Introduction

Thermal methods, such as steam-assisted gravity drainage (SAGD) (Butler 1991; Vander et al. 2007; Al-Bahlani and Babadagli 2009; Miura and Wang 2012; Yang et al. 2016) and steam huff-puff (Marx and Langenheim 1959; Willman and Valleroy 1961; Boberg and Lanz 1966; Hou and Chen 1997; Sandler et al. 2014; Sun et al. 2017a) have been proved successful for heavy oil recovery. Precisely predicting thermophysical parameters of thermal fluid along wellbores is the first important task for oilfield engineers. However, the prediction of thermophysical properties of thermal fluid in wellbores is never easy due to the complexity of thermal fluid flow in wellbores (Sun et al. 2017b, c, d).

In 1960s, conventional oil resources are relatively abundant and the development of heavy oil was subjected to many restrictions (Ramey 1962; Holst and Flock 1966; Willhite 1967; Orkiszewski 1967; Beggs and Brill 1973). However, with the change of energy demand and with the progress of technology, the recovery of heavy oil has come more into focus. Predicting thermal parameters of thermal fluid along the wellbores is significant for optimizing heavy oil recovery (Cheng et al. 2011, 2012, 2013, 2014; Dong et al. 2014; Gu et al. 2014, 2015a, b; Wei et al. 2015; Fan et al. 2016).

Based upon the energy balance equation, Satter (1965) gave a model for predicting steam quality in wellbores. Pacheco and Ali (1972) proposed a comprehensive mathematical model for predicting pressure in wellbores taking friction losses into consideration. Farouq Ali (1981) developed a model to analyze the pressure profiles for both upward and downward flow. Adopting iteration technique, Durrant and Thambynayagam (1986) gave an improved model for quick estimation of transient heat transfer rate along the wellbores, which laid a solid foundation for follow-up studies (Ejiogu and Fiori 1987; Tortike and Ali 1989; Sagar et al. 1991; Alves et al. 1992; Bahonar et al. 2010, 2011).

Several studies have been done by Hasan and Kabir (1991, 1994, 2007, 2009, 2012), Hasan (1995) and Hasan et al. (2007a, b, 2010) on the flow and heat transfer characteristics of wet steam in wellbores, which serve as the foundation for later studies (Chiu and Thakur 1991; Cheng et al. 2011, 2012, 2013, 2014). However, all of these previous studies were focused upon the simple-structured single-tubing wells.

Field tests have shown that the application of single-tubing wells is easier to cause non-uniform steam suction (Hight et al. 1992; Griston and Willhite 1987; Liu 2009; Gu et al. 2014). Therefore, DTW was proposed to alleviate these shortcomings, and fortunately, it has been proved to be effective (Barua 1991; Hight et al. 1992).

Caetano (1985)developed an early model for predicting pressure drop in annuli, which presented a basic reference for follow-up studies (Brill 1987; Hasan and Kabir 1992; Brill and Mukherjee 1999; Lage and Time 2000, 2002; Yu et al. 2010; Wu et al. 2011). Based on the new concept of equivalent radius, Griston and Willhite (1987) presented an improved model for predicting pressure drop in annuli, which was proved reliable later by Kaya et al. (2001). In recent years, Gu et al. (2014) presented an improved model to estimate the pressure drop in annuli. It their work, a new radius calculation method for steam flow in annuli was proposed.

However, all of these studies were focused on saturated steam flow and they did not cover the subject of SHS flow in DTW. Zhou et al. (2010), Xu et al. (2013), Gu et al. (2015a, b), Fan et al. (2016) and Sun et al. (2017e, f) presented basic models for predicting thermophysical properties of SHS in conventional single-tubing wells for both onshore and offshore conditions. But the heat transfer characteristics in single-tubing wells are very different from that in DTW (Sun et al. 2017g). Sun et al. (2017g) presented a numerical model for SHS flow in CDTW. However, their model cannot be used directly to PDTW. Besides, an essential comparison between CDTW and PDTW is needed. More researches are needed to reveal the flow characteristics of SHS in DTW.

In this paper, a new model is developed for predicting thermophysical properties of SHS in DTW. The thermophysical properties of SHS along the wellbores are obtained by finite difference method on space and solved with iteration technique. Moreover, thermal radiation and convection between MT and AT is also accounted for in the model. The effect of injection temperature in each tubing on the profiles of thermophysical properties of SHS in wellbores is analyzed. This study unravels some intrinsic flow characteristics of SHS in DTW, which has a significant impact on the optimization of SHS injection parameters and analysis of heat transfer law in DTW.

