Slip Behavior of Gelled Oil and Restart-Pressure Prediction in High-Water-Cut Inclined Pipelines

Abstract

In high-water-cut inclined gathering pipelines, shutdown may lead to blockage due to the flotation and wall slip of gelled crude oil. This study investigates the slip behavior and critical conditions of gelled crude oil particles on an inclined pipe wall through visualization experiments, mechanical analysis, and the Extended Derjaguin–Landau–Verwey–Overbeek (XDLVO) theory and develops a method for calculating pipeline restart pressure. Experiments conducted using four crude oil samples identified three post-floating behaviors: adhesion, adhesion–slip, and bulk floating aggregation. The measured critical slip temperatures of samples 1#–4# were 33, 29, 36, and 37 °C, respectively, corresponding to 1–2 °C below their gel points. Adhesion represents a safe shutdown condition, whereas adhesion–slip and bulk floating aggregation indicate a risk of oil accumulation and blockage. When the temperature falls below the critical slip temperature, the gelled oil remains stably adhered to the pipe wall. A critical slip temperature prediction model was established using the gel point, oil–water density difference, and wax content as input parameters. The model produced absolute errors of 0.1–0.6 °C, with a mean absolute error of 0.4 °C. By coupling the Sukhov temperature-drop equation with the mechanical equilibrium equation, a restart-pressure calculation tool was developed. For two field restart cases, the predicted pressures showed relative errors of 6.45% and 7.70%, with a maximum absolute error of 0.06 MPa. The proposed method provides a preliminary quantitative tool for shutdown risk assessment, low-temperature transportation boundary determination, and restart-pressure estimation in high-water-cut inclined gathering pipelines.

1. Introduction

Currently, most onshore oilfields in China have entered the late stage of high-water-cut development, with comprehensive water cut generally ranging from 70% to 90% [1,2]. Conventional heating gathering processes face prominent challenges such as high energy consumption and operating costs. Consequently, unheated gathering technology has become a key development direction for surface engineering in high-water-cut oilfields due to its significant energy-saving benefits [3]. However, the operational safety of inclined pipelines under high-water-cut conditions remains a severe challenge [4]. During shutdown, as the temperature drops below the wax appearance temperature, the crude oil gels. Owing to its lower density compared to the water phase, the gelled oil floats to the upper pipe wall. Driven by the tangential component of the net buoyant force, it slips towards the high point and accumulates, easily causing blockages [5]. Therefore, revealing the slip mechanism of gelled crude oil particles in inclined pipelines and establishing a corresponding restart-pressure calculation method are of great significance for ensuring the safe operation of gathering systems in high-water-cut oilfields.

Early investigations into unheated gathering primarily confirmed the feasibility of reducing pipeline operating temperatures through field-scale flow simulator tests [6]. As water cut has progressively risen toward and beyond 90%, the governing physics of low-temperature transport have undergone a qualitative shift: the continuous water phase becomes the dominant heat carrier, while the dispersed oil-phase transitions from an emulsified liquid to a semi-solid gel as the temperature falls below the WAT [7,8]. The operational envelope of unheated gathering is therefore constrained by the coupled effects of ambient temperature, pipeline geometry, water cut, and crude oil composition [2,8]. Among the physical phenomena governing this envelope, wall adhesion has received the most concentrated research attention. A consistent mechanistic picture has emerged: adhesion is triggered when gelation-driven viscosity escalation overcomes the hydrodynamic stripping forces exerted by the flowing water phase [1,9,10]. Zhang et al. [11] formalized this balance within an adhesion force theory relating the wall sticking occurrence temperature to interfacial free energy, water cut, and substrate surface properties. Subsequent work has extended this framework along three directions: Lyu et al. [3] incorporated shear flow effects on the oil/brine/substrate interface; Yin R et al. [12] demonstrated that non-metallic pipelines exhibit adhesion characteristics systematically different from carbon steel; and Qin Y et al. [13] applied XDLVO theory to integrate Lifshitz–van der Waals, Lewis acid–base, and electrostatic double-layer interactions into a unified force balance defining the safety temperature boundary after shutdown. Despite this progressive refinement from empirical observation to interfacial thermodynamics, a critical gap persists: all of the aforementioned adhesion studies were conducted in horizontal configurations. In inclined pipes, the tangential component of buoyancy introduces a driving force for deposited gel to slip along the wall—a coupled adhesion–slip problem that has not been systematically addressed.

