Coupling Changes in Pressure and Flow Velocity in Oil Pipelines Supported by Structures

Abstract

To investigate the time-varying influence of oil viscosity and water content on flow behavior in crossing pipelines, we developed a three-dimensional finite element/CFD model using advanced simulation software with fluid dynamics capabilities. Simulations were performed under varying viscosity and water-cut conditions, and the analyses covered fluid velocity, pressure distribution, and secondary flow characteristics. The results show clear quantitative trends: in the horizontal span, the stabilized centerline velocity reached 2.46 m/s (+23.0% versus the 2.00 m/s inlet). At Node 10, increasing viscosity from 0.306 to 0.603 Pa·s reduced the mean pressure by 11.2 kPa (−11.2% relative to a 0.10 MPa baseline), and a further increase to 1.185 Pa·s produced an additional 4.5 kPa (−4.5%) drop. At Node 1, the low-viscosity case yielded a centerline velocity 1.1× higher than the high-viscosity case (+10.0%). Consistent with these observations, higher viscosity and water cut decreased the average flow velocity and lengthened the duration of pressure fluctuations. These findings provide quantitative insight into the dynamic behavior of multiphase flow and offer a basis for understanding fluid–structure interaction phenomena in crude oil pipeline transport systems.

1. Introduction

Pipeline transportation is a reliable and economical mode in the oil and gas industry. In structure-supported (“crossing”) pipelines, however, unsteady wall loads can trigger FSI responses [1,2]. Conventional design often treats the fluid as added mass and uses lumped hydrodynamics, obscuring how oil viscosity and water cut shape pressure–velocity coupling, pressure losses, and bend-induced secondary flows. To address this gap, we develop a 3-D FE/CFD framework that, under controlled variations of viscosity and water cut, quantifies the spatio-temporal coupling and extracts design-relevant indicators—pressure drop, velocity peaks, and secondary-flow transitions—for FSI-aware operation. Recent advances increasingly adopt digital-twin and data-driven methods (fusing high-fidelity CFD/FEM with streaming data and using ML/PIML surrogates to predict pressure drop, velocity profiles, and transition markers), yet crossing geometries remain underexplored, and time-resolved evidence on fluctuation duration and the two velocity transitions is scarce.

Numerous theoretical and experimental studies, both domestic and international, have been conducted on the flow characteristics of mediums in pipes. Regarding theoretical research on medium flow, Brauner N [3,4] summarized experiments, models, flow pattern classification criteria, and pressure drop calculation methods for liquid–liquid two-phase flow. Shamsul [5] considered the differences between crude oil and mineral oil in real-world production processes, as well as variations in pipeline inclination angles, providing a detailed review of oil–water two-phase pipe flow experiments, flow patterns, liquid holdup, pressure drop, and reverse flow. SA Ahmed et al. [6] comprehensively reviewed studies on liquid–liquid flow in horizontal pipelines, discussing flow modes, mode transitions (including reverse flow), flow pressure gradients, effective viscosity, and heat transfer. Xiao Bin et al. [7] examined the impact of flow rate on pipeline vibration characteristics by introducing additional mass and considering unidirectional fluid–structure coupling to address pipeline excitation vibrations. Li Jishi et al. [8] discussed the effects of changes in flow rate, pressure, and fluid density on pipeline wet mode characteristics by introducing equivalent additional stiffness, damping, and mass effects. Meng Dan et al. [9] used the Kane method to derive the pipeline vortex-induced vibration equation under the combined effects of internal and external flow, analyzing the influence of flow velocity, fluid viscosity, and pipe span on pipeline stability. Tan LQ et al. [10] established a nonlinear Timoshenko model for coupled vibrations of pipeline transport fluids, studying the amplitude–frequency response of forced vibrations in viscoelastic pipes and verifying the necessity of the coupled Timoshenko model. Uczko J et al. [11] examined the effects of flow rate and pulse frequency on pipeline vibration characteristics, confirming the presence of parametric resonance phenomena. Weng G Y [12] proposed and developed a bidirectional pipe–crude oil FSI dynamic finite element (FEC) model, calculating the displacement and acceleration responses of the pipe wall under FSI and analyzing the first six natural frequencies and principal vibration modes of the FEC model. Zhao J et al. [13] investigated the dynamic characteristics of pipelines under varying fluid velocities and wall thicknesses, finding that vibrations caused by pipeline valves could not be mitigated by reducing fluid inlet velocity, but could be significantly reduced by increasing wall thickness. Liu J B et al. [14] investigated vortex-induced vibration characteristics of flexible tubes at various positions in a cylindrical fluid domain, showing that as the deviation angle between the flexible tube and inlet velocity increased, fluid elastic instability and vibration intensity were more likely to occur. Andrade De M et al. [15] extended the quasi-two-dimensional flow model for fluid transmission in elastic pipes, analyzing the energy transfer and dissipation effects in fluid pipeline systems. Liu Z [16] used ANSYS Fluent to study the impact of crude oil viscosity on liquid dynamic noise characteristics, finding that crude oil viscosity is the primary factor affecting static pressure distribution and transient characteristics in centrifugal pumps. Lower viscosity crude oil leads to greater static pressure variation. Wu et al. [17] investigated the flow characteristics of oil–water central annular flow in a 90° bend pipe, examining the effects of oil–water content, oil viscosity, oil density, and curvature ratio on these characteristics. Tan C [18] proposed a modified correlation between oil viscosity and pressure difference, utilizing 3D computer fluid dynamics simulations and oil–water two-phase flow experiments, calculating flow characteristics of other oil–water mixtures by adjusting the tuning factor. Maklakov [19] developed a mathematical model for the cross-sectional upflow field of stratified oil–water flow under laminar flow conditions, incorporating factors such as pipeline inclination and capillary forces. Li Y [20] applied the two-dimensional gas–liquid flow model to oil–water liquid–liquid flow for the first time, studying the non-Newtonian characteristics of oil–water laminar flow in horizontal pipelines. M Elfaki [21] numerically studied the effects of different crude oil grades on slug characteristics, finding that increased crude oil density leads to more frequent slugs, promoting the formation of liquid slugs with higher translational velocities and shorter wavelengths further upstream near the inlet.