Model description

Basic assumptions

A schematic of the DTW is shown in detail in Fig. 1. In order to study the flow characteristics of SHS in DTW, some basic assumptions are made as follows:

Fig. 1
figure 1

A schematic of SHS flow in DTW: a SHS flow in CDTW. b SHS flow in PDTW

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  1. 1.

    The injection parameters at wellhead are assumed to be constant.

  2. 2.

    SHS flow in DTW is assumed to be steady state.

  3. 3.

    The momentum balance of the SHS flow process is assumed to be steady state.

  4. 4.

    Heat transfer rate inside the wellbores is assumed to be steady state.

  5. 5.

    Heat transfer rate in formation is assumed to be transient (Sun et al. 2017h).

  6. 6.

    Thermophysical properties of formation are assumed to be independent of well depth (Sun et al. 2017h).

Mathematical modeling of SHS flow in IJT and MT

Firstly, based on the mass conservation law in the IJT (Sun et al. 2017g) and MT, the mass balance equation of SHS flow in IJT (Sun et al. 2017g) and MT can be given as:

$$\frac{{{\text{d}}w_{\text{ij,MT}} }}{{{\text{d}}z}} = \pi r_{\text{ij,MTi}}^{2} \frac{{{\text{d}}\left( {\rho_{\text{ij,MT}} v_{\text{ij,MT}} } \right)}}{{{\text{d}}z}} = 0$$
(1)

where \(w_{\text{ij,MT}}\) denotes the mass flow rate of SHS in IJT or MT, kg/s; \(z\) denotes the vertical depth of the wellbore, m; \(r_{\text{ij,MTi}}\) denotes the inner radius of the IJT or MT, m; \(\rho_{\text{ij,MT}}\) denotes the SHS density in IJT or MT, kg/m3; \(v_{\text{ij,MT}}\) denotes the flow velocity of SHS in IJT or AT, m/s.

Secondly, the energy balance equation of SHS in the IJT (Sun et al. 2017g) can be expressed as:

$$\frac{{{\text{d}}Q_{\text{ij}} }}{{{\text{d}}z}} = - \, w_{\text{ij}} \frac{{{\text{d}}h_{\text{ij}} }}{{{\text{d}}z}} - w_{\text{ij}} \frac{\text{d}}{{{\text{d}}z}}\left( {\frac{{v_{\text{ij}}^{2} }}{2}} \right) + w_{\text{ij}} g\cos \theta$$
(2)

where \(Q_{\text{ij}}\) denotes the heat exchange rate between IJT and annuli, J/s; \(h_{\text{ij}}\) denotes the enthalpy of SHS in IJT, J/kg; \(\theta\) denotes the included angle between wellbore and the vertical line, rad.

The energy balance equation of SHS in MT can be expressed as:

$$\frac{{{\text{d}}Q_{\text{MTout}} }}{{{\text{d}}z}} - \frac{{{\text{d}}Q_{\text{MTin}} }}{{{\text{d}}z}} = - w_{\text{MT}} \frac{{{\text{d}}h_{\text{MT}} }}{{{\text{d}}z}} - w_{\text{MT}} \frac{\text{d}}{{{\text{d}}z}}\left( {\frac{{v_{\text{MT}}^{2} }}{2}} \right) + w_{\text{MT}} g\cos \theta$$
(3)

where \(Q_{\text{MTout}}\) denotes heat loss rate of the MT (Wei. 2015), J/s; \(Q_{\text{MTin}}\) denotes the heat absorption rate of the MT (Wei. 2015), J/s; \(h_{\text{MT}}\) denotes the enthalpy of SHS in MT, J/kg.

Finally, the steady-state momentum balance equation of SHS in the IJT (Sun et al. 2017g) and MT can be expressed as:

$$\frac{{{\text{d}}p_{\text{ij,MT}} }}{{{\text{d}}z}} - \rho_{\text{ij,MT}} g\cos \theta + \frac{{\tau_{\text{f}} }}{{\pi r_{\text{ij,MTi}}^{2} {\text{d}}z}} + \frac{{{\text{d}}\left( {\rho_{\text{ij,MT}} v_{\text{ij,MT}}^{2} } \right)}}{{{\text{d}}z}} = 0$$
(4)

where \(p_{\text{ij,MT}}\) denotes the SHS pressure in IJT and MT, Pa; \(\tau_{\text{f}}\) denotes the shear stress in IJT and MT, N.