The transport of gelled oil particles in a water-filled inclined pipe draws on the broader physics of liquid–liquid two-phase flow yet extends beyond it in important respects. The foundational hydrodynamic framework for dispersed-phase transport was established by Clift et al. [14], whose drag coefficient correlations remain the reference for terminal velocity calculations. Building on this, Xiao [15] developed theoretical models for oil droplet rise in quiescent water, and Liu et al. [16] extended the analysis to gelled crude oil particles, demonstrating through numerical simulation that particle deformation under hydraulic transport is governed by the competing effects of internal yield stress and external hydrodynamic shear. Han et al. [17] further showed that oil droplet spreading dynamics on solid substrates vary strongly with surface wettability, implying that the post-collision behavior of a gelled particle at the pipe wall—adhesion, slip, or rebound—depends on the coupled physicochemical properties of the gel and the substrate. For inclined pipes specifically, the phase distribution becomes asymmetric: water preferentially wets the lower wall while oil accumulates near the upper generatrix [18], yet existing inclined-pipe studies have focused almost exclusively on steady flowing conditions, leaving the quiescent cooldown period—during which gravity-driven phase segregation and wax gelation occur simultaneously—insufficiently characterized.

The restart difficulty of a gelled pipeline is ultimately governed by the mechanical strength of the wax gel that forms during shutdown. As comprehensively reviewed by Yin X N et al. [19], the causal chain from wax molecular precipitation through crystal network formation to a space-spanning yield stress is now qualitatively well established, though quantitative linkages between successive stages remain under active investigation. A convergent finding across multiple studies is that the gel yield stress is controlled not by the total precipitated wax mass alone, but by the coupled influence of wax molecular composition—particularly the long-chained n-alkane fraction—and the thermal history experienced during cooling [20,21]. The presence of a dispersed water phase, characteristic of high-water-cut conditions, introduces additional complexity: water droplets serve as nucleation sites that alter the wax crystal network topology [22], and the resulting gel yield stress varies non-monotonically with water volume fraction and droplet size distribution [23]. Li Y H and Zhao [24] further showed that crude oils differing in asphaltene content develop gels of markedly different strength under identical cooling protocols, implicating interfacial active species in mediating wax–water connectivity. On the rheological modeling front, the classical description treats gelled oil as a viscoplastic material with a well-defined yield stress [25]; however, real waxy gels deviate from this ideal in that yielding is a progressive structural degradation process rather than a singular event [26,27]. Zhao et al. [28] captured the axial stress localization accompanying gel breakdown through a strain-dependent viscosity correlation, and Bao [29] incorporated viscoelastic and thixotropic effects into a modified constitutive framework, which Bao and Zhang [30] subsequently applied to numerically simulate the spatial propagation of gel breakdown during pipeline restart. Fakroun and Benkreira [31] provided the experimental linkage between laboratory rheological parameters and field-scale restart pressure through dimensionless scaling.

From an engineering perspective, Zhang et al. [32] developed a restart-pressure calculation method integrating hydrostatic head, frictional pressure drop, and thixotropic contributions, validated against field data. Jing et al. [5] and Li et al. [4] independently demonstrated through pipeline-scale experiments that shutdown duration and pipeline inclination govern not only the degree of gelation but also the extent of gravitational oil–water-phase segregation. Yin et al. [33] further showed that even for the idealized water-ring transport geometry, the restart-pressure drop evolves with time in a manner dependent on the gel’s prior shear and thermal history. On the numerical side, Kumar and Paso [34] established that gel compressibility and thermal contraction are the dominant factors controlling pressure propagation velocity during non-isothermal restart. Tikariha and Sanyal [35] extended the restart analysis to vertical riser geometries, and Ventura and Guimarães [36] recently quantified the sensitivity of restart pressure to cooling rate and minimum shutdown temperature. A consistent insight from these numerical studies—with direct significance for the present work—is that the restart pressure is most sensitive to the gel yield stress in the near-wall region where flow first initiates, rather than to the volume-averaged gel strength.

The aforementioned studies clearly demonstrate the achievements that have been made to date, as summarized in

Table 1. Summary of representative research progress.