The above-mentioned studies indicate that significant progress has been made both domestically and internationally in understanding flow characteristics; however, research specifically focused on fluid flow dynamics within crossing pipelines remains insufficient. Therefore, it is of significant value to investigate the flow characteristics of the medium within crossing pipelines. Focusing on an oil–water mixed medium, this study develops a three-dimensional finite element model for a trans-oil pipeline to investigate the time-varying effects of oil viscosity and water content on the flow characteristics. The findings provide crucial theoretical insights for understanding the dynamic behavior of crude oil pipelines and fluid–structure interaction vibrations.

2. Theoretical Analysis and Simulation

2.1. Fluid Flow Equation and Model Theory

2.1.1. Flow Control Equation

Fluid flow inside pipelines adheres to the fundamental conservation laws of physics, including the conservation of mass, momentum, and energy [22,23]. In computational fluid dynamics (CFD), the governing equations of the flow field are discretized to obtain the key physical quantities of the flow, such as pressure and velocity. The governing equations of the flow field are expressed as follows:

The mass conservation equation for flow without chemical reaction and source terms states that, over a unit of time, the mass flowing into a control volume is equal to the mass flowing out. The governing equation is given by Equation (1).

$${\displaystyle \frac{\partial \rho }{\partial t}}+\nabla \cdot \left(\rho v_i\right)=0$$

(1)

For incompressible steady flows, the mass conservation equation can be expressed as follows (2):

$$\nabla \cdot v_i=0$$

(2)

The momentum conservation equation represents the sum of all external forces acting on a particle, which is equal to the rate of change of the momentum of the internal fluid with respect to time. The momentum conservation equation in each direction is expressed as a tensor and can be written as Equation (3).

$${\displaystyle \frac{\partial \rho v_i}{\partial t}}+{\displaystyle \frac{\partial }{\partial x_j}}\left(\rho v_i{v}_j\right)=-{\displaystyle \frac{\partial \rho }{\partial x_i}}+{\displaystyle \frac{\partial {\tau }_{ij}}{\partial x_j}}+\rho f_i$$

(3)

The momentum conservation equation for an incompressible fluid under steady flow conditions is expressed as follows (4):

$${\displaystyle \frac{\partial }{\partial x_j}}\left(\rho v_i{v}_j\right)=-{\displaystyle \frac{\partial p}{\partial x_j}}+{\displaystyle \frac{\partial {\tau }_{ij}}{\partial x_j}}+\rho f_i$$

(4)

where the velocity components in the three directions of ν_i are denoted as u, v, and w, respectively. p is the pressure on the outside of the cell. τ_ij represents the nine components of surface shear stress in different directions. fi stands for the volume forces F_x, F_y, and F_z acting on the micro body. The only volume forces considered in this paper are gravity; therefore, F_x = 0, F_y = 0, F_z = −p g.

2.1.2. Turbulence Model

The Reynolds number in this paper exceeds the critical Reynolds number, indicating that the flow enters the turbulent state; therefore, the influence of turbulence should be considered. Turbulent flow has an irregular, multi-scale, and complex structure and strong characteristics of dissipation and diffusion [24]. The selection of the turbulence model is extremely important in the simulation of oil–water two-phase flow. The k-ε model contains many variables and unmeasurable factors. The standard k-epsilon model in Fluent is widely used in the petrochemical industry due to its good convergence speed and relatively low memory requirements.