Thus, the governing hydrodynamic equations of SHS flow in IJT and MT are established.

Mathematical modeling of SHS flow in annuli and AT

Firstly, based on the mass conservation law in annuli, the mass balance equation of SHS flow in the IJT can be given as (Sun et al. 2017g):

$$\frac{{{\text{d}}w_{\text{an}} }}{{{\text{d}}z}} = \pi \left( {r_{\text{ai}}^{2} - r_{\text{ijo}}^{2} } \right)\frac{{{\text{d}}\left( {\rho_{\text{an}} v_{\text{an}} } \right)}}{{{\text{d}}z}} = 0$$
(5)

where \(w_{\text{an}}\) denotes the mass flow rate in annuli in CDTW, kg/s; \(r_{\text{ai}}\) denotes the inner radius of the inner tubing, m; \(r_{\text{ijo}}\) denotes the outer radius of the IJT, m; \(\rho_{\text{an}}\) denotes the SHS density in annuli, kg/m3; \(v_{\text{an}}\) denotes the flow velocity of SHS in annuli, m/s.

The mass balance equation of SHS flow in AT can be expressed as:

$$\frac{{{\text{d}}w_{\text{AT}} }}{{{\text{d}}z}} = \pi r_{\text{ATi}}^{2} \frac{{{\text{d}}\left( {\rho_{\text{AT}} v_{\text{AT}} } \right)}}{{{\text{d}}z}} = 0$$
(6)

where \(w_{\text{AT}}\) denotes the mass flow rate of SHS in AT, kg/s; \(r_{\text{ATi}}\) denotes the inner radius of the AT, m; \(\rho_{\text{AT}}\) denotes the SHS density in AT, kg/m3; \(v_{\text{AT}}\) denotes the flow velocity of SHS in AT, m/s.

Secondly, based on the energy conservation law in annuli, the energy balance equation of SHS in annuli can be expressed as (Sun et al. 2017g):

$$\frac{{{\text{d}}Q_{\text{an}} }}{{{\text{d}}z}} - \frac{{{\text{d}}Q_{\text{ij}} }}{{{\text{d}}z}} = - w_{\text{an}} \frac{{{\text{d}}h_{\text{an}} }}{{{\text{d}}z}} - w_{\text{an}} \frac{\text{d}}{{{\text{d}}z}}\left( {\frac{{v_{\text{an}}^{2} }}{2}} \right) + w_{\text{an}} g\cos \theta$$
(7)

where \(Q_{\text{an}}\) denotes the heat transfer rate from annuli to formation, J/s; \(w_{\text{an}}\) denotes the mass flow rate of SHS in annuli, kg/m3; \(h_{\text{an}}\) denotes the SHS enthalpy in annuli, J/s; \(v_{\text{an}}\) denotes the flow velocity of SHS in annuli, m/s.

The energy balance equation of SHS flow in AT can be expressed as:

$$\frac{{{\text{d}}Q_{\text{ATout}} }}{{{\text{d}}z}} - \frac{{{\text{d}}Q_{\text{ATin}} }}{{{\text{d}}z}} = - w_{\text{AT}} \frac{{{\text{d}}h_{\text{AT}} }}{{{\text{d}}z}} - w_{\text{AT}} \frac{\text{d}}{{{\text{d}}z}}\left( {\frac{{v_{\text{AT}}^{2} }}{2}} \right) + w_{\text{AT}} g\cos \theta$$
(8)

where \(Q_{\text{ATout}}\) denotes the AT heat loss rate (Wei. 2015), J/s; \(Q_{\text{ATin}}\) denotes the AT heat absorption rate (Wei. 2015), J/s; \(w_{\text{AT}}\) denotes the mass flow rate of SHS in AT, kg/m3; \(h_{\text{AT}}\) denotes the SHS enthalpy in AT, J/kg; \(v_{\text{AT}}\) denotes the flow velocity of SHS in AT, m/s.

Finally, the momentum balance equation of SHS flow in annuli can be expressed as (Sun et al. 2017g):

$$\frac{{{\text{d}}p_{\text{an}} }}{{{\text{d}}z}} - \rho_{\text{an}} g\cos \theta + \frac{{\tau_{\text{f}} }}{{\pi \left( {r_{\text{ai}}^{2} - r_{\text{ijo}}^{2} } \right){\text{d}}z}} + \frac{{{\text{d}}\left( {\rho_{\text{an}} v_{\text{an}}^{2} } \right)}}{{{\text{d}}z}} = 0$$
(9)

where \(p_{\text{an}}\) denotes the SHS pressure in annuli, Pa.