Research Category	Representative References	Core Findings & Contributions	Main Limitations
Low-temperature gathering and wall adhesion behavior	[1,3,9,10,11,12,13]	Verified the technical feasibility of unheated gathering in high-water-cut oilfields; determined the wall sticking occurrence temperature under different working conditions; established a theoretical framework for wall adhesion force calculation based on XDLVO theory; extended the application scope of adhesion model to non-metallic pipeline materials.	Most studies are carried out for horizontal pipeline scenarios; the slip behavior, initiation mechanism and critical conditions of gelled oil on inclined pipe walls under buoyancy drive lack systematic mechanical analysis and quantitative evaluation methods.
Oil–water two-phase flow and particle dynamics	[14,15,16,17,18]	Established classical drag coefficient and terminal velocity calculation systems for bubbles, droplets and solid particles; revealed the phase distribution law of oil–water flow in inclined pipes; clarified the influence of surface wettability on oil droplet spreading and particle–wall interaction.	Existing studies mainly focus on liquid oil droplet systems and have not fully classified and characterized the slip behavior of gelled crude oil particles with yield stress properties in inclined pipeline environments.
Wax precipitation, gelation and rheological properties	[19,20,21,22,23,24,25,26,27,28,29,30,31]	Revealed the formation mechanism of wax crystal networks and the evolution law of gel structure; developed composition-based correlations for yield stress prediction; constructed viscoelastic–thixotropic constitutive models of gelled crude oil; clarified the influence of thermal history and emulsified water on gel mechanical properties.	Rheological studies are mostly based on homogeneous oil-phase systems under uniform cooling; the quantitative mapping relationship between microscopic interfacial interaction and macroscopic slip behavior of gelled oil particles under high-water-cut and radial temperature gradient conditions remains to be established.
Pipeline shutdown and restart-pressure prediction	[4,5,32,33,34,35,36,37,38,39]	Clarified the key influencing factors such as shutdown duration, pipeline topography and cooling rate on restart pressure; proposed restart-pressure calculation methods considering hydrostatic head, frictional resistance and thixotropic degradation; established numerical simulation methods for transient restart process of gelled pipelines.	There is a lack of integrated engineering calculation methods coupling gelled oil slip accumulation mechanism, pipeline thermal hydraulic conditions and inclined pipe geometric characteristics, which cannot directly support the blockage risk assessment and restart-pressure quantitative calculation of high-water-cut inclined gathering pipelines.

. Meanwhile, they also reveal several unresolved challenges that require further investigation. Wall adhesion is now reasonably well understood for horizontal pipes [1,3,10,11,12,13], and restart pressure can be estimated with engineering accuracy for single-phase gelled pipelines through combined yield stress measurement and numerical simulation [5,30,31,32,33,34,35,36]. The critical lacuna—and the focal point of the present study—is the behavior of gelled oil in the inclined, water-filled pipe segments that constitute the operational reality of onshore gathering networks. Specifically, the mechanical condition governing slip initiation of a gelled oil mass on an inclined pipe wall has not been reduced to a quantitative criterion: the constituent forces—net buoyancy along the pipe axis, gel–wall adhesion, and lubrication-mediated friction—have each been studied in isolation, but their synergistic action at an inclined boundary remains an unsolved mechanics problem. Furthermore, the post-initiation behavior—whether the gelled oil creeps, slides intermittently, or accelerates toward the pipe high point—has not been classified into distinct dynamical regimes, making it impossible to predict whether a given shutdown scenario will result in tolerable accumulation or complete blockage. Compounding these particle-scale uncertainties, none of the existing restart models accounts for the localized gel concentration at pipeline high points resulting from buoyancy-driven slip because they assume, without exception, a spatially uniform gel distribution. Bridging this multi-scale gap—from particle-level force balance through slip regime classification to system-level restart-pressure prediction—demands an integrated experimental, mechanical, and computational approach that has not been previously attempted.

Therefore, this paper takes gelled crude oil particles in high-water-cut inclined pipelines as the research object. Through laboratory experiments, mechanical analysis, and numerical calculation, it systematically studies the floating and slip behavior of gelled oil, establishes a prediction model for critical slip temperature, and develops a software tool for restart-pressure calculation. The research results aim to provide theoretical support and engineering tools for the safe shutdown and restart of gathering systems in high-water-cut oilfields.

The main innovations of this study are summarized as follows:

(1)

A mechanistic innovation in assessing the slip behavior of gelled crude oil on inclined pipe walls. Through visualized experiments, this study systematically identifies three typical behaviors of gelled oil after floating to the inclined pipe wall, adhesion, adhesion–slip and bulk floating aggregation, and reveals the mechanical mechanism whereby the tangential component of net buoyancy drives slip while interfacial adhesion force and static friction resist slip.