Turbulent kinetic energy k can be calculated using Equation (5), and turbulent kinetic energy dissipation ε can be calculated using Equation (6):

$$\begin{array}{c}{\displaystyle \frac{\partial }{\partial t}}(\rho k)+{\displaystyle \frac{\partial }{\partial x_i}}\left(\rho k{u}_i\right)={\displaystyle \frac{\partial }{\partial x_j}}\left[\left(\mu +{\displaystyle \frac{{\mu }_t}{{\sigma }_k}}\right){\displaystyle \frac{\partial k}{\partial x_j}}\right]\\ +G_k+G_b-\rho \epsilon -Y_M+S_k\end{array}$$

(5)

$$\begin{array}{c}{\displaystyle \frac{\partial }{\partial t}}(\rho \epsilon )+{\displaystyle \frac{\partial }{\partial x_i}}\left(\rho \epsilon u_i\right)={\displaystyle \frac{\partial }{\partial x_j}}\left[\left(\mu +{\displaystyle \frac{{\mu }_t}{{\sigma }_{\epsilon }}}\right){\displaystyle \frac{\partial \epsilon }{\partial x_j}}\right]\\ +C_{1\epsilon }{\displaystyle \frac{\epsilon }{k}}\left(G_k+C_{3\epsilon }{G}_b\right)-C_{2\epsilon }\rho {\displaystyle \frac{{\epsilon }^2}{k}}+S_{\epsilon }\end{array}$$

(6)

where G_k represents the turbulent kinetic energy generated by the average velocity gradient; G_b represents the turbulent kinetic energy generated by buoyancy; Y_M represents the contribution of wave expansion in compressible turbulence to the total dissipation rate, calculated in terms of the effect of compressibility on turbulence in the k-ε model. C_1ε, C_2ε, and C_3ε are constants; σ_k and σ_ε represent the turbulent Prandtl numbers of k and ε; S_k and S_ε represent user-defined source entries.

2.1.3. VOF Model

The VOF (Volume of Fluid) model [25,26,27] is a surface-tracking method based on a fixed Euler grid. The VOF model is applied to numerical simulations of the interface of two or more immiscible fluids. In the VOF model, the volume fraction a_q represents the proportion of the Qth-phase fluid in the cell volume. The value of a_q determines the composition of the fluid in the cell. When a_q is equal to 1, the cell is completely occupied by the Qth-phase fluid, and when a_q is 0, the cell is occupied by another fluid. An a_q value between 0 and 1 indicates that the cell contains the Qth-phase fluid and other fluids. To solve the continuity Equation (7) for the volume fraction:

$${\displaystyle \frac{\partial }{\partial t}}\left({\alpha }_q{\rho }_q\right)+\left({\alpha }_q{\rho }_q{\overline{v}}_q\right)=S_{a_q}+\sum _{p=1}^n\left(m_{pq}-m_{qp}\right)$$

(7)

where m_pq and m_qp are mass transfers between the p phase and the q phase.

Compared with other models, the VOF model is suitable for dealing with highly complex free surfaces and has better convergence. Therefore, the VOF model was selected in this paper to determine the coupling degree of oil–water two phases and track the changes of the interface.

2.2. Finite Element Calculation Model

In crude oil transportation, the oil is typically transported in an oil–water mixed phase. Upon entering the horizontal section of a crossing pipeline, stable two-phase stratified flow becomes the dominant transport mode. The pressure gradient along the pipeline, the height of the oil–water interface, and the velocity field are crucial in establishing the flow model for complex fluids. This study develops a three-dimensional finite element model for the crossing pipeline.

The model is built upon the mass conservation equation, momentum conservation equation, and turbulence model, with the following assumptions: (1) the fluid in the pipeline is steady and incompressible; (2) heat transfer during the flow process is negligible.

2.2.1. Geometric Model and Meshing

As an example, a finite element calculation model for a crossing pipeline with an inner diameter of 760 mm, a span length of 214 m, and a wall thickness of 14.5 mm was developed. The pipeline model consists of four sections: the ascending section, the arc bending section, the horizontal section, and the descending section. The length of both the ascending and descending sections is 10 m, the arc bending section is a 1/4 arc with a radius of 2 m, and the longitudinal span of the horizontal section is 214 m.

To facilitate the analysis and summarization of the medium flow characteristics in the straddle oil pipeline, the midpoint of the cross section of the ascending section near the pipeline entrance was chosen as the starting node (Node 1). A total of 21 nodes were selected from left to right, with the positions of each node symmetrically distributed in the form of Node 11. The origin O is located at the center of rotation of the bend’s curvature, where θ represents the polar angle. θ = 0 is defined at the entrance of the bend section, and θ = 90° is defined at the exit section.

The three-dimensional model of the crossing pipeline was developed using advanced large-scale finite element simulation software, with the model grid divided into tetrahedral elements. The specific modeling process diagram is shown in Figure 1.