The momentum balance equation of SHS flow in AT can be expressed as:

$$\frac{{{\text{d}}p_{\text{AT}} }}{{{\text{d}}z}} = \rho_{\text{AT}} g\cos \theta - \frac{{\tau_{\text{f}} }}{{\pi r_{\text{ATi}}^{2} {\text{d}}z}} - \frac{{{\text{d}}\left( {\rho_{\text{AT}} v_{\text{AT}}^{2} } \right)}}{{{\text{d}}z}}$$
(10)

where \(p_{\text{AT}}\) denotes the SHS pressure in AT, Pa.

The above equation models SHS flow in annuli.

Numerical solution of the model

The proposed model is solved with finite difference method on space. Both IJT and annuli or MT and AT are evenly divided into equal segments. Then, the temperature and pressure at the interface between the neighboring segments can be obtained in DTW. The pressure and temperature in the IJT and annuli or in the MT and AT are solved sequentially. Based on Eq. (5), the difference equation can be expressed as:

$$\frac{{p_{\text{ij,MT,out}} - p_{\text{ij,MT,in}} }}{\Delta z} - g\cos \theta \frac{{\rho_{\text{ij,MT,out}} + \rho_{\text{ij,MT,in}} }}{2} + \frac{{\tau_{\text{f}} }}{{\pi r_{\text{iji}}^{2} \Delta z}} + \frac{{\rho_{\text{ij,MT,out}} v_{\text{ij,MT,out}}^{2} - \rho_{\text{ij,MT,in}} v_{\text{ij,MT,in}}^{2} }}{\Delta z} = 0$$
(11)

Similarly, based on Eq. (9), the difference equation for pressure solving in annuli can be obtained:

$$\frac{{p_{\text{an,out}} - p_{\text{an,in}} }}{\Delta z} - g\cos \theta \frac{{\rho_{\text{an,out}} + \rho_{\text{an,in}} }}{2} + \frac{{\tau_{\text{f}} }}{{\pi \left( {r_{\text{ai}}^{2} - r_{\text{ijo}}^{2} } \right)\Delta z}} + \frac{{\rho_{\text{an,out}} v_{\text{an,out}}^{2} - \rho_{\text{an,in}} v_{\text{an,in}}^{2} }}{\Delta z} = 0$$
(12)

The difference equation for pressure solving in AT can be obtained:

$$\frac{{p_{\text{AT,out}} - p_{\text{AT,in}} }}{\Delta z} - g\cos \theta \frac{{\rho_{\text{AT,out}} + \rho_{\text{AT,in}} }}{2} + \frac{{\tau_{\text{f}} }}{{\pi r_{\text{ATi}}^{2} \Delta z}} + \frac{{\rho_{\text{AT,out}} v_{\text{AT,out}}^{2} - \rho_{\text{AT,in}} v_{\text{AT,in}}^{2} }}{\Delta z} = 0$$
(13)

In order to solve Eqs. (11–13), we must assume a pair of the outlet temperature. Taking CDTW as an example, this will leave two unknown numbers (\(p_{\text{ij,out}}\) and \(p_{\text{an,out}}\)) with two equations, and the outlet pressure of the studied segment can be calculated. In this paper, function zero method is adopted, as shown below:

$$\begin{aligned} & p(p_{\text{ij,MT,out}} ) \\ & = \;\frac{{p_{\text{ij,MT,out}} - p_{\text{ij,MT,in}} }}{\Delta z} - g\cos \theta \frac{{\rho_{\text{ij,MT,out}} + \rho_{\text{ij,MT,in}} }}{2} \\ & + \;\frac{{\tau_{\text{f}} }}{{\pi r_{\text{iji}}^{2} \Delta z}} + \frac{{\rho_{\text{ij,MT,out}} v_{\text{ij,MT,out}}^{2} - \rho_{\text{ij,MT,in}} v_{\text{ij,MT,in}}^{2} }}{\Delta z} = 0 \\ \end{aligned}$$
(14)
$$\begin{aligned} & f(p_{\text{an,out}} ) \\ & = \;\frac{{p_{\text{an,out}} - p_{\text{an,in}} }}{\Delta z} - g\cos \theta \frac{{\rho_{\text{an,out}} + \rho_{\text{an,in}} }}{2} + \frac{{\tau_{\text{f}} }}{{\pi \left( {r_{\text{ai}}^{2} - r_{\text{ijo}}^{2} } \right)\Delta z}} + \frac{{\rho_{\text{an,out}} v_{\text{an,out}}^{2} - \rho_{\text{an,in}} v_{\text{an,in}}^{2} }}{\Delta z} = 0 \\ \end{aligned}$$
(15)
$$\begin{aligned} & f(p_{\text{AT,out}} ) \\ & = \;\frac{{p_{\text{AT,out}} - p_{\text{AT,in}} }}{\Delta z} - g\cos \theta \frac{{\rho_{\text{AT,out}} + \rho_{\text{AT,in}} }}{2} + \frac{{\tau_{\text{f}} }}{{\pi r_{\text{ATi}}^{2} \Delta z}} + \frac{{\rho_{\text{AT,out}} v_{\text{AT,out}}^{2} - \rho_{\text{AT,in}} v_{\text{AT,in}}^{2} }}{\Delta z} = 0 \\ \end{aligned}$$
(16)

Similarly, we can solve for temperature, which can be derived from Eqs. (2, 3, 7, 8).

$$\begin{aligned} & f\left( {T_{\text{ij,out}} } \right) \\ & = \;\frac{{q_{\text{ij,out}} + q_{\text{ij,in}} }}{2} + w_{\text{ij}} \frac{{\left( {h_{\text{ij,out}} - h_{\text{ij,in}} } \right)}}{\Delta z} + w_{\text{ij}} \frac{\text{d}}{\Delta z}\left( {\frac{{v_{\text{ij,out}}^{2} }}{2} - \frac{{v_{\text{ij,in}}^{2} }}{2}} \right) - w_{\text{ij}} g\cos \theta = 0 \\ \end{aligned}$$
(17)
$$\begin{aligned} & f(T_{\text{MT,out}} ) \\ & = \;\frac{{q_{\text{MTout,out}} + q_{\text{MTout,in}} }}{2} - \frac{{q_{\text{MTin,out}} + q_{\text{MTin,in}} }}{2} \\ & + \;w_{\text{MT}} \frac{{h_{\text{MT,out}} - h_{\text{MT,in}} }}{\Delta z} + w_{\text{MT}} \frac{1}{\Delta z}\left( {\frac{{v_{\text{MT,out}}^{2} - v_{\text{MT,in}}^{2} }}{2}} \right) - w_{\text{MT}} g\cos \theta = 0 \\ \end{aligned}$$
(18)
$$\begin{aligned} & f\left( {T_{\text{an,out}} } \right) \\ & = \;\frac{{q_{\text{an,out}} + q_{\text{an,in}} }}{2} - \frac{{q_{\text{ij,out}} + q_{\text{ij,in}} }}{2} \\ & + \;w_{\text{an}} \frac{{\left( {h_{\text{an,out}} - h_{\text{an,in}} } \right)}}{\Delta z} + w_{\text{an}} \frac{\text{d}}{\Delta z}\left( {\frac{{v_{\text{an,out}}^{2} - v_{\text{an,in}}^{2} }}{2}} \right) - w_{\text{an}} g\cos \theta = 0 \\ \end{aligned}$$
(19)
$$\begin{aligned} & f(T_{\text{AT,out}} ) \\ & = \;\frac{{q_{\text{ATout,out}} + q_{\text{ATout,in}} }}{2} - \frac{{q_{\text{ATin,out}} + q_{\text{ATin,in}} }}{2} \\ & + \;w_{\text{AT}} \frac{{h_{\text{AT,out}} - h_{\text{AT,in}} }}{\Delta z} + w_{\text{AT}} \frac{\text{d}}{\Delta z}\left( {\frac{{v_{\text{AT,out}}^{2} }}{2} - \frac{{v_{\text{AT,in}}^{2} }}{2}} \right) - w_{\text{AT}} g\cos \theta = 0 \\ \end{aligned}$$
(20)

In order to get accurate results of the outlet values of the studied segment, iteration method is then adopted.

Finally, the accurate values of pressure and temperature at outlet of the ith segment are used as inputs for the inlet of the (i + 1)th segment. Then, another iteration begins. Finally, the pressure and temperature along the wellbores can be obtained. The detailed mathematical model solving process is shown as a flowchart in Fig. 2.