(2)

Engineering method and tool innovation for restart-pressure calculation. Coupling the Sukhov temperature-drop formula with the mechanical equilibrium equation of the gelled oil plug, this study proposes a complete calculation method for the restart pressure of high-water-cut inclined pipelines and develops dedicated calculation software integrating thermal calculation, yield stress determination, restart-pressure calculation and slip risk assessment.

2. Material and Methods

2.1. Experimental Section

2.1.1. Experimental Oil Sample

Four oil samples (numbered 1# to 4#) were collected from a high-water-cut oilfield in China. All samples were pretreated following standard dehydration procedures: they were sealed, heated to 80 °C, held at this temperature for 2 h, and then left to stand in the dark for 48 h. The physical property parameters and SARA analysis of the oil samples are listed in

Table 2. Basic physical properties and SARA analysis of oil samples.

Specification	1#	2#	3#	4#
Gel point (°C)	35	31	37	38
Density (20 °C), kg/m3	866	868	869	869
Wax appearance temperature, °C	36.70	35.69	44.63	52.24
Peak wax precipitation temperature, °C	20.67	17.81	24.43	26.92
Wax content, wt%	17.33	17.35	16.39	21.68
Saturates/wt%	53.64	52.00	52.59	48.83
Aromatics/wt%	12.38	12.00	13.58	17.97
Resins/wt%	12.38	16.00	12.84	17.19
Asphaltenes/wt%	1.94	4.36	3.46	0.39

Note: 1#, 2#, 3# and 4# denote four different crude oil samples.

. The viscosity–temperature characteristics were measured using an Anton Paar rheometer (Anton Paar GmbH, Graz, Austria). Following the thermal pretreatment described above, each sample was cooled at a rate of 0.5 °C min⁻¹ to the specified test temperature and maintained at that temperature for 30 min before measurement. Apparent viscosities were recorded at shear rates of 10, 20, 30, 40, and 50 s⁻¹, as presented in Figure 1. The gel point and density were determined following SY/T 0541-2009 and GB/T 1884-2020 [40,41], respectively. Wax content, wax appearance temperature, and peak wax precipitation temperature were measured using a TA Q20 differential scanning calorimeter [42].

2.1.2. Experimental Apparatus and Instrumentation

The gelled oil slip apparatus is designed to investigate the buoyancy-driven upward migration and wall adhesion behavior of gelled oil during static pipeline shutdown, thereby providing a scientific basis for ensuring safe pipeline restart. This apparatus consisted of a HAAKE-AC200 thermostatic water-bath system (Thermo Electron (Karlsruhe) GmbH, Karlsruhe, Germany; temperature-control accuracy: ±0.01 °C), transparent acrylic circulation and observation tanks, a high-speed camera, and a polished 304 stainless-steel plate mounted on an adjustable support (Figure 2). The plate inclination was set according to the specific experimental condition. Contact-angle measurements were performed using a Biolin Scientific optical surface tensiometer equipped with a temperature-controlled chamber with an accuracy of 0.01 °C [43].

2.1.3. Experimental Methods

Experimental Steps for Gelled Oil Floating.

(1)

Based on the wall sticking characteristics of the oil, select three temperature points above the wall sticking occurrence temperature and three points below the gel point so that the selected points cover the expected transition range from no-slip to slip conditions.

(2)

During preheating, evenly coat the plate wall with the same oil sample and allow it to solidify.

(3)

After preheating, take 1 g of oil sample using a sample spoon and place it at the bottom of a beaker. Then place the stainless-steel plate in the beaker.

(4)

To avoid a temperature drop during the waiting period, adjust the circulating water bath to the desired test temperature and conduct the experiment inside the bath.

(5)

Place the beaker in the circulating water bath and wait for 15 min to observe whether floating occurs.

(6)

If floating occurs, check for slip at that test temperature. Slip is determined from videos captured by the camera: after the oil adheres to the plate, mark its front edge. If the gelled oil covers the mark, it is considered as slip. If slip occurs, continue the experiment at lower temperatures; if no slip occurs, continue at higher temperatures until the boundary temperature for slip is identified.

To evaluate the repeatability of the visual observations, repeated experimental runs were conducted at each selected temperature. According to the original experimental records, 28, 25, 20, and 25 runs were performed for oil samples 1#–4#, respectively. For oil sample 1#, four to seven runs were conducted at each temperature, whereas five runs were conducted at each temperature for oil samples 2#–4#. Each run was evaluated separately using the recorded video and was classified as no floating, adhesion, adhesion–slip, or floating aggregation. The critical slip temperature was defined as the lowest tested temperature at which adhesion–slip behavior was observed in the repeated runs.