2.2.2. Calculation Settings and Parameter Values

This study investigates oil–water mixed two-phase flow using gravity acceleration set at 9.81 m/s². The standard k-epsilon model and the Volume of Fluid (VOF) model were employed for multiphase flow simulations to model the fluid dynamics within a pipeline. The primary phase in the pipeline was crude oil, with liquid water as the secondary phase. The density of crude oil was set to 889 kg/m³, and its dynamic viscosity was treated as a variable. The density of the water phase was 998.2 kg/m³, with a dynamic viscosity of 1.003 mPa·s. The inlet was defined as a velocity inlet, with the direction perpendicular to the riser section, and the inlet velocity was set to 2 m/s. The outlet was treated as a free-flow boundary, while the walls were modeled using the standard wall function with no-slip conditions. The SIMPLE algorithm was applied for pressure–velocity coupling, and the PRESTO method was used for pressure dispersion. The momentum equation was solved using a second-order upwind scheme, while the turbulent kinetic energy and dissipation rate were solved using a first-order upwind scheme. To ensure stable convergence and reliable simulation data, the time step was set to 0.02 s, with a total of 10,000 steps to guarantee that the fluid fully traversed the pipeline and maintained a stable flow state. A three-level grid refinement was performed (coarse/medium/fine). Key quantities of interest, including the peak centerline velocity at Node 11, pressure drop between Nodes 7 and 15, and cross-sectional velocity skewness, changed by ≤1–3% between the medium and fine grids; therefore, the medium grid was adopted for all reported cases. The Grid Convergence Index (GCI) values and wall y⁺ statistics are provided to support reproducibility.

Based on the actual engineering background [28], five different calculation conditions were simulated to investigate the effects of varying crude oil viscosity and water content. The parameter values for each condition are presented in the

Table 1. Parameter values.

Operating Condition	Crude Oil Viscosity/(Pa·s)	Moisture Content/%
Operating condition 1	0.603	10
Operating condition 2	1.185	10
Operating condition 3	0.306	10
Operating condition 4	0.603	30
Operating condition 5	0.603	50

.

3. Results

3.1. Pressure and Velocity Distribution Cloud Image

3.1.1. Longitudinal Distribution of Cloud Image Along the Pipeline

Through finite element simulations of the flow characteristics of the oil–water mixture in the crossing oil pipeline, the pressure and flow rate under five different operating conditions, as shown in Table 1, were calculated. The results reveal that the distribution of the calculated values in the form of cloud images exhibited similar characteristics across the various conditions. Working condition 3 was selected for detailed analysis. The pressure and flow velocity distributions in the longitudinal arc bending section of the pipeline under working condition 3 are presented in Figure 2.

As shown in Figure 2, the fluid medium flows from the pipe inlet to the bend inlet (Node 4), and the pipe pressure gradually decreases in the direction of fluid flow. The fluid velocity reaches its maximum at the center of the cross section, and the maximum velocity distribution is located at the front end of the circular bend inlet. Due to the friction between the fluid and the pipe wall, the flow velocity near the pipe wall is reduced.

After the fluid enters the left bend section (from Node 4 to Node 7), the pressure near the inside of the bend decreases, and the flow velocity increases. Conversely, the pressure near the outside of the bend increases, and the flow velocity decreases compared to the inlet of the bend section. At the bend outlet (from Node 7 to Node 8), the pressure in the inner area increases, and the flow velocity decreases, while the pressure in the outer area decreases, and the flow velocity increases. The distribution of pressure and flow velocity is reversed compared to the inlet conditions.

After the fluid enters the bend section at the right end (from Node 15 to Node 18), the changes and distribution characteristics of pressure and flow velocity at the bend are similar to those at the left end. However, the pressure at the center section of the right end of the pipeline is significantly lower than that at the left end. Due to the effect of gravity, the maximum flow velocity at the right end is higher than that at the left end, with the maximum flow velocity distribution appearing near the outside of the pipeline.

Based on the results of the theoretical analysis, Figure 2 illustrates the mechanisms of fluid flow characteristics within the pipeline. At the inlet of the circular bend, the fluid near the outer area, influenced by inertial and centrifugal forces, moves along the region with a larger curvature radius, pushing against the outer wall surface. This results in an increase in pressure in the outer area and a decrease in pressure in the inner area [29]. Due to the pressure gradient difference formed at the bend and the frictional resistance at the outer wall, the flow velocity in the outer area decreases, while the flow velocity in the inner area increases. At the outlet of the circular bend, the impact between the fluid and the outer wall of the pipe generates a reaction force on the fluid, causing the pressure and flow velocity near the bend and outer area to change in the opposite direction. This results in a flow phenomenon that is the reverse of the pressure and flow velocity distribution at the inlet.

Taking working condition 3 as an example, two sections—Node 7 to Node 11 and Node 11 to Node 15—along the horizontal section of the pipeline were selected to analyze the cloud image distribution of fluid pressure and flow velocity, as shown in Figure 3.