Fig. 2
figure 2

Calculation flowchart for the mathematical model

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Results and discussion

Validation of the model

In order to verify the accuracy of the proposed model, the model was validated by testing it against field data. Basic tubing parameters used for calculation can be found in Table 1. And the simulated results for SHS flow in CDTW are presented in Fig. 3. Besides, simulated results for SHS flow in PDTW are also added for comparison.

Table 1 Basic parameters used for calculation for CDTW
Full size table
Fig. 3
figure 3

Comparison of the calculated results with field data: a pressure. b Temperature

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Figure 3 shows the temperature and pressure profiles in DTW. For CDTW, the injection pressure, temperature and mass flow rate at wellhead in IJT and annuli are 4.5 MPa, 650 K, 293 t/d, and 3.5 MPa, 600 K, 259 t/d, respectively. It is observed that the predicted results from our model are in good agreement with field data. What is to stress that some additional factors (pipe contact, formation heterogeneity, etc.) that are neglected in the mathematical model causes more temperature drop (Fig. 3b). This is the main reason why the predicted temperature is a little higher than field data. However, the maximum relative error for both pressure and temperature predictions is still smaller than 5%.

Characteristic analysis of SHS flow in CDTW

From Fig. 3a, it is observed that: (1) SHS pressure in IJT and annuli or in MT and AT decreases with well depth; (2) pressure gradient in the IJT is larger than in annuli. This is because the relatively larger injection rate in the IJT, which causes more friction loss; (3) pressure gradient in MT is larger than in AT. This is because the flow radius of SHS in MT is smaller and the mass flow rate of SHS in MT is larger, which causes more friction loss.

It can be observed from Fig. 3b that: (1) Temperature in the IJT decreases rapidly from wellhead to the depth of 50 m; after 50 m, the temperature decrease is gradual; (2) Temperature in annuli has an increase from wellhead to the depth of 50 m, after which it gradually decreases. This is because \(U_{\text{ijo}}\) is much higher than \(U_{\text{ao}}\). For our particular case, \(U_{ijo}\) has a value of 1560 W/(m2 K). Thus, the heat exchange rate between IJT and annuli is large even if there exists a small temperature difference between IJT and annuli; (3) Temperature gradient in IJT and annuli tends to be equal from the depth of 50 m to well bottom; (4) the heat exchange rate between MT and AT is much smaller compared with that between IJT and annuli. This is because the MT and AT are not contact to each other. The channels of heat transfer between MT and AT are thermal radiation and convection. As a result, SHS with a higher temperature in MT decreases slowly compared with that in IJT. And the SHS with a lower temperature in AT increases slowly compared with that in annuli.

Figure 4a shows the profiles of heat transfer rate in CDTW. It can be observed that (1) heat transfer rate from IJT to annuli has a decrease of 79% from wellhead to the depth of 50 m; (2) the net heat loss rate in annuli increases rapidly from −132,298 to −2751 W/m from wellhead to the depth of 50 m, and turns positive at the depth of 165 m. SHS in IJT begins to absorb heat from SHS in annuli. This is because the SHS temperature in IJT drops faster than SHS temperature in annuli.

Fig. 4
figure 4

Type curves of SHS flow in DTW: a heat transfer rate in wellbores. b Temperature profiles. c Profiles of superheat degree

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From Fig. 4b, we can find that the temperature in annuli is much higher than the temperature at the interface of the cement sheath and formation, and of course, much higher than the initial formation temperature. Consequently, CDTW heat losses cannot be avoided due to this temperature difference.

Figure 4c shows the profiles of superheat degree in the IJT and annuli and in the MT and AT. It is observed that (1) superheat degree in annuli increases at first and then begins to decrease with well depth. (2) Superheat degree in the IJT decreases rapidly from wellhead to the depth of 100 m, and then, the superheat degree decrease is gradual. In fact, in the case of CDTW, superheat degree in IJT and annuli is influenced not only by the injection conditions at wellhead, but also by the heat transfer between the IJT and annuli. For example, when the SHS in annuli absorbs heat from SHS in the IJT, the latter is just like a heat resource that can transfer heat to annuli, causing the superheat degree in annuli to increase with well depth. (3) The decrease rate of superheat degree in MT from wellhead to the depth of 100 m is smaller than that in IJT. This is because the heat transfer rate from MT to AT is smaller than that from IJT to annuli. As a result, superheat degree in AT has a small increase from wellhead to the depth of 50 m compared with that in annuli.

