1. Introduction
Driven by the progressive exploration of offshore oil fields, the construction speed and total mileage of submarine crude oil pipelines have been dramatically accelerated per year [
1,
2]. For new submarine hot oil pipelines that transport waxy and heavy crude oil, pipeline preheating is needed at the initial stage of transportation [
3,
4,
5]. By transmitting a heated preheating medium before crude oil, submarine pipelines and subsea mud are preheated to a predesigned temperature. It avoids the heated crude oil contacting with cold submarine pipes directly after oil injection, which prevents a dramatic temperature drop and viscosity rise of crude oil [
6]. Because of high capacity, low viscosity, and easy accessibility, water can be selected as a preheating medium [
7].
The preheating process is a three-dimensional unsteady flow and heat transfer coupled process in which the pipeline and the surrounding soil continuously absorb heat from the preheating medium [
8]. Before being preheated, fluid in pipelines remains stagnant, while the temperature of the fluid and pipe walls is consistent with the cold subsea environment. Once the hot preheating medium is injected, the internal fluid starts to change continuously in both hydraulic and thermal aspects, until a new steady status forms. Inadequate preheating duration and temperature may weaken the required preheating effect, causing an unexpected increase of viscosity and pressure drop after hot oil transmission. Hence, for a new submarine hot oil pipeline, preheating is indeed a key procedure to decide whether it can be put into production safely and smoothly [
6].
Experimental [
9,
10] and modeling investigations [
11,
12] have been carried out on the thermal change of submarine hot oil pipelines. Based on the similarity theory, Qi [
13] built experimental equipment of submarine oil pipelines to investigate the effect of crude oil temperature, seawater temperature, seawater flow rate, and water content on temperature drop after a pipeline shutdown. Thusyanthan [
14] presented a novel laboratory setup using the thermal imaging technique to simulate the heat loss of offshore pipelines and determine the thermal conductivity of offshore clay. Oh [
15] created a laboratory-scale experimental apparatus to imitate a submarine pipeline partially or fully buried into the seabed. By analyzing the temperature of a pipe’s external surface, the effect of buried depth on the steady-state heat transfer coefficient was investigated. Through a series of research, Chakraborty [
9,
10,
16,
17] illustrated the mechanism of heat loss from offshore buried pipelines by experimental modeling.
Compared with experimental investigations, modeling is more convenient. For solving thermal change during the preheating process, earlier studies have deduced the analytical solution in the theory of linear heat resources or equivalent cylinders [
18,
19,
20]. In recent years, with the development of computing technology [
21,
22], numerical simulation has become the dominating method for studying the unsteady process of buried oil pipelines [
23,
24,
25]. Based on the adopted numerical heat transfer theories, the algorithms commonly used can be divided into the finite difference method (FDM) [
26], the finite element method (FEM) [
27] and the finite volume method (FVM) [
7,
28,
29]. The most serious defect of FDM lies in its poor adaptability to complex regions, and that the conservation of discretization equations is difficult to be guaranteed [
30]. In comparison, benefiting from unstructured meshing technology, FEM and FVM are more adaptable to handle the irregular computational regions of heat transfer problems, which are applied to buried oil pipelines widely [
31,
32]. By combining mesh generation and the numerical discretization scheme, numerical simulation is helpful to rapidly predict the parameter changes under unsteady process. In this way, researchers have modeled the preheating of hot oil pipelines to investigate potential tendencies. The most common issues studied include the change of fluid temperature, the soil temperature field, and heat storage as well as oil injection time [
12,
27]. For example, Wheeler [
33] researched the rule of thermal transmission and dissipation of buried oil pipelines with FEM and FDM. Patience [
20] studied the starting of a compressible Newtonian fluid in small-distance pipes. In the hypothesis of constant physical properties, Vedeneev [
26] presented a thermal analysis of long-distance pipe start-ups with FDM from the aspects of momentum and energy. Xu [
34] treated the soil as a porous medium and built a heat-moisture model for waxy crude oil pipelines. The model was verified by experimental data and used to evaluate the operation safety of another pipeline. Lu [
35] investigated the influence of ambient temperature and preheating water temperature on the soil temperature field of buried oil pipelines via FVM. With the employment of annular thermal influence regions, Xing [
7] established a coupled heat transfer model for crude oil pipeline commissioning and proposed an FDM and FVM combined methodology to solve the temperature of fluid, pipe walls, and soil. An optimization for preheating the Niger crude oil pipeline was performed, too. To accelerate the computational efficiency, Li [
28] adopted the parallel computing and matrix method. In addition, Yu [
36] established a POD–Galerkin reduced-order model based on FVM for the thermal process of buried oil pipelines, including batching transportation and commissioning.