As shown in Figure 3, the pressure at the left end (Node 7) of the horizontal section of the pipeline is the highest, while the pressure at the right end (Node 15) is the lowest. Due to the resistance losses along the pipeline, the pressure gradually decreases in the longitudinal direction, with pressure changes remaining steady throughout the section from Node 7 to Node 15, without any abnormal pressure distribution. The inlet of the horizontal section (Node 7) is influenced by the pressure change at the outlet of the bend, causing turbulence in the fluid over a certain distance, with significant variations in velocity. As the flow distance along the pipeline increases, the flow state stabilizes, and the region with the maximum flow velocity appears within the horizontal section. The maximum flow velocity stabilizes at 2.46 m/s. The flow of crude oil in the horizontal section of the crossing pipeline is primarily influenced by static and dynamic pressures, with the effect of gravity being relatively small. Additionally, considering the stable pressure variation within the pipe and the interaction between the fluid and the frictional resistance of the pipe wall, the flow state of the oil–water mixture achieves relative stability in the horizontal section.

3.1.2. Cloud Image of Time-Varying Distribution of Cross-Sectional Pressure and Velocity at Key Parts

The distribution of fluid pressure and velocity along the longitudinal direction of the pipeline provides insights into the changes at both ends of the circular bend and the horizontal section. For the horizontal section with relatively stable flow, time-varying cloud images of velocity and pressure distribution over a 5 s period were extracted from the cross section where Node 11 of the pipeline model in working condition 3 is located, in order to further highlight the distribution characteristics of fluid pressure and velocity within the pipe. These results are shown in Figure 4.

As shown in Figure 4, with the continuous inflow of the oil–water mixture, the pressure and velocity distribution of the fluid medium through the key cross section of the pipeline changes significantly over time. At 55 s, just before the oil–water mixture flows through this cross section, the pressure within the pipe shows an irregular distribution. At this point, the flow velocity at the center of the cross section is high and decreases in a linear step shape along the radius toward the pipe wall. From 56 s to 57 s, as the fluid medium passes through the cross section at Node 11, the fluid pressure accumulates with a chaotic distribution, causing the maximum pressure area to shift down the wall, while the region with the highest flow velocity moves toward the middle and lower parts. The gradient of flow velocity reduction no longer follows a linear pattern. Between 58 s and 59 s, the pressure and velocity distribution in the pipeline gradually stabilizes. The pressure decreases progressively from the lower wall to the upper wall, and the flow velocity is higher in the lower part of the cross section. Once the fluid flows stably through the section, the pressure concentrates on the lower wall of the pipeline due to density stratification and the effect of gravity. This pressure difference between the upper and lower walls drives the fluid downward, resulting in an increased flow velocity in the lower half of the pipe.

3.2. Pipe Fluid Velocity Flow Diagram

3.2.1. Velocity Flow Diagram at the Arc Bend

When crude oil flows through a bend in the pipeline, the pipe wall experiences elastic deformation, and the fluid is subjected to both centripetal and centrifugal forces. This leads to changes in secondary flow and induces fluid–structure coupling effects. The secondary flow generates shear forces and vortices within the bend, which increase the frictional losses and energy dissipation of the fluid.

To investigate the formation and development of secondary flow at the curved section, pipeline models consisting of four horizontal sections and eight curved sections were selected for analysis. The entrance of the bending section of the ascending arc pipe was designated as Section 1, with a 30° interval between adjacent sampling sections. Since the secondary flow velocity cloud map extracted from the sampling sections shows similar patterns across different crude oil–water contents and viscosities, the working condition with 10% crude oil–water content and 0.603 Pa·s viscosity was selected as a case study. The velocity distribution at the arc bend is shown in Figure 5.

As shown in Figure 5, the formation and development process of secondary flow in the circular bend pipe is as follows: At the inlet of Section 1 (θ = 0°) of the circular bend, the flow lines bulge towards the right side of the section. Due to the large radial pressure gradient in this region, the radial velocity of the fluid points towards the inner wall. In Section 2 (θ = 30°) of the bend, the pressure near the outer wall is greater than that near the inner wall, causing the fluid near the outer wall to flow both upward and downward, while the fluid near the inner wall flows inward. This generates secondary flow and forms a secondary vortex pair. The flow velocity near the inner wall is high, and the flow lines are densely packed. In Section 3 (θ = 60°), influenced by the pipe’s bending radius, the centrifugal and centripetal forces on the fluid reach their maximum, resulting in extreme flow instability and the highest intensity of secondary flow. Since the flow of incompressible fluid is strongly elliptical, secondary flow is still observed at θ = 90° (Section 4), consistent with findings in the literature [30]. This leads to fluid flow disorder within a certain range of the transverse length of the straight pipe section and alters the flow field distribution.

