Introduction

Thermal energy is widely adopted in engineering (Sheikholeslami et al. 2013, 2017, 2018; Sheikholeslami and Ganji 2014, 2016; Sheikholeslami and Bhatti 2017; Sheikholeslami and Rokni 2017a, b, 2018; Sheikholeslami and Sadoughi 2017, 2018; Sheikholeslami and Seyednezhad 2017, 2018; Sheikholeslami and Shehzad 2017a, b, 2018a, b, c). In the petroleum industry, the recovery of heavy oil can be mainly divided into two types: thermal recovery (Dong et al. 2015; Rego et al. 2017; Nian and Cheng 2017; Mullakaev et al. 2017; Akhmedzhanov et al. 2017; Telmadarreie and Trivedi 2017) and the cold recovery (Coskuner et al. 2015; Zhou et al. 2018). The thermal recovery is mainly conducted by injecting thermal fluid (e.g., wet steam, superheated steam, multi-component thermal fluid) into oil layers (Marx and Langenheim 1959; Sun et al. 2017a). This is because the viscosity of heavy oil is extremely sensitive to temperature. A small increase in temperature can cause great drop of viscosity of heavy oil. One may find that when these thermal methods are adopted, the heat loss estimation from wellhead to well-bottom must be conducted in order to increase the economic performance (Jacobson 2009; Sun et al. 2017b). What is worth to stress is that the wet steam (a mixture of steam and water) is always selected as the thermal carrier due to low cost and high economic efficiency (Sun et al. 2017c).

Ramey (1962) proposed an equation for wet steam temperature estimation based on the energy balance equation. However, their energy balance equation was built upon incompressible fluid. Raymond (1969) presented an improved model that can be used for temperature estimation. Then, a series of works were done on modeling of wet steam flow in the wellbores (Eickmeier et al. 1970; Alves et al. 1992; Hasan and Kabir 1994, 1996, 2012; Hasan et al. 2009; Pourafshary et al. 2009; Livescu et al. 2010; Bahonar and Azaiez 2011a, b; Mao and Harvey 2013; Gu et al. 2014; Sivaramkrishnan et al. 2015).

However, the pressure and temperature of the wet steam are in function relationship. That is to say the temperature can be obtained when the pressure is known, which is different from supercritical water. Zhou (2010) and Xu (2011) proposed models for estimating superheated steam flow in the vertical wells and obtained the pressure and temperature profiles by numerical methods (de Almeida et al. 2017). Xu et al. (2013) added some oil displacement mechanisms of superheated steam to previous models (Fan et al. 2016). Sun et al. (2017c) proposed an novel model by taking the effect of frictional work on fluid temperature into consideration, which laid a solid foundation for following studies (Sun et al. 2017d, e). Gu et al. (2015) proposed a basic model for single-phase fluid flow in the horizontal section of the wellbores (Dong et al. 2014, 2016). Sun et al. (2017f) improved Dong et al.’s model by modifying the energy balance equation. As a result, the scope of application of the multi-component thermal fluid flow model in horizontal wells was extended to a wider range of injection rate. Then, flow behaviors of saturated steam in parallel or concentric dual-tubing wells were revealed by Wei (2015) and Gu (2016). Based mainly on Wei and Gu et al.’s works, Sun et al. (2017g, h, i) conducted a series of studies on superheated fluid (superheated steam or multi-component thermal fluid) flow in parallel or concentric dual-tubing wells. They found out that heat exchange inside the wellbores has an obvious influence on thermophysical properties of superheated fluid in the integral joint tubing and annuli or in the main tubing and auxiliary tubing.

However, these previous works were focused on superheated steam (or multi-component thermal fluid), which cannot be used to analyze the flow behaviors of SCW in wellbores. At present, the study on SCW flow in wellbores was very limited. This paper moves one step forward to develop a numerical model for simulating SCW flow in offshore wellbores. There are mainly three contributions of this paper to the existing body of the literature: (1) a numerical model is developed for SCW flow in offshore wells with consideration of turbulent flow of seawater. (2) Type curves of SCW flow in offshore wells were obtained by finite difference method on space and iteration technique. (3) Effect of injection parameters on the profiles of thermophysical properties of SCW in offshore wellbores was discussed in detail.

Model description

General assumptions

The flow channel of SCW from wellhead to well-bottom is shown in Fig. 1. Besides, some basic assumptions are made in order to establish the model, as shown below (Sun et al. 2017j):

Fig. 1
figure 1

A schematic of SCW flow in offshore wells

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  1. 1.