Gelled Oil Diameter Measurement.

The gelled oil particle diameter was determined from experimental images using ImageJ software (version 1.54f, National Institutes of Health, Bethesda, MD, USA). After the gelled oil particle floated to the plate surface, the water-bath temperature was reduced to 10 °C below the gel point and maintained for 10 min to ensure stable adhesion. The stainless-steel plate was then removed, the residual surface water was carefully wiped off, and an image of the adhered particle was acquired. Before each measurement, the image was spatially calibrated using a known dimension of the stainless-steel plate located in the same imaging plane as the gelled oil particle. A line was drawn along the known plate dimension using the straight-line tool, and the corresponding actual length and unit were entered through the “Analyze–Set Scale” function to establish the pixel-to-length conversion.

For images with a clear contrast between the particle and the background, the image was converted to 8-bit grayscale and the particle region was identified using the “Image–Adjust–Threshold” function. For images with irregular or indistinct boundaries, the particle contour was manually traced using the polygon-selection tool. The projected area was then obtained using the “Analyze–Measure” function under the calibrated spatial scale. For each oil sample, two separately acquired images were analyzed, and each image was measured three times using the same image-processing criteria. Thus, six diameter determinations were obtained for each oil sample. The particle diameter was reported as the mean ± standard deviation (n = 6).

Contact-Angle Measurement.

(1)

Contact angles of water, glycerol, and diiodomethane on the solid surface were measured by the sessile drop method at ambient pressure (Figure 3). The three probe liquids were chosen according to the Hollander criterion [44]. The surface-tension parameters of the probe liquids were checked using reference values before the surface free energy calculation. At 20 °C and atmospheric pressure, the surface tensions of deionized water, glycerol, and diiodomethane were taken as 72.8 ± 0.1, 64.0 ± 0.1, and 50.8 ± 0.1 mN/m, respectively. The corresponding temperature-dependent surface-energy components were used at the actual test temperatures.

(2)

Prior to each measurement series, the solid substrate and sample cell were sequentially cleaned with petroleum ether, anhydrous ethanol, and distilled water. The sample stage was leveled, and the camera focus and aperture were adjusted until a clear and stable droplet profile was obtained. The temperature-controlled chamber had an accuracy of 0.01 °C and was allowed to reach the prescribed test temperature before measurement.

(3)

A computer-controlled dispensing unit was used to generate droplets of constant volume. After each droplet stabilized, its profile was recorded using the magnified camera, and the droplet contour was extracted to determine the contact angle. For each combination of solid surface, probe liquid, and test temperature, at least three separately generated droplets were measured. The contact-angle values are reported as the mean ± standard deviation, with the standard deviation representing the between-droplet repeatability of the measurement.

2.2. Theoretical Calculation Methods

The theoretical framework adopted in this study was selected by considering physical interpretability, parameter availability, computational efficiency, and engineering applicability. The shutdown and restart of waxy crude oil pipelines may involve transient heat transfer, pressure-wave propagation, thixotropic structural evolution, oil–water redistribution, and progressive yielding of the gelled oil. Comprehensive models that incorporate all these processes generally require numerous rheological parameters and detailed transient boundary conditions. In contrast, the present study focuses on the local flotation and wall-slip behavior of gelled oil and the rapid estimation of restart pressure in short, small-diameter, high-water-cut gathering pipelines. Therefore, a simplified framework combining force balance, interfacial adhesion, temperature-drop calculation, yield stress estimation, and overall plug yielding was adopted. The physical framework can be transferred to other oil–water–wall systems after the corresponding material and interfacial parameters are remeasured, whereas the empirical correlations used in the framework require independent validation or recalibration when the crude oil properties or operating conditions change.

2.2.1. Force Analysis of Gelled Oil Particles

The force-balance model was selected because the observed slip behavior is mainly governed by the competition between the tangential component of the net buoyant force and the resistance generated by wall adhesion and static friction. Compared with a complete multiphase transient-flow or computational fluid dynamics model, the present formulation directly describes the local slip-initiation condition and requires fewer input parameters. The required parameters, including particle diameter, phase densities, wall inclination, adhesion force, and friction coefficient, can be obtained experimentally or from basic physical property measurements. Therefore, the model is suitable for rapid assessment of gelled oil slip under quasi-static shutdown conditions.