The secondary flow is also generated by the pressure gradient change and fluid inertia in the curved pipe on the right side of the horizontal section. As shown in Figure 5, Sections 5–8 illustrate the formation and development process of secondary flow. The characteristics of the secondary flow formation and development are similar to those observed in the circular bend pipe on the left side of the horizontal section. However, since the direction of fluid flow aligns with the direction of gravity, the secondary flow intensity at the right bend is weaker.

3.2.2. Velocity Flow Diagram of Horizontal Section of Pipeline

It can also be observed in Figure 6 that the flow area near Section 4 and Section 5 is relatively smooth. To verify whether secondary flow persists over a certain distance along the length of the horizontal section of the pipeline, a flow line cloud image of the section between Node 8 and Node 11 was extracted, as shown in Figure 6.

As shown in Figure 6, since Node 8 is located near the exit of the bend at the left end of the horizontal section, secondary flow still exists due to the turbulence at the exit. The vortex at the section between Nodes 9 and 11 gradually dissipates, and the flow line distribution becomes more uniform. As the fluid flows along the length of the pipeline, the pressure within the pipe becomes evenly distributed, and the fluid state far from the bend exit stabilizes. The intensity of the secondary flow weakens progressively until it disappears, eventually transitioning into primary flow.

3.3. Influence of Viscosity on Flow Rate

In this study, a finite element calculation model was developed for an oil pipeline with a total length of 240 m. The inlet flow rate of the oil–water mixture was set to 2 m/s, with the oil and water mixed in the pipeline modeled as a two-phase fluid. After 200 s of simulation, the oil–water mixture reached a steady state, and the relationship between the fluid flow rate and time was analyzed for the first 200 s. The water content of crude oil was set to 10%, with crude oil viscosities of 0.306 Pa·s, 0.603 Pa·s, and 1.18 Pa·s used for simulation and comparison. These conditions were selected to study the changes in fluid flow characteristics under varying crude oil viscosities.

3.3.1. Influence of Horizontal Pipe Section Viscosity on Flow Rate

Under the influence of three different viscosities of crude oil, the oil–water two-phase mixture flows through the inlet section of the pipeline for 200 s. The 200 s time history curve of flow rate changes at Nodes 8, 9, 10, 12, 13, and 14 is shown in Figure 7.

For the oil–water mixed two-phase flow, the boundary condition was set with crude oil as the main phase, and the initial condition for the calculation assumed the pipeline was filled with crude oil at 0 s. The VOF algorithm for homogeneous oil–water two-phase flow was applied. Therefore, the two-phase flow under various working conditions in Figure 7 all share the same initial flow rate. As shown in Figure 7, when the oil–water mixture flows through the pipeline, it is influenced by the effect of crude oil viscosity on the fluid flow rate. As the flow time increases, the time history curve for the fluid with low viscosity fluctuates significantly, indicating an unstable fluid flow state. As the crude oil viscosity increases, the velocity fluctuations of the fluid medium decrease. In the horizontal section of the pipeline, the greater the crude oil viscosity, the smaller the maximum flow velocity of the medium fluid.

When the oil–water two-phase fluid flows into the pipeline from 0 to 200 s, the maximum velocity of the fluid at each node in the horizontal section occurs around 10 s and increases as viscosity decreases. Over the course of 200 s, the time history curve of the flow velocity at each node varies significantly depending on the node’s location. Nodes farther from the fluid inlet cross section, such as Node 14, first maintain crude oil flow for a period, with the flow velocity reaching its maximum around 10 s; this maximum persists until 90 s. Similarly, the maximum flow velocities at Nodes 13, 12, 10, and 9 are sustained until 80 s, 75 s, 40 s, and 30 s, respectively. Afterward, the velocity of the oil–water mixture flowing through the nodes in the pipe changes twice, with the average velocity decreasing as viscosity increases. Node 8, being close to the outlet of the bend, is influenced by the change in the curvature radius, leading to turbulent flow. As a result, the velocity fluctuates significantly, and the fluid movement characteristics become more complex. It is clear that the effect of crude oil viscosity on fluid velocity within the pipeline should not be ignored.

3.3.2. Influence of Viscosity in Rising/Falling Sections of an Elbow on Flow Rate

The time history curves of Nodes 1, 2, and 3 in the ascending section, as well as Nodes 19, 20, and 21 in the descending section, were extracted over 200 s under the influence of different viscosities, as shown in Figure 8.

As shown in Figure 8, when the oil–water mixture starts to flow at an initial velocity of 2 m/s from the inlet of the pipeline, the viscosity significantly affects the flow velocity, with crude oil of lower viscosity exhibiting higher velocity. When the viscosity is 0.306 Pa·s, the fluid velocity at Node 1 is approximately 1.1 times greater than that at a viscosity of 1.185 Pa·s. As the oil–water mixture flows through Nodes 1, 2, and 3 in the ascending section, the flow velocity fluctuates slightly within a certain range. As the flow distance along the pipeline increases from left to right, the curvature radius of the pipeline changes near Node 3, which is close to the inlet of the bend. This causes the flow velocity fluctuations of the fluid medium to become more pronounced. The velocity fluctuations are more significant in the medium with higher viscosity.