    The injection parameters of SCW at platform are constant during the whole injection process.

  2. 2.

    Heat transfer rate from SCW to the outside wall of riser/cement sheath is steady state.

  3. 3.

    Thermophysical properties of seawater are independent from well depth.

  4. 4.

    Heat transfer rate from the outside wall of the cement sheath to formation is transient state.

  5. 5.

    Thermophysical properties of SCW remain unchanged as it flows from the platform to the sea surface.

Governing equations of the mathematical model

The development of the governing equations of SCW flow in wellbores is based on the theory proposed by previous researchers (Gu et al. 2014; Dong et al. 2016; Wei 2015; Sun et al. 2018a, b, c, d, e, f, g, h).

Firstly, an equation describing the mass change process is developed. The gradient of mass flow rate in the vertical offshore tube is equal to zero.

$$ \frac{{{\text{d}}w_{\text{SCW}} }}{{{\text{d}}z}} = \pi r_{ai}^{2} \frac{{{\text{d}}\left( {\rho_{\text{SCW}} v_{\text{SCW}} } \right)}}{{{\text{d}}z}} = 0 $$
(1)

Based on the energy conservation law (Sun et al. 2018i), the heat transfer rate from wellbore to seawater/formation should be equal to the total energy change in SCW in wellbores. The energy balance equation can be expressed as:

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

where \( Q_{\text{SCW}} \) denotes the heat transfer rate from SCW to seawater/formation (Willhite 1967; Cheng et al. 2012; Liu 2013; Huang et al. 2015) (J/s).

The flowing process of SCW in the vertical offshore tube is subjected to the law of momentum conservation.

$$ \pi r_{ai}^{2} {\text{d}}p_{\text{SCW}} = \rho_{\text{SCW}} \pi r_{ai}^{2} g\cos \theta {\text{d}}z - f - \pi r_{ai}^{2} {\text{d}}\left( {\rho_{\text{SCW}} v_{\text{SCW}}^{2} } \right) $$
(3)

where \( f \) denotes the shear force (Yuan 1982) (N).

Numerical solution of the mathematical model

In order to obtain the numerical solutions of the model, the governing equations are expressed in the form of difference equations, as expressed below:

$$ \begin{aligned} f\left( {T_{\text{SCW,out}} } \right) & = \frac{{q_{\text{SCW,out}} + q_{\text{SCW,in}} }}{2} + w_{\text{SCW}} \frac{{\left( {h_{\text{SCW,out}} - h_{\text{SCW,in}} } \right)}}{\Delta z} \\ & \quad + w_{\text{SCW}} \frac{d}{\Delta z}\left( {\frac{{v_{\text{SCW,out}}^{2} }}{2} - \frac{{v_{\text{SCW,in}}^{2} }}{2}} \right) - w_{\text{SCW}} g\cos \theta \\ \end{aligned} $$
(4)
$$ \begin{aligned} f(p_{\text{SCW,out}} ) & = \pi r_{\text{ai}}^{2} \left( {p_{\text{SCW,out}} - p_{\text{SCW,in}} } \right) - \pi r_{\text{ai}}^{2} g\cos \theta \frac{{\rho_{\text{SCW,out}} + \rho_{\text{SCW,in}} }}{2}\Delta z \\ & \quad + \tau_{f} + \pi r_{\text{ai}}^{2} \left( {\rho_{\text{SCW,out}} v_{\text{SCW,out}}^{2} - \rho_{\text{SCW,in}} v_{\text{SCW,in}}^{2} } \right) \\ \end{aligned} $$
(5)

The model is solved under the boundary condition. The boundary condition (injection pressure and temperature conditions) at the platform is shown below:

$$ \left\{ {\begin{array}{*{20}c} {p\left( {\text{wellhead}} \right) = p_{0} } \\ {T\left( {\text{wellhead}} \right) = T_{0} } \\ \end{array} } \right. $$
(6)

For a small segment, the injection values at the inlet surface is assumed to be given. Therefore, the outlet values are calculated based on the difference equations shown above. Next, the calculated values at the outlet surface are regared as new input values of the following segment, and the same calculation method is adopted. Finally, the entire distributions of pressure and temperature values along the offshore tube are obtained. In conclusion, the model is solved with straight forward numerical method.

A calculation flowchart for the above discussion is presented in Fig. 2.