Assuming that the gelled oil particle can be represented by an equivalent sphere, the net buoyant force was calculated as the difference between the buoyancy and gravity of the particle, following the conventional force formulation for particles immersed in fluids [17].

$$F_g=F_f-G={\displaystyle \frac{\pi }{6}}\left({\rho }_w-{\rho }_o\right)d^{3}g$$

(1)

where ρ_w is the density of water, kg/m³; ρ_o is the density of oil, kg/m³; g is the acceleration due to gravity, 9.8 m/s²; and d is the diameter of the gelled crude oil particle, m.

For the calculation of adhesion force, the surface free energy components of the gelled oil layer and the stainless-steel plate are first determined using the van Oss method. Based on these components, the work of adhesion between the gelled oil and the pipe wall is then calculated using the Extended Derjaguin–Landau–Verwey–Overbeek (XDLVO) theory [16,45,46,47].

The XDLVO theory was employed because it relates the adhesion interaction to measurable surface-energy components and distinguishes the Lifshitz–van der Waals and Lewis acid–base contributions. Compared with assigning a constant empirical adhesion force, this approach provides greater physical interpretability and allows the adhesion interaction to be recalculated when the crude oil, aqueous medium, or wall material changes. Nevertheless, its application requires new contact-angle measurements for each oil–water–wall system.

The support force from the wall on the gelled oil block is the normal reaction force, perpendicular to the pipe wall. When in equilibrium, its magnitude should be the sum of the adhesion force and the radial component of the net buoyant force.

$$F_n=F_a+F_{g}cos\alpha$$

(2)

where θ is the angle between the pipe wall and the horizontal plane.

The limiting static friction at the onset of slip was calculated using the classical Coulomb friction relation [48].

$$f=\mu F_n$$

(3)

where μ is the coefficient of static friction. The magnitude of the static friction depends on the normal force, F_n, which is in turn determined by the adhesion force and the radial component of the net buoyant force.

Equations (1)–(3) are based on general buoyancy, mechanical-equilibrium, and Coulomb friction principles. Therefore, their mathematical forms are not restricted to a specific crude oil. However, the model is not parameter-free. The values of particle diameter, oil and water densities, adhesion force, static friction coefficient, wall inclination, and surface condition are system-dependent and must be determined for the specific crude oil, aqueous phase, and wall material. Accordingly, the force-balance framework has physical transferability, whereas direct use of the parameters obtained in the present experiments for other oilfields or pipe-wall conditions is not recommended.

2.2.2. Restart-Pressure Calculation

To estimate the restart pressure under the engineering conditions considered in this study, the critical slip temperature and the minimum temperature along the pipeline are first determined. The yield stress of the gelled oil in the blocked section is then estimated, followed by calculation of the pressure difference required to initiate the overall yielding of the gelled oil plug. This sequential approach was selected because its required inputs, including crude oil properties, pipeline geometry, operating temperature, mass flow rate, and blockage length, are generally available in field operations. Compared with complete transient restart models, it requires fewer time-dependent rheological parameters and is more convenient for rapid engineering assessment.

First, the critical slip temperature calculation formula for the oil was used to obtain the predicted critical slip temperature. Then, under the one-dimensional steady heat-transfer assumption, the axial temperature distribution was calculated using the Sukhov temperature-drop equation [49].

$$T_L=T_0+\left(T_R-T_0\right)e^{-{\displaystyle \frac{K\pi D}{G\mathrm{c}}}L}$$

(4)

where T_L is the oil temperature at distance L from the starting point, °C; T₀ is the natural ground temperature at the center burial depth, °C; T_R is the oil temperature at the pipeline starting point, °C; K is the overall heat transfer coefficient of the pipeline, W/(m²·°C); D is the outer diameter of the pipeline, m; G is the mass flow rate, kg/s; c is the specific heat capacity of the oil at the average transportation temperature, J/(kg·°C); and L is the length of the pipeline for heating transportation, m.

The Sukhov temperature-drop equation was selected because it provides an analytical relationship between the axial temperature distribution and routinely available parameters, including the overall heat-transfer coefficient, pipe diameter, mass flow rate, specific heat capacity, and environmental temperature. Compared with transient numerical heat-transfer models, it has a lower computational cost and requires fewer thermal boundary conditions, which facilitates its integration into an engineering calculation tool.