Similarly, when the oil–water mixture flows into the pipeline, the fluid at each node in the descending section experiences a large initial flow rate. For the next 90 s, the flow rate of the fluid at each node remains relatively stable. However, as the oil–water mixture passes through each node, the flow rate changes twice.

3.3.3. Influence of Viscosity on Flow Rate in an Arc Bend Section

The time history curves of Nodes 4, 5, 6, and 7 at the left end, as well as Nodes 15, 16, 17, and 18 at the right end of the arc bend section, were extracted over 200 s, as shown in Figure 9.

As shown in Figure 9, the time history curve of the fluid flow rate in the circular bend section is relatively complex. When the oil–water mixture flows through the left bend section (Node 4 to Node 7), the angle between the flow direction of the fluid and the direction of gravity gradually decreases from 180° to 90°. Under the influence of gravity, the flow velocity of the medium with a viscosity of 0.306 Pa·s continuously decreases and is also affected by the bend’s curvature radius. The velocity of the fluid at this viscosity fluctuates most significantly at Node 6. When the oil–water mixture flows through the right bend section (Node 15 to Node 18), the angle between the flow direction and the direction of gravity decreases from 90° to 0°. Simultaneously, the flow velocity of the medium with a viscosity of 0.603 Pa·s first decreases and then increases, influenced by both the bend’s curvature radius and the viscosity. Additionally, the flow characteristics of the fluid with the same viscosity differ significantly at both ends of the bend. The flow rate in the left bend section fluctuates more frequently, making the flow state more unstable compared to the right bend section.

3.4. Influence of Viscosity on Pressure

To analyze the influence of different viscosities on pressure changes within the pipeline, the oil–water mixture with a working pressure of 0.1 MPa and a water content of 10% was selected as the research object, with an initial velocity of 2 m/s set at the inlet. The fluid pressure under Working Conditions 1, 2, and 3 was calculated, and the pressure time history curves for a total of 20 nodes were extracted. Since the absolute pressure values in the pipeline are large, but the pressure fluctuations are relatively small, the pressure time history curves relative to the working pressure were plotted to clearly illustrate the effect of viscosity on pressure, as shown in Figure 10.

As shown in Figure 10, when the oil–water mixture flows through the pipeline, the pressure at each node exhibits similar characteristics of change, with the initial pressure increasing as the fluid viscosity decreases. The fluid with a viscosity of 0.306 Pa·s experiences the maximum initial pressure at each node. When the oil–water mixture with different viscosities flows through the nodes, the pressure values also change significantly due to the viscosity differences. For example, at Node 10, when the viscosity increases from 0.306 Pa·s to 0.603 Pa·s, the average pressure decreases by 11,200 Pa; when the viscosity increases from 0.603 Pa·s to 1.185 Pa·s, the average pressure decreases by 4500 Pa. The total time history range of the fluid pressure in the pipeline increases as the viscosity increases. The lower the viscosity of the crude oil, the greater the pressure change over a short period of time. It is clear that the viscosity of the fluid medium has a non-negligible effect on the pressure change within the pipeline.

3.5. Influence of Moisture Content on Flow Velocity

In this study, crude oil with a viscosity of 0.603 Pa·s was used as an example. When the inlet velocity of the oil–water mixture was 2 m/s, the water content of the crude oil was set to 10%, 30%, and 50%, respectively, to simulate the flow characteristics of the two-phase oil–water mixture in the pipeline.

3.5.1. Influence of Water Content on Flow Velocity in a Horizontal Section

Under the influence of three types of crude oil fluids with different water content, the two-phase oil–water mixture flowed through the inlet section of the pipeline for 200 s. The time history curves of the flow rate changes over 200 s at Nodes 8, 9, 10, 12, 13, and 14 in the horizontal section are shown in Figure 11.

As shown in Figure 11, when the viscosity of crude oil is constant, the maximum velocity reached by the fluid at each node in the horizontal section increases as the water content decreases during the 0–200 s time period, when the oil–water two-phase mixture flows into the pipeline inlet. The maximum velocity for oil–water mixtures with 30% and 50% water content is similar. As the oil–water mixture flows through the nodes, the fluid velocity changes twice, and the average velocity decreases with the increase in water content. The fluid velocity decreases significantly with 50% water content and stabilizes at a lower value. This is due to the increase in water content, which alters the interaction forces between water and oil molecules in the mixture. The force between water molecules becomes stronger, while the interaction between water and oil molecules weakens, leading to this flow characteristic [31]. It is evident that the flow velocity in the pipeline for crude oil with different water contents cannot be ignored.

