Thermophysical properties estimation and performance analysis of superheated-steam injection in horizontal wells considering phase change

Energy Conversion and Management 99 (2015) 119–131

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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 b

Department of Petroleum Engineering, China University of Petroleum, Beijing, 18 Fuxue Road, Changping, Beijing 102249, China 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

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

1. Introduction Thermal recovery methods [1], such as CSS (cyclic steam stimulation), steamflooding and SAGD (steam-assisted gravity drainage) [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 steam into pay zones can release a large amount of latent heat and sensible 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 characterized by high steam quality, high temperature and low pressure [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 ⇑ Corresponding author. Tel.: +86 10 89733726. E-mail address: [email protected] (H. Gu). http://dx.doi.org/10.1016/j.enconman.2015.04.029 0196-8904/Ó 2015 Elsevier Ltd. All rights reserved.

improve flow environment in porous media [4] and promote aquathermolysis of heavy oils [5]. At present, cyclic superheatedsteam 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 productive than a vertical well due to its larger reservoir contact area. As superheated steam flows along a horizontal wellbore, its thermophysical properties, such as mass flow rate and the degree of superheat, always change with horizontal well length, more importantly, superheated steam may undergo phase change and be cooled to wet steam 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 wellbore/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 heatconduction time function, which was further improved by

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Nomenclature Ac Ad B dp=dL Dci f f ci f ðtÞ g h H HL I Ir J0 J1 J pi K Kr L DL Mr nperf N p p Dp Qc Q in Q rad;i Q out r ci r co rh r ph rw Rei s Sw Swi t T T deg T ei

cross-sectional area of casing, m2 drainage area, m2 volume factor, m3/m3 pressure drop gradient, Pa/m inside diameter of casing, m friction factor, dimensionless forced-convection heat transfer coefficient on inside of casing, W/(m2 K) transient heat-conduction time function, dimensionless gravitational acceleration, m/s2 specific enthalpy, J/kg thickness of oil layer, m liquid holdup, dimensionless volumetric outflow rate of fluid injected into the formation, m3/s injectivity ratio, dimensionless first kind Bessel functions of zero order first kind Bessel functions of first order productivity index, m3/(s Pa) permeability, lm2 relative permeability, dimensionless horizontal well length, m length of perforation unit, m volumetric heat capacity of pay zone, J/(m3 K) perforation density, m1 total number of perforations or perforation units pressure, Pa average pressure, Pa pressure drop, Pa heat conduction rate, W energy carried by hot fluid at the inlet, W energy transferred to the formation due to radial outflow, W energy carried by hot fluid at the outlet, W inside radius of casing, m outside radius of casing, m heated radius, m radius of perforation hole, m radius of horizontal wellbore, m Reynolds number, dimensionless skin factor, dimensionless average water saturation, dimensionless initial water saturation, dimensionless injection time, s temperature, K degree of superheat, K initial temperature of the formation, K

Cheng et al. [8] who considered the effect of wellbore heat capacity on 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] suggested 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 thermophysical 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 was corrected by Wang et al. [13]. Dong et al. [14] created a predictive

T interf T DT u U co

Du=u

m mr msg w x Dx Y0 Y1

cement/formation interface temperature, K average fluid temperature, K temperature drop, K dummy variable for integration, dimensionless over-all heat transfer coefficient between fluid and cement/formation interface, W/(m2 K) roughness function velocity, m/s radial velocity, m/s superficial gas velocity, m/s mass flow rate, kg/s steam quality, dimensionless steam quality drop, dimensionless the second kind Bessel functions of zero order the second kind Bessel functions of first order

Greek letters a thermal diffusivity of formation, m2/h b unit conversion factor, dimensionless e roughness of casing wall, m h well angle from horizontal kcas 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 s q density, kg/m3 sD dimensionless time x ratio of the formation heat capacity to the wellbore heat capacity, dimensionless / porosity of oil layer, dimensionless

Subscripts acc acceleration h horizontal m mixture ns no-slip o oil perf perforation roughness pot potential energy r reservoir s dry steam superh superheated steam v vertical w saturated water i; j; k index

model aimed at thermophysical properties of multi-thermal fluid in 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 experiments 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], concentric 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

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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 wellbore 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 temperature 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 pressure; 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 comprehensive 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 field data, the effects of some important factors on the profiles of thermophysical properties are analyzed in detail. 2. Mathematical model Fig. 1 shows a simplified schematic of superheated-steam injection in a horizontal well. In this paper, to simplify the model calculation, several major assumptions are made:

(1) The injection conditions, including injection pressure, temperature 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 wellbore to the cement sheath is steady-state, while heat conduction 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 to be wi1 ; pi1 and T i1 , respectively, and the corresponding values at the outlet are wi ; pi and T i , respectively, also, the volumetric outflow rate of hot fluid injected into the formation through the perforation is assumed to be Ii . 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 

i X

ðqj Ij Þ;

16i6N

ð1Þ

j¼1

where w0 denotes the mass flow rate of superheated steam at the heel-position of the horizontal well; qj represents the average density 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

Overburden Perforation

Toe

Oil layer

rci

rco

Heel

Casing

Horizontal well

Cement

Formation

Underburden

(a) Schematic offluid flow in a horizontal injection well.

(b) Structure of a horizontal wellbore.

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

rw

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

qsuperh ¼ qðp; TÞ ¼

25:822345p ; 0:019266ðT  273:15Þ  0:203202p þ 1:017353

4 MPa 6 p 6 9 MPa

qw ¼ 3786:31  37:2487T þ 0:196246T 2  5:04708  104 T 3 þ 6:29368  107 T 4

ð2Þ

 3:08480  1010 T 5 ; 273:15 K 6 T 6 640 K

ð6Þ

16:251218p qsuperh ¼ qðp; TÞ ¼ ; 0:017255ðT  273:15Þ  0:151339p  0:904339 9 MPa < p 6 14 MPa ð3Þ

Based on the above discussion, the expression of qj in Eq. (1) can be summarized as follows:

where p; T and qsuperh are the pressure, temperature and density of superheated steam, respectively. Table 1 shows the maximum absolute residual (Max AR), mean absolute residual (MAR) and the maximum 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/water mixture fluid. Usually, it is defined as

qj ¼

qm ¼ qðHL ; TÞ ¼ qs ð1  HL Þ þ qw HL

ð4Þ

where qm represents the density of mixture fluid; HL is the liquid holdup, which can be calculated by using the classic method presented 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 calculated by empirical correlations, and in the computer procedure, they are given as follows [25]:

(

ð7Þ

where pj and T j are the average pressure and average temperature of hot fluid in the j-th perforation unit, respectively, pj ¼ ðpj1 þ pj Þ=2, T j ¼ ðT j1 þ T j Þ=2. In addition, in Eq. (1), the volumetric outflow rate of hot fluid injected into the formation, Ij , can be estimated by [12]

Ij ¼ J pi;j Ir;j ðpj  pr Þ

ð8Þ

where pr is the average reservoir pressure, which can be calculated with the method suggested by Chen [26]; J pi;j and Ir;j are the productivity index and the injectivity ratio for the j-th perforation unit, respectively, which can be calculated from [12,27]

J pi;j ¼ b

2p

qffiffiffiffi

ln



Kh K DL BKo rol Kv v o

0:571

pffiffiffiffiffi Ad;j

rw

þ BKwrw l

w

 ð9Þ

þ s  0:75

A

Ir;j ¼

ln qs ¼ 93:7072 þ 0:833941T  0:00320809T 2

qsuperh;j ¼ qðpj ; T j Þ; superheated steam qm;j ¼ qðHL;j ; T j Þ; wet steam

2 ln rd;j 2  3:86 w

A

ln rd;j 2  2:71

ð10Þ

w

þ 6:57652  106 T 3  6:93747  109 T 4 þ 2:97203  1012 T 5 273:15 K 6 T 6 645 K

ð5Þ

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) Eq. (3)

1.224 3.532

0.301 0.778

4.204 4.228

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 and l are the relative permeability, the volume factor and the viscosity, respectively, and the subscripts 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 determines K ro and K rw

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Sw ¼ Swi þ

w0 t

qhw Ad H/

ð11Þ

where Sw and Swi 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

dpt;i dppot;i dpwall;i dpperf;i dpacc;i ¼    dL dL dL dL dL

ð12Þ

where dpt;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), dppot;i =dL, represents the pressure drop due to potential energy change, which can be written as

dppot;i ¼ qi g sin h dL

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 dppot;i =dL is equal to zero. The second term of the right side in Eq. (12), dpwall;i =dL, represents the pressure drop due to casing wall friction and a general expression for it is

dpwall;i q0 mi 2 ¼ f wall;i i dL Dci 2

ð14Þ

where Dci denotes the inside diameter of casing; mi is the average velocity of fluid in the i-th perforation unit, which can be estimated by

mi1 þ mi

Table 2 Empirical correlations for estimating sing-phase friction factor of pipe flow [29]. Flow pattern Laminar flow Critical region Turbulent flow Smooth pipe Transition region Rough pipe