For safety, the yield stress corresponding to the minimum temperature of the blocked section is usually used for calculation. The yield stress of gelled oil at different temperatures can be obtained using a regression formula [50,51]:

$${\tau }_y=a{(T_n-T_x)}^b\exp \left[c{({\displaystyle \frac{T_n-T_y}{T_n}})}^d{({\displaystyle \frac{{\delta }_{\mathit{p}\mathit{r}\mathit{e}\mathit{c}\mathit{i}\mathit{p}\mathit{i}\mathit{t}\mathit{a}\mathit{t}\mathit{i}\mathit{o}\mathit{n}}}{{\delta }_{Total}}})}^e\right]$$

(5)

where τ_y is the yield stress of gelled oil, Pa; T_n is the gel point temperature of crude oil, °C; T_x is the wax peak temperature of crude oil, °C; T_y is the yield stress test temperature, °C; δ_{precipitation} is the cumulative wax precipitation amount at the test temperature, wt%; δ_Total is the total wax content of crude oil, wt%; and a, b, c, d, and e are fit parameters from experimental results, with values shown in

Table 3. Fitted parameter values.

Parameter	a	b	c	d	e
Parameter value	2.04	0.93	6.75	0.60	0.36

.

The yield stress correlation was selected because it estimates the gel strength using temperature- and wax-related properties that are more readily available than a complete set of time-dependent rheological data. This reduces the amount of testing required for preliminary engineering calculations. However, the fitted parameters in Equation (5) are empirical. Therefore, although the form of the correlation can be applied as an engineering framework, its coefficients should be independently validated or recalibrated before application to crude oils whose physicochemical properties differ substantially from those used to establish the original correlation.

Unlike the complex restart process in long-distance pipelines, gathering pipelines have the characteristics of small diameter, short blockage length, and low flow rate, and the gelled oil plug forms a continuous gel structure during shutdown. Therefore, when calculating the extrusion pressure, the pressure-wave propagation process and the thixotropic structural degradation of the gelled oil section are not considered. It is assumed that the yield of the gelled oil blockage occurs simultaneously at the same instant, and the following formula is used to calculate the extrusion pressure:

$$\Delta p={\displaystyle \frac{4L_b}{D}}{\tau }_y$$

(6)

where ΔP is the pressure difference across the two ends of pipeline section L, Pa; D is the actual inner diameter of the pipeline, m; and L_b is the length of the pipeline blockage, m.

Overall, the main advantage of the proposed framework is not that it replaces more comprehensive transient models but that it provides a physically interpretable and computationally efficient method for preliminary engineering assessment using parameters that are generally available in high-water-cut gathering systems. Its applicability depends on the validity of the quasi-static force-balance assumption, the accuracy of the interfacial and rheological input parameters, and the similarity between the target pipeline conditions and the conditions considered in this study.

Based on the restart-pressure calculation method, the “High Water Cut Inclined Pipeline Restart Pressure Calculation Software” was developed using the Python programming language. This software integrates four modules: thermal calculation, yield stress determination, restart-pressure calculation, and slip risk assessment, realizing the complete function from inputting basic parameters to outputting calculation results. The software calculation process is shown in Figure 4.

Figure 4. Program block diagram of the restart-pressure calculation software for high-water-cut inclined pipelines.

Figure 4. Program block diagram of the restart-pressure calculation software for high-water-cut inclined pipelines.

3. Results

3.1. Gelled Oil Slip Experiment Results and Mechanical Analysis

Floating and slip experiments were conducted on four oil samples (1#–4#) at temperatures below their respective gel points. The observed phenomena, classified as adhesion, adhesion–slip, or floating aggregation, are summarized in

Table 4. Floating and slip experiment results of gelled oil for four oil samples.

Oil Sample	Test Temperature/°C	Experimental Phenomenon
1#	32	Adhesion
	33	Adhesion–Slip
	34	Adhesion–Slip
	35	Floating Aggregation
2#	28	Adhesion
	29	Adhesion–Slip
	30	Floating Aggregation
3#	35	Adhesion–Slip
	36	Adhesion–Slip
	37	Floating Aggregation
4#	36	Adhesion–Slip
	37	Adhesion–Slip
	38	Floating Aggregation

Note: 1#, 2#, 3# and 4# denote four different crude oil samples.

.

Table 4 shows that the critical slip temperatures for each oil sample are: 33 °C for oil 1# (2 °C below its gel point), 29 °C for oil 2# (2 °C below its gel point), 36 °C for oil 3# (1 °C below its gel point), and 37 °C for oil 4# (1 °C below its gel point). Statistics indicate that the critical slip temperature is generally in the range of 1–3 °C below the gel point.