The studies mentioned are all about onshore pipelines. There is relatively less corresponding research on preheating submarine hot oil pipelines. For submarine pipelines, Barletta [
37] transformed the unsteady two-dimensional heat conduction equations into a steady dimensionless form. Solving by FEM software numerically, the rule of heat loss under periodic thermal boundary conditions was performed. Furthermore, he investigated two different start-up cases: The step-rising case and the smooth-rising case [
11]. Different preheating effects and transient operation rules were discussed consequently. By studying the effect of the velocity of bottom seawater on the preheating effect in the Bohai sea, Qi [
38] demonstrated that the upper boundary of the computational region for subsea mud could be simplified to the Dirichlet boundary at the temperature of seawater. Bai [
39] performed a parametric study to evaluate the effects of the thermal power loss, burial depth, pipe diameter, and soil thermal conductivity on the thermal field in the near field of a subsea buried pipeline.
By combining the fluid temperature model with differential equations of unsteady heat conduction, the studies mentioned have established the basic resolving thoughts for preheating submarine hot oil pipelines. It can be concluded that the methods adopted are basically the same as those used to model other unsteady heat transfer problems of buried pipelines, which go through the development procedure from analytical to numerical solution. However, in current simulation research for pipeline preheating processes, one of the biggest problems is the neglect of transient hydraulics due to the main focus on temperature change during preheating. Without the coupled solution of hydrodynamic equations and change in the physical property of the preheating medium [
26], most research are regarding preheating as a quasi-steady state process—only presenting the thermal equations of transported fluid [
27,
38]. In this way, the unsteady hydrodynamics caused by flow and pressure fluctuation cannot be solved simultaneously. The thermal simulation results relevant to hydraulic calculation would also consequently be affected. For example, in reference [
11,
12], the pressure was absent in the fluid heat dissipation model. In reference [
7] and [
36], although hydraulics were involved in the fluid model, the hydro–thermal coupling algorithm was not presented. Hence, it is what is very needed to improve the preheating model for submarine hot oil pipelines and to propose corresponding numerical algorithms.
Besides fluid temperature, the overall heat transfer coefficient is another key indicator for hot oil pipelines to reflect its operation status, such as preheating, shutdown, and restarting [
40,
41]. But it has been not discussed in the above research for preheating operations. Although Zakarian [
40], Sund [
42], Chakraborty [
9] and Magda [
43] et al. have studied the overall heat transfer coefficient and shape factor from partially and fully buried submarine pipelines successively, the studies were all in the view of a steady-state model, which cannot be employed for unsteady preheating directly. Given this, it is necessary to establish a computing expression of the overall heat transfer coefficient for submarine hot oil pipelines based on an unsteady-state model. In this way, the operational status of subsea pipelines and transient thermal propagation can be described.
Considering these facts, this paper establishes a coupled mathematical model for preheating new submarine hot oil pipelines. The numerical algorithm for the coupling resolution of hydrodynamics, thermodynamics, and unsteady heat transfer is provided. Field data was used to verify its accuracy. Furthermore, for analyzing the pipeline operational status during preheating, the temperature, overall heat transfer coefficient, and heat storage are discussed. Based on the designed scenario, an optimization of preheating parameters is performed at the end of this paper.
2. Mathematical Model
The thermal system of submarine pipelines contains the convective heat transfer of fluid in the pipe, the heat conduction of pipe walls and subsea mud. They are affected by a lot of parameters such as temperature and properties of pipe, subsea mud, and seawater. In proposing the mathematical model, the following reasonable assumptions are made in this paper [
7,
36].
(1) The fluid parameter on a fixed pipeline cross-section is assumed to be uniform. Hence, the pressure, flow rate, and temperature in pipelines only depend on the time and axial location of the pipeline.
(2) By neglecting the axial temperature drop of subsea mud, the heat transfer outside pipelines is assumed to be two dimensional.
(3) The computational region of heat conduction outside the pipeline is semi-infinite in nature. For solving the problem, a finite thermal influence region is adopted, as shown in
Figure 1. According to the published literature, the thermal influence region along buried pipelines can be simplified to 20 × 10 m rectangles [
23,
44].