Rei Rei 6 2000

f wall;i 64 f wall;i ¼ Re i

2000 < Rei 6 3000

–

8

3000 < Rei 6 59:7=e7 8 7

59:7=e < Rei 6 Rei >

665765 lg e

e

665765 lg e

e

p ffiffiffiffiffi f wall;i ¼ 0:3164 4 Rei   1:11  2 e f wall;i ¼ 1:8lg 6:9 Rei þ 3:7Dci  2 ci f wall;i ¼ 2lg 3:7D e

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 side in Eq. (12), dpperf;i =dL, represents the pressure drop due to perforation roughness, which can be calculated as

dpperf;i q0 mi 2 ¼ f perf;i i dL Dci 2

ð17Þ

where the friction factor of perforation roughness, f perf;i , is determined by the following implicit equation [15,16]

0 sffiffiffiffiffiffiffiffiffiffiffi1 sffiffiffiffiffiffiffiffiffiffiffi 8 Rei f perf;i A Du ¼ 2:5 ln @ þ B    3:75 f perf;i u 2 8

ð18Þ

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

0 sffiffiffiffiffiffiffiffiffiffiffi1 sffiffiffiffiffiffiffiffiffiffiffi 8 Rei f wall;i A B¼ þ 3:75  2:5 ln @ f wall;i 2 8   2rph nperf  Du ¼ 7:0 u Dci 12

ð19Þ

ð20Þ

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 Ac Þ (k ¼ i  1 or

where r ph is the radius of perforation hole. The fourth term of the right side in Eq. (12), dpacc;i =dL, represents the pressure drop due to acceleration. For superheated-steam single-phase flow, dpacc;i =dL can be expressed as

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

dpacc;i qi m2i  qi1 m2i1 ¼ dL DL

mi ¼

2

(

q0i ¼

qsuperh;i ¼ qðpi ; T i Þ ; superheated steam qns;i ; wet steam

ð15Þ

ð16Þ

where qns;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 correlations 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 empirical 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 method proposed by Beggs and Brill [23]. In their study, the two-phase

ð21Þ

While for steam/water two-phase flow, dpacc;i =dL can be calculated by

qm;i mm;i msg;i dpt;i dpacc;i dmm;i ¼ qm;i mm;i ¼ dL dL 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,

qsuperh;i mi 2 dpt;i ¼ qsuperh;i g sin h  ðf wall;i þ f perf;i Þ dL Dci 2 

qi m2i  qi1 m2i1 DL

ð23Þ

However, for steam/water two-phase flow, dpt;i =dL can be obtained by substituting Eqs. (13), (14), (16), (17) and (22) into Eq. (12),

124

dpt;i qm;i g sin h  ðf wall;i þ f perf;i Þ ¼ dL 1  qm;i mm;i msg;i =pt;i

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

qns;i mi 2 Dci

2

Q rad;i þ Q c;i ¼ Q in;i  Q out;i

ð25Þ

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 surroundings due to heat conduction. Applying energy balance to the i-th perforation unit yields

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 Ii qi hi þ

m2r;i

!

2

Fig. 3. Calculation flowchart for the mathematical model.

ð26Þ

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

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 Ii =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,

  m2 Q in;i ¼ wi1 hi1 þ i1 2 Q out;i

ð27Þ

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 thermophysical parameter. In the following, how to obtain the governing 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

  m2 ¼ wi hi þ i 2

ð28Þ

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 hðp; TÞ ¼ 2588:296398 þ 3:670906  ðT  273:15Þ 186852:258072 3835:954647  38:997194p þ T  273:15 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 ) Thickness of oil layer (H) Initial reservoir pressure (pr;i ) Initial oil saturation (Soi ) Porosity of oil layer (/) Initial formation temperature (T ei ) Oil viscosity at T ei (lo ) Volume factor of oil (Bo ) Volume factor of water (Bw ) Drainage area for the horizontal well (Ad )

m m MPa – – K mPa s m3/m3 m3/m3 m2

287 15 2.38 0.75 0.32 291.29 1366 1.05 1.01 29,700

Length of the horizontal wellbore (L) Perforation density (nperf ) Radius of perforation hole (r ph ) Inside radius of casing (r ci ) Outside radius of casing (rco ) Radius of wellbore (r w ) Roughness of casing (e) Thermal conductivity of the cement (kcem ) Thermal conductivity of the formation (ke ) Thermal diffusivity of the formation (a)

m m1 m m m m m W /(m K) W /(m K) m2/h

195.3 12 0.0075 0.0807 0.0889 0.12 0.0000457 0.933 1.73 0.00037

1.2

630 Measured field data

1.0

Simulated results 610

Steam quality

Fluid temperature (K)