3.5.2. Influence of Rising Pipe/Falling Pipe Moisture Content on Flow Rate

The time history curves of Nodes 1, 2, and 3 in the ascending section, as well as Nodes 19, 20, and 21 in the descending section of the pipeline, over 200 s are shown in Figure 12, when extracting crude oil with different water contents.

As shown in Figure 12, when the oil–water mixture starts to flow at an initial velocity of 2 m/s from the inlet of the pipeline, the water content significantly affects the flow velocity. As the oil–water mixture flows through Nodes 1, 2, and 3 in the ascending section, the flow velocity fluctuates slightly within a certain range. As the flow distance along the pipeline from left to right increases, the curvature radius of the pipeline changes near Node 3, which is close to the inlet of the bend. This causes the flow velocity fluctuations of the fluid to become more pronounced. The fluctuation of fluid velocity is more noticeable in mixtures with lower water content.

Similarly, when the oil–water mixture enters the pipeline, the fluid at each node in the descending section experiences a larger initial flow rate. Over the next 90 s, the flow rate at each node remains relatively stable until it changes twice as the oil–water mixture flows through each node. After the oil–water mixture passes through the node, the average velocity of the fluid decreases as the water content increases.

The time history curves of Nodes 4, 5, 6, and 7 at the left end, as well as Nodes 15, 16, 17, and 18 at the right end of the arc bend section over 200 s are shown in Figure 13.

As shown in Figure 13, the time history curve of the fluid flow rate in the circular arc bend section is relatively complex under different water content conditions. When the oil–water mixture flows through the left bend section (Node 4 to Node 7), the flow velocity of the fluid with this viscosity fluctuates most violently at Node 6, influenced by the curvature radius of the bend. When the oil–water mixture flows through the right circular bend section (Node 15 to Node 18), the fluid velocity of the medium with 10% water content first decreases and then increases. The flow characteristics of the fluid with the same water content differ significantly at both ends of the bend in the horizontal section. The flow velocity in the left bend section fluctuates more frequently, and the flow state is more unstable compared to the right.

3.6. Influence of Moisture Content on Pressure

To analyze the influence of different water contents on the pressure change within the pipeline, the oil–water mixture with a working pressure of 0.1 MPa and a viscosity of 0.603 Pa·s was selected as the research object, with an initial velocity of 2 m/s set at the inlet. The fluid pressure under Working Conditions 3, 4, and 5 was calculated, and the pressure time history curves for a total of 20 nodes were extracted, as shown in Figure 14.

As shown in Figure 14, when the oil–water mixture begins to flow into the inlet of the pipeline, the pressure at each node exhibits similar characteristics of change. Initially, the fluid near the pipeline inlet experiences a large pressure. As the fluid flows, the pressure rapidly decreases and stabilizes within a small range. After the oil–water mixture with different water contents flows through each node, the pressure undergoes a second decrease. The higher the water content of the fluid, the greater the pressure reduction over the 200 s period. Due to pipeline losses, the total pressure reduction at nodes near the pipeline inlet is greater than that at nodes near the outlet, and the higher the water content, the larger the range of secondary pressure change.

4. Conclusions

This study investigated the fluid–flow characteristics of an oil–water two-phase mixture in a structure-supported (crossing) pipeline, focusing on velocity and pressure time history responses under varying crude oil viscosities and water cut. The main findings are as follows.

Sensitivity-wise, viscosity exerts the dominant first-order effect on mean velocity and pressure loss, whereas water cut primarily governs interface redistribution and the duration/intensity of pressure fluctuations, with their interaction becoming more evident near bends and structural supports. In crossing geometries, bend-induced secondary flows intensify pressure–velocity coupling and produce two distinct velocity transitions before reorganizing downstream; as viscosity and water cut increase, the average velocity decreases while pressure losses and fluctuation durations grow. Low-viscosity conditions consistently yield a larger initial flow rate and higher wall-pressure levels at monitoring locations.

In the ascending section, the maximum velocity occurs near the entrance of the bend and pressure decreases along the flow direction. Within the circular bends, higher velocity appears along the inner curvature while the outer curvature experiences elevated pressure; the inner region exhibits comparatively lower pressure. In the horizontal segment, the flow tends to be faster near the pipe invert, maintaining a relatively stable profile as pressure gradually decreases along the span. In the descending section, the maximum velocity shifts toward the outer curvature and is accompanied by a pressure drop along the path.

Decreasing viscosity or water cut increases the initial flow velocity and elevates the wall pressure at representative locations. Conversely, higher viscosity or water cut is associated with a lower average velocity and a broader time window of pressure fluctuations. These consistent trends provide a qualitative basis for design and operation and inform FSI-related risk assessment in crude oil pipeline transport. Practically, the amplified and prolonged pressure fluctuations, together with the two distinct velocity transitions—particularly near bends and structural supports—suggest greater susceptibility to fatigue and vibration-induced damage. Targeted monitoring and shorter inspection intervals are therefore recommended at these locations.