(4) The subsea mud outside the pipeline is modeled as an isotropic medium.
The mathematical model of pipeline preheating consists of hydraulic and thermal coupling, in addition to the thermal coupling of the fluid and the submarine environment. With the simplification mentioned above, the mathematical model is established as follows.
1. Transient flow equation
Equations of continuity, momentum, and energy for liquid are used to model the hydrodynamic and thermodynamic parameters during the preheating operation [
44].
where
a is the velocity of pressure wave and calculated as Equation (2) [
45]
2. Heat transfer equation of submarine pipeline walls.
The submarine pipeline is built in the form of pipe-in-pipe, which is composed of a carrier pipe, insulation, air space, a casing pipe and anticorrosive coating [
34,
46].
In Equation (3), i = 1, 2, 3, 4, 5 represents the inner steel wall, insulation wall, air space, outer steel wall, and anticorrosive coating, respectively.
3. Heat transfer equation of subsea mud.
The differential equation of unsteady heat conduction is applied to model the thermal diffusion in pipe walls and subsea mud [
11].
4. Matching conditions [
34,
36].
The interface between the fluid and the inner steel wall of submarine pipes:
The interface between the adjacent walls of the submarine pipe:
The interface between the outer wall and the subsea mud:
5. Thermal coupling equation
The heat loss from the fluid to the inner wall for the unit area is selected to couple the preheating medium and the external environment.
6. Boundary conditions
During preheating in this case, the inlet flow rate and temperature of the preheating medium are specified, and the outlet pressure is set to be higher than a fixed value so as to ensure the equipment on the downstream platform operates smoothly, as shown below.
Meanwhile, only the right half of the computational region for subsea mud is taken into consideration because of symmetry. So the boundary conditions of the thermal influence region can be given as follows [
7,
36]:
External surfaces of risers in submarine pipelines contact air and seawater directly. Hence, for risers, the Robin condition is employed to model the convective heat transfer between the outer wall and the external environment, as expressed below.
5. Conclusions
For better numerical simulation of the preheating of submarine pipelines, a mathematical model including equations of continuity, momentum, energy and unsteady heat transfer for submarine pipeline preheating was established. Then, a novel model that combined double MOC and FEM, namely DMOC-FEM, was proposed to solve the model numerically. Its accuracy is verified by comparing simulated values with field data and the OLGA simulator. By discussing the temperature, heat storage of subsea mud, heat transfer coefficient and preheating plans, the main conclusions can be summarized as follows:
(1) Compared with the initial status, the temperature of pipe walls after being preheated obviously increases while that of the surrounding subsea mud rises weakly. Meanwhile, the change of heat storage of subsea mud keeps in accordance with fluid temperature during the preheating process. Combining the temperature and heat storage calculation results, it can be concluded that a hot preheating medium mainly warms every pipe wall rather than subsea mud.
(2) It is pointed out in this work that the inverse-calculation method is not suitable for determining the overall heat transfer coefficient during the preheating process. Hence, considering the computational section, temperature of the fluid, pipe walls and subsea mud as well as the heat loss flux, three equivalent overall heat transfer coefficients named K1, K2, and K3 are proposed. Being closely related to the fluid temperature, the equivalent values deduced make up for the deficiency of the inverse-calculation method, and can be an indicator in representing the detailed unsteady fluid and thermal propagation during the preheating process.
(3) Subject to a rated heating power and an extreme preheating flow rate, the preheating parameter at a lower fluid temperature combined with a higher flow rate can produce better preheating effects. Considering the outlet temperature, preheating time and water consumption, the operational parameter of 250 m3/h and 65 °C is the optimal preheating plan among the modeled 11 preheating plans. In comparison with the preheating parameter of 150 m3/h and 70 °C which is adopted in practice, the optimal plan will save the preheating time of 9.2 hours and water consumption of 50 m3
The proposed mathematical model has been proven to be able to provide an accurate simulation for preheating submarine hot oil pipelines. With reasonable boundary conditions, the model is applicable for onshore crude oil pipelines, too. However, preheating is just the first step in pipeline commissioning which is followed by oil transportation. Due to the difference in temperature, the flow rate and physical properties between crude oil and the preheating medium, after oil injection, the parameters along pipelines will be redistributed. The transient process after oil injection can be further studied based on the proposed methodology in this work.