620

600 590

0.8

0.6 Measured field data Simulated results

0.4

580

0.2

570 0

50

100

150

Horizontal well length (m)

(a)

200

0

50

100

150

Horizontal well length (m)

(b)

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

200

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

m2r;i

!

2pr co U co ke DL þ ðT i  T ei Þ r co U co f ðtÞ þ ke 2     m2 m2 ¼ wi1 hi1 þ i1  wi hi þ i 2 2

Ii qi hi þ

ð30Þ

Table 4 Basic parameters used for the performance analysis of superheated-steam injection in horizontal wells. Parameter

Unit

Value

The degree of superheat Injection rate Injection pressure Horizontal and vertical permeabilities Oil viscosity

K t/h MPa lm2 mPa s

40 8 12 1500  103 1500

where hi ¼ hðpi ; T i Þ; hk ¼ hðpk ; T k Þ (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 temperature. By polynomial interpolation, Tortike et al. [25] proposed an important empirical correlation to describe this relationship, which is given by  p p 2 T ¼ f ðpÞ ¼ 280:034 þ 14:0856 ln þ 1:38075  ln 1000 1000   p 3 p 4  0:101806 ln þ 0:019017 ln 611 Pa 1000 1000 7 6 p 6 2:212  10 Pa ð31Þ Thus, the degree of superheat at the outlet of the i-th perforation unit (T deg;i ) is

T deg;i ¼ T i  f ðpi Þ

ð32Þ

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 the horizontal wellbore.

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

2.3.2. Implicit equation for steam quality After superheated steam is cooled to wet steam, the fluid temperature 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 hðx; TÞ ¼ xhs þ ð1  xÞhw

ð33Þ

127

Combing Eqs. (33) and (30) results in an implicit equation for steam quality inside the horizontal wellbore,

m2r;i

!

2pr co U co ke DL ðT i  T ei Þ r co U co f ðtÞ þ ke     m2 m2 ¼ wi1 hm;i1 þ i1  wi hm;i þ i 2 2

Ii qi hm;i þ

2

þ

ð36Þ

where 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]

where hm;i ¼ hðxi ; T i Þ; hm;k ¼ hðxk ; T k Þ (k ¼ i  1 or k ¼ i) and Dxi ¼ xi  xi1 : Similarly, Dxi or xi can be obtained by solving Eq. (36) with an iterative technique.

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

3. Calculation flowchart for the mathematical model

 1:33437  105 T 4 þ 1:26913  108 T 5  4:96880  1012 T 6 ; 273:15 K 6 T 6 640 K

ð34Þ

As stated above, it is necessary to adopt iterative method to solve the mathematical model. The main steps are given as follows:

hw ¼ 23665:2  366:232T þ 2:26952T 2  0:00730365T 3 þ 1:30241  105 T 4  1:22103  108 T 5 þ 4:70878  1012 T 6 ; 273:15 K 6 T 6 645 K

ð35Þ

(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, iterative 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 horizontal 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 wellbore. It is clearly observed that the oil layer is heterogeneous, and the permeability in each segment can be estimated with arithmetic 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 temperature 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 horizontal 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 analysis 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 provided 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 horizontal 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

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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 viscosity and inject larger volume of superheated steam into the oil layer per unit of time, the difference in the volumetric outflow rate 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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Fig. 10. Effects of oil viscosity on the profiles of mass flow rate (a) and the degree of superheat (b) in the horizontal wellbore.

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, therefore, 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 distance 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 131.14 m, 154.81 m and 178.05 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. From Fig. 8(a) and (b), it is observed that in the same position of the horizontal wellbore, both mass flow rate and the degree of superheat decrease with injection pressure. The reason can be given as follows: 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 is obviously 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 thermophysical 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 estimate the thermophysical properties of superheated steam in horizontal injection wells, and phase change behavior of superheated 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

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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.

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