Fig. 2
figure 2

Numerical solution for the mathematical model

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Results and discussion

Type curve analysis

Based on the discussion above, the obtained values are shows as curves for discussion. The injection pressure, temperature and mass flow rate of SCW at wellhead are 23 MPa, 700 K and 216 t/d. The other basic parameters used for calculation are shown in Table 1. The predicted results from the model are shown in Fig. 3.

Table 1 Basic parameters used for calculation
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Fig. 3
figure 3

Type curve of SCW flow in offshore wellbores

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It is observed from Fig. 3a that: (a) The value of pressure gradient in the seawater section of the wellbores (from 0 to 150 m) is close to the that in the formation section of the wellbores (from 150 to 1350 m). Therefore, it is concluded that the effect of seawater on SCW pressure in wellbores is negligible. (b) SCW pressure increases with well depth. This is because the SCW density increases with well depth, as shown in Fig. 3b.

It is observed from Fig. 3b that: (a) SCW density increases with well depth. This is because there always exists heat loss from SCW to seawater/formation, which causes the decrease in SCW volume. Therefore, the SCW density increases with decreasing of SCW temperature. (b) There exists a turning point where SCW reaches the seabed (the depth of 150 m). The gradient of density curve in the seawater section of the wellbores is larger than that in the formation section of the wellbores. This is because the SCW temperature drops faster in the seawater section of the wellbores, as shown in Fig. 3c.

It is observed from Fig. 3c that: (a) SCW temperature always decreases with well depth no matter in the seawater section of the wellbores or in the formation section of the wellbores. This is because there always exists a temperature difference between SCW in wellbores and seawater/formation, which leads to heat conduction of the wellbore/seawater system. (b) SCW temperature decreases rapidly in the seawater section of the wellbores. This is because the seawater is always flowing, which breaks the temperature field around the wellbores. Therefore, the temperature difference between wellbore and seawater is larger than that between wellbore and formation, which causes a higher heat loss rate in the seawater section of the wellbores. In conclusion, in order to obtain a better oil recovery effect, high-quality insulation material should be adopted to decrease the heat loss rate in the seawater section of wellbores.

It is observed from Fig. 3d that: (a) SCW enthalpy decreases with well depth. This is because there always exists heat loss from SCW to seawater/formation due to temperature difference. (b) The enthalpy gradient in the seawater section of the wellbores is larger than that in the formation section of the wellbores.

In conclusion, effect of seawater on the profiles of SCW pressure in wellbores is negligible. However, the flow of seawater results in a rapid decline in the temperature/enthalpy of SCW in wellbores.

Sensitivity analysis

Injection rate

In this section, effect of injection rate on the profiles of thermophysical properties of SCW is discussed in detail. Different injection rates (90, 140, 190, 240, 290 and 340 t/d) input into the model under the condition that the injection pressure and temperature are kept unchanged. The predicted results are shown in Fig. 4.

Fig. 4
figure 4

Effect of injection rate on the profiles of thermophysical properties of SCW in offshore wellbores: a SCW pressure; b SCW density; c SCW temperature; d SCW enthalpy

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It is observed from Fig. 4a that SCW pressure decreases with increasing of injection rate. In fact, the form of SCW pressure curve is the comprehensive effect of injection rate and SCW density. The increase in injection rate leads to pressure drop, while the increase in SCW density leads to pressure increase.

It is observed from Fig. 4b that (a) the density gradient in the seawater section of wellbores is always larger than that in the formation section of wellbores. (b) SCW density increases rapidly when the injection rate is small. Taken 90 t/d as an example, the SCW density has an increase of 72.03% from wellhead to well-bottom. However, it is 13.45% when the injection rate is 340 t/d. This is because the SCW temperature drops rapidly when the injection rate is small, which causes the rapid decrease in SCW volume.

It is observed from Fig. 4c that (a) the temperature gradient in the seawater section of wellbores is always larger than that in the formation section of wellbores under various values of injection rate. (b) SCW temperature increases with increasing of injection rate. (c) When the injection rate is large enough (larger than 240 t/d), the rate of temperature rise decreases.

It is observed from Fig. 4d that (a) the enthalpy gradient in the seawater section of wellbores is always larger than that in the formation section of wellbores under various values of injection rate. (b) SCW enthalpy increases with increasing of injection rate. (c) When the injection rate is large enough (larger than 240 t/d), the rate of enthalpy rise decreases.