Effect analysis of the injection temperature in IJT and MT

In this section, effect of injection temperature in IJT on the profiles of thermophysical properties of SHS in wellbores is studied. Temperatures were tested (550, 570, 590, 610, 630, 650 K) with no change in values of injection pressure and rate. Besides, injection temperatures in MT with same values were tested for comparison. The predicted results are presented in Fig. 5.

Fig. 5
figure 5

Effect of injection temperature in IJT and MT under various injection temperatures: a SHS temperature in IJT and MT; b SHS temperature in annuli and AT; c superheat degree in IJT and MT; d superheat degree in annuli and AT; e heat transfer rate from the IJT to annuli; f CDTW heat loss rate

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Figure 5a shows that (1) when \(T_{\text{ij}}\) and \(T_{\text{MT}}\) are relatively small, SHS temperature in IJT and MT increases from wellhead to the depth of 50 and 150 m, respectively, and then turns to decrease. This is because when \(T_{\text{ij}}\) and \(T_{\text{MT}}\) are smaller than \(T_{\text{an}}\) and \(T_{\text{AT}}\), SHS in IJT and MT absorbs heat from SHS in annuli and AT, which causes the increase in SHS temperature near wellhead. (2) SHS temperature starts to decrease when \(T_{\text{ij}}\) reaches 600 K, and the temperature starts to decrease when \(T_{\text{MT}}\) reaches 600 K. The reason is that when \(T_{\text{ij}}\) and \(T_{\text{MT}}\) are smaller than \(T_{\text{an}}\) and \(T_{\text{AT}}\), respectively, SHS in IJT and MT absorbs heat from SHS in annuli and AT, respectively. (3) From wellhead to the depth of 50 m, temperature in the IJT changes drastically due to the fact that there exists a large temperature difference between the IJT and annuli. Similarly, from wellhead to 150 m, temperature in MT changes rapidly with a large temperature difference between MT and AT. For depths greater than 50 m in CDTW, temperature in IJT and annuli decreases at a same rate. This is because the temperature difference between IJT and annuli is very small after 50 m. For depths greater than 150 m in PDTW, temperature in MT and AT decreases with a same rate. This is because the temperature difference between MT and AT becomes small when it is deeper than 150 m.

Figures 5b shows that: (1) When SHS injection temperature in IJT and MT is smaller than 600 K, SHS temperature in annuli and AT decreases from wellhead to the depth of 50 and 150 m, respectively. This is because when \(T_{\text{ij}}\) and \(T_{\text{MT}}\) are smaller than \(T_{\text{an}}\) and \(T_{\text{AT}}\), respectively, SHS in annuli and AT transfers heat to SHS in IJT and MT, respectively. (2) When SHS injection temperature in IJT and MT is larger than 600 K, SHS in annuli and AT begins to absorb heat from SHS in IJT and MT. As a result, SHS temperature in annuli and AT increases from wellhead to the depth of 50 and 150 m, respectively.

Figure 5c, d is important from an engineering point of view as they show the superheat degree at well bottom. It is observed that (1) the superheat degree in both CDTW and PDTW increases with the increase in \(T_{\text{ij}}\) and \(T_{\text{MT}}\), respectively. But the practicing engineers are suggested to take the boiler capacity into consideration in order to achieve maximum economic benefits. (2) Gradient of the superheat degree in IJT with depth greater than 50 m is larger than that in MT. This is because the MT and AT are not contacted to each other. There exists thermal radiation and convection between MT and AT. Therefore, the effect of injection temperature in MT on AT is weaker compared with the effect of IJT on annuli.

Figure 5e shows heat transfer rate from IJT to annuli. It is observed that for depths greater than 100 m, heat transfer rate inside the wellbore gradually becomes zero. Figure 5f shows the profiles of net heat loss rate in annuli (CDTW heat loss rate) under various injection temperatures in IJT; it can be observed that the net heat loss rate is large when there exists a large temperature difference between IJT and annuli, but the curve becomes smooth with the increase in \(T_{\text{ij}}\) and turns negative when \(T_{\text{ij}}\) reaches 600 K.

Analysis of the injection temperature in annuli

In the following section, we will study the effect of the injection temperature in annuli on temperature profiles, degree of superheat and heat transfer rate in CDTW. Different injection temperatures in annuli were tested (550, 570, 590, 610, 630, 650 K) with no change in values of injection pressure and rate. The results are presented in Fig. 6.

