,b,c,d,e,f):

dQ IT = 2π rIToU ITo (TIT − Tan ) dL

U ITo

r r r 1  =  ITo ln ITo + ITo +   λIT riITi h fITi rITi h fITo 

(B-1) −1

(B-2)

Appendix C. Enthalpy of saturated steam. During the solving process of the above model, equations for steam enthalpy calculation are needed. In this paper, accurate empirical formulas are adopted for the iterative calculation. The expressions for steam enthalpy are expressed below (Gu 2016):

hIT , an = xLv + hw 22

(C-1)

ACCEPTED MANUSCRIPT Lv =

7184500 + 11048.6 × T − 88.405 × T 2 + 0.162561 × T 3 − 1.21377 × 10 −4 × T 4 (C-2)

hw = 23665.2 − 366.232 × T + 2.26952 × T 2 − 0.00730365 × T 3

(C-3)

+1.30241×10−5 × T 4 − 1.22103 × 10−8 × T 5 + 4.70878 × 10−12 × T 6 Appendix D. Shear force in IT and annuli.

as (Yuan et al., 1982; Gu 2016; Sun et al., 2018a,b,c,d,e,f):

1 fτ = π rITi f wall ρ IT vIT2 dL 4

(D-1)

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The f wall is calculated based on the Table D-1 below:

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According to the two-phase flow theory in the fluid mechanics, the shear force can be expressed

Table D-1. Calculation equations for shear force based on flow pattern (Yuan et al., 1982; Gu 2016; Sun et al., 2018a,b,c,d,e,f). Reynolds number

laminar flow

Rei ≤ 2000

Transition flow

2000 < Rei ≤ 3000

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Flow pattern

8

8

59.7 ε 7 < Rei ≤

Mixed friction

Rei >

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Hydraulic roughness

64 Rei

665 − 765lg ε

ε

665 − 765lg ε

ε

f wall =

f wall = {−1.8lg[

6.9 ε 1.11 −2 +( ) ]} Rei 3.7 Dci

f wall = (2lg

3.7 Dci

ε

)−2

needed. In this paper, accurate empirical formulas are adopted for the iterative calculation. The expressions for steam viscosity are expressed below (Gu 2016):

µw = −12.3274 +

µ ws = xµ s + (1 − x ) µ w

(D-2)

2.71038 ×104 2.35275 ×107 1.01425 ×1010 2.17342 ×1012 2.35275 ×1014 − + − + T T2 T3 T4 T5 (D-3)

µs = −0.546807 + 6.8949 ×10−3T − 3.39999 ×10−5 T 2 + 8.29842 ×10−8 T 3

23

−2

0.3164 4 Re i

During the solving process of the above model, equations for steam viscosity calculation are

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nt flow

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turbule

f wall =

−0.9 f wall = 1.14 − 2lg ( ∆ + 21.25N Re )

3000 < Rei ≤ 59.7 ε 7

Hydraulic smooth

Moody friction coefficient

(D-4)

ACCEPTED MANUSCRIPT Appendix E. Volume absorption rate of saturated steam from annuli to oil layer. The injection process from annuli to oil layer is expressed as (Williams et al. 1980; Chen et al. 2007; Gu 2016; Sun et al., 2018a,b,c,d,e,f):

I an = J an I r ( pan − pr )

(E-1)

2π K h K v K v dL ( K ro Bo µ o + K rw Bw µ w )

J an = β

2ln Ad rwi2 − 3.86 Ir = ln Ad rwi2 − 2.71

(E-3)

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Appendix F. Wellbore heat loss rate.

(E-2)

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0.571 Ad ln + S − 0.75 rwi

The energy balance equation is composed of heat conduction and energy change of fluid in

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wellbores. The heat conduction rate from annuli to oil layer can be expressed as (Liu 2013; Gu 2016; Sun et al. 2017b, 2018a,g,h,i,j,k,l,m,n):

dQan = 2π rwoU wo (Tan − Th ) dL

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U wo

 r r r r r  =  wo + wo ln wo + wo ln cemo  rwo   f wi rwi λcas rwi λcem

(F-1)

−1

(F-2)

Equation (F-2) can be reduced to (Liu 2013; Gu 2016; Sun et al. 2017b, 2018a,g, h,i,j,k,l,m,n):

U wo

 r r  =  wo ln cemo  rwo   λcem

−1

(F-3)

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Another useful model proposed by Hasan et al. (1995) was expressed as:

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dQan T −T = 2πλe h ei dz f (t )

(F-4)

f ( t ) is the function of injection time, which can be expressed as (Cheng et al. 2012): f (t ) =

16ω 2

π2

∫

∞

1 − exp ( −τ D u 2 ) u 3∆ ( u, ω )

0

(F-5)

du

∆ ( u, ω ) can be expressed as (Cheng et al. 2012): ∆ ( u , ω ) =  uJ 0 ( u ) − ω J1 ( u )  +  uY0 ( u ) − ωY1 ( u )  2

f ( t ) can be reduced to:

24

2

(F-6)

ACCEPTED MANUSCRIPT

(

)

f ( t ) = ln 2 τ D −

0.5772 1 + 2 4τ D

  1  1 +  1 − ω  ln ( 4τ D ) + 0.5772     

(F-7)

Appendix G. Perforation Moody friction coefficient During the calculation process of friction loss caused by perforation, the equations are shown

f perf A=

= 2.5 ln( 8 f perf

f perf

Rei 2

8

− 2.5 ln(

B = 7(

f perf

Rei 2

8

rph n perf )( ) rwi 12

) + 3.75

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References

) + A − B − 3.75

SC

8

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below (Su et al. 1993, 1994; Gu 2016): (G-1)

(G-2)

(G-3)

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EP

TE D

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Zheng JL, Zhu YH, Zhu MQ, Wu HT, Sun RC. Bio-oil gasification using air - Steam as gasifying agents in an entrained flow gasifier. Energy, Volume 142, 1 January 2018, Pages 426-435.

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ACCEPTED MANUSCRIPT A two-phase model was proposed for steam flow in IT and annuli. Calculation accuracy in steam quality and enthalpy was improved. Multi-point injection technique does benefit to uniform steam absorption.

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The effect of frictional heating on saturated steam temperature is weak.