In conclusion, heat loss is the dominant factor of physical parameter distribution in wellbores. When the injection rate is relatively small, heat loss results in a rapid decrease in temperature and enthalpy, which causes the rapid increase in SCW density. As a result, SCW pressure increases rapidly with well depth.

Injection pressure

In this section, effect of injection pressure on the profiles of thermophysical properties of SCW is discussed in detail. Different injection pressure (23, 24, 25, 26 and 27 MPa) is input into the model under the condition that the injection rate and temperature are kept unchanged. The predicted results are shown in Fig. 5.

Fig. 5
figure 5

Effect of injection pressure on the profiles of thermophysical properties of SCW in offshore wellbores: a SCW pressure; b SCW density; c SCW temperature; d SCW enthalpy

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It is observed from Fig. 5a that effect of seawater on SCW pressure is negligible under various values of injection pressure. Pressure gradient in the seawater section of wellbores is almost equal to that in the formation section of wellbores.

It is observed from Fig. 5b that (a) the density gradient in the seawater section of wellbores is always larger than that in the formation section of the wellbores under various values of injection pressure. (b) SCW density increases with increasing of SCW pressure. This is because the volume becomes smaller under a higher pressure. As a result, the flow velocity decreases with increasing of density.

It is observed from Fig. 5c that (a) the temperature gradient in the seawater section of wellbores is always larger than that in the formation section of wellbores under various values of injection pressure. (b) SCW temperature increases with increasing of injection pressure. This is because when the injection pressure is small, the SCW density is small. At this point, heat loss has an obvious influence on temperature drop of SCW with a small density.

It is observed from Fig. 5d that (a) the enthalpy gradient in the seawater section of wellbores is always larger than that in the formation section of wellbores under various values of injection pressure. (b) The SCW enthalpy decreases with increasing of injection pressure. This is because the pressure is decreasing while the injection temperature is kept unchanged at wellhead.

In conclusion, heat loss has an obvious influence on temperature drop when SCW is sparse in volume. In order to bring more heat to well-bottom (reservoir condition), a smaller injection pressure is recommended.

Injection temperature

In this section, effect of injection temperature on the profiles of thermophysical properties of SCW is discussed in detail. Different injection temperature (690, 700, 710, 720, 730 and 740 K) is input into the model under the condition that the injection rate and pressure are kept unchanged. The predicted results are shown in Fig. 6.

Fig. 6
figure 6

Effect of injection temperature on the profiles of thermophysical properties of SCW in offshore wellbores: a SCW pressure; b SCW density; c SCW temperature; d SCW enthalpy

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It is observed from Fig. 6a that the SCW pressure decreases with increasing of injection temperature. This is because SCW density decreases with increasing of injection temperature, as shown in Fig. 6b.

It is observed from Fig. 6b that (a) the density gradient in the seawater section of wellbores is always larger than that in the formation section of the wellbores under various values of injection temperature. (b) The SCW density decreases with increasing of injection temperature. This is because the SCW volume per unit mass increases under the condition that the SCW pressure is kept unchanged.

It is observed from Fig. 6c, d that (a) the temperature/enthalpy gradient in the seawater section of wellbores is always larger than that in the formation section of wellbores under various values of injection temperature. (b) The SCW temperature increases with increasing of injection temperature. As a result, SCW enthalpy increases with increasing of temperature, as shown in Fig. 6d.

Conclusions

In this paper, a series of works were done to study the effect of seawater on SCW flow in offshore wellbores. Besides, effect of injection parameters of the profiles of thermophysical properties of SCW in wellbores was discussed in detail. Some main findings are summarized below:

  1. (a)

    Effect of seawater on the profiles of SCW pressure in wellbores is negligible. However, the flow of seawater results in a rapid decline in the temperature/enthalpy of SCW in wellbores.

  2. (b)

    Heat loss is the dominant factor of physical parameter distribution in wellbores. When the injection rate is relatively small, heat loss results in a rapid decrease in temperature and enthalpy, which causes the rapid increase in SCW density. As a result, SCW pressure increases rapidly with well depth.

  3. (c)

    Heat loss has an obvious influence on temperature drop when SCW is sparse in volume. In order to bring more heat to well-bottom (reservoir condition), a smaller injection pressure is recommended.

  4. (d)

    The SCW pressure decreases with increasing of injection temperature. This is because SCW density decreases with increasing of injection temperature.