Fig. 6
figure 6

Effect of injection temperature in annuli and AT on flow behavior of SHS in CDTW and PDTW: a temperature in IJT and MT; b temperature in annuli and AT; c superheat degree in IJT and MT; d superheat degree in annuli and AT; e heat transfer rate from the IJT to annuli; f CDTW heat loss rate

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Figure 6a shows the influence of injection temperature in annuli and AT on temperature profiles in IJT and AT, respectively. It is observed that (1) the SHS temperature at well bottom in both IJT and MT increases with the increase in injection temperature in annuli and AT, respectively. (2) SHS temperature in IJT and MT decreases rapidly from wellhead to the depth of 100 m when injection temperature in annuli and AT is smaller than that in IJT and MT, respectively. This is because the heat transfer rate between the IJT and annuli or between MT and AT wells is large when there exists a large temperature difference between the IJT and annuli or between MT and AT, as shown in Fig. 6e. Moreover, Fig. 6f also illustrates the characteristics of heat transfer in CDTW, and we can observe that the net heat loss rate in annuli increases rapidly with the increase in \(T_{\text{an}}\). (3) Temperature gradient in IJT is larger than that in MT. This is because the effect of heat transfer in PDTW on temperature profiles in MT and AT is weak compared with that in CDTW.

Figure 6b shows the effect of injection temperature in annuli and AT on the profiles of temperature in annuli and AT. Firstly, we can find that the temperature near wellhead in both annuli and AT increases when the injection temperature in annuli and AT is smaller than that in IJT and MT. This is because SHS in annuli and AT absorbs heat from SHS in IJT and AT due to temperature difference between IJT and annuli or between MT and AT. Secondly, the increase in temperature near wellhead in annuli and AT gradually becomes smaller with the increase in injection temperature in annuli and AT, respectively. This is because the heat transfer rate inside the wellbore decreases and the net heat loss rate in annuli increases with the increase in \(T_{\text{an}}\), as shown in Fig. 6e, f.

Figure 6c shows the effect of injection temperature in annuli and AT on the profiles of superheat degree in IJT and AT. It is observed that (1) superheat degree in both IJT and MT at a certain place in wellbores increases with the increase in injection temperature in annuli and AT. (2) Superheat degree decreases rapidly near wellhead when injection temperature in annuli and AT is smaller than that in IJT and MT, respectively.

Figure 6d shows the effect of injection temperature in annuli and AT on the profiles of superheat degree in annuli and AT. Firstly, it is observed that the superheat degree increases near wellhead when the injection temperature in annuli and AT is smaller than that in IJT and MT, respectively. Secondly, the increase in temperature near wellhead in annuli and AT gradually decreases with the increase in injection temperature in annuli and AT, respectively. In conclusion, in order to obtain a satisfactory oil recovery rate, a higher injection rate in annuli and AT is recommended. Besides, the effect of injection temperature on superheat degree profiles becomes weak when PDTW is selected.

Conclusions

In this paper, a novel model is proposed to analyze the flow and heat transfer characteristics of SHS in DTW. The model was validated using field data. We can conclude:

  1. 1.

    In CDTW, temperature difference between IJT and annuli has a strong influence on the profiles of temperature and superheat degree in both IJT and annuli. A relatively small difference in temperature at wellhead is able to cause a large amount of heat to flow between IJT and annuli. Besides, the effect of injection temperature on superheat degree profiles becomes weak when PDTW is selected.

  2. 2.

    For depth less than 50 m, SHS temperature in IJT decreases rapidly near wellhead, while SHS temperature in annuli increases rapidly. For depth greater than 50 m, the heat exchange rate between IJT and annuli is small. For depth less than 150 m, SHS temperature in MT decreases rapidly near wellhead, while SHS temperature in AT increases rapidly. For depth greater than 150 m, the heat exchange rate between MT and AT is small.

  3. 3.

    Superheat degree in IJT and annuli or in MT and AT increases with the increase in injection temperature in IJT and MT, respectively.

  4. 4.

    Gradient of superheat degree near wellhead in IJT is larger than that in MT. This is because the MT and AT are not contacted to each other. Thermal radiation and convection are the main ways of heat exchange between MT and AT.

  5. 5.

    In order to obtain a satisfactory oil recovery rate, a higher injection temperature for both CDTW and PDTW is recommended. However, practicing engineers are suggested to take the boiler capacity into consideration.