The experimental results show that the occurrence of gelled oil floating and slipping requires two basic conditions: first, that the oil temperature is below its gel point, providing the basis for gelation; second, that the oil temperature is above a certain critical value, so that the gelled oil has not yet developed a sufficiently rigid structure and can still float and slip under buoyancy. In other words, gelled oil floating and slipping occur in the temperature range where the oil temperature is below the gel point but above the critical slip temperature. For a blocked section, if the actual temperature falls within this range, the gelled oil is prone to slip and accumulate, leading to a high blockage risk. If the actual temperature drops below the critical slip temperature, the gelled oil adheres stably on the inclined section, and the pipeline can be safely shut down.

The behavior of gelled oil after floating to the pipe wall can be divided into three types (see Figure 5): In the adhesion behavior, the gelled oil firmly adheres to the pipe wall without displacement, corresponding to a safe shutdown condition. In the adhesion–slip behavior, the gelled oil slowly and intermittently slips along the pipe wall, potentially converging towards the high point, corresponding to a risky condition. In the floating aggregation behavior, the gelled oil rapidly slips as a whole to the high point of the pipeline, forming a concentrated plug, corresponding to a risky condition [52].

To reveal the mechanical mechanism of gelled oil slip, a force analysis is performed on the gelled crude oil particle adhering to the pipe wall (see Figure 6). The gelled oil is assumed to be a standard sphere, and deformation is neglected. The upward direction along the pipe wall is taken as the positive x-axis, and the direction perpendicular to the pipe wall upward is taken as the positive y-axis.

The gelled oil is subjected to gravity G, buoyancy F_f, adhesion force F_a, normal support force F_n, and static friction f. The mechanical analysis indicates that the driving force for gelled oil slip originates from the tangential component of the net buoyant force, F_g, while the resistance comes from the support force and adhesion force from the pipe wall.

3.1.1. Surface Free Energy and Adhesion Force Calculation

To quantify the interfacial interactions governing gelled oil adhesion and slip, the adhesion force between the gelled oil particle and the pipe wall was systematically analyzed. The calculation procedure involves three steps: (1) measurement of contact angles of probe liquids on the test surfaces, (2) determination of surface free energy components using the van Oss method, and (3) calculation of the work of adhesion and adhesion force based on the XDLVO theory. To calculate the adhesion force for oil samples 1#–4#, the contact angles of three probe liquids—deionized water, glycerol, and diiodomethane—on the surface of the gelled oil layer and on the stainless-steel plate were measured at the critical slip temperature using the sessile drop method. The measured contact angles are presented in

Table 5. Contact-angle measurements.

Test Solid Surface	Test Temperature/°C	θw/°	θg/°	θd/°
1#	33	80.43 ± 1.33	71.83 ± 0.82	32.47 ± 1.01
2#	29	83.26 ± 1.64	78.06 ± 0.88	39.52 ± 1.57
3#	36	86.28 ± 1.46	75.58 ± 1.03	35.12 ± 0.88
4#	37	85.33 ± 1.02	73.26 ± 0.85	34.32 ± 0.97
Stainless-Steel Plate	29	83.99 ± 1.07	68.89 ± 1.19	36.14 ± 0.97
	33	82.38 ± 0.97	67.68 ± 0.92	36.41 ± 1.1
	36	80.94 ± 1.31	66.79 ± 1.17	37.19 ± 0.99
	37	80.43 ± 1.07	66.45 ± 0.87	38.20 ± 1.04

Note: 1#, 2#, 3# and 4# denote four different crude oil samples.

.

Based on the measured contact angles, the surface free energy components of each test solid surface were determined using the van Oss–Chaudhury–Good method. According to the van Oss–Chaudhury–Good surface-energy theory [53], the total surface free energy of a solid is decomposed into the Lifshitz–van der Waals and Lewis acid–base components:

$${\gamma }^{Total}={\gamma }^{LW}+{\gamma }^{AB}={\gamma }^{LW}+2\sqrt{{\gamma }^{+}{\gamma }^{-}}$$

(7)

where γ⁺ and γ⁻ are the electron-acceptor and electron-donor parameters, respectively. The calculated surface free energy components for the gelled oil layer and the stainless-steel plate are summarized in

Table 6. Surface free energy components of test solid surfaces.

Test Solid Surface	Test Temperature/°C	γLW/mN/m	γ+/mN/m	γ−/mN/m
#1	33	6.12	0.10	2.60
#2	29	5.89	0.00	2.76
#3	36	6.07	0.03	1.99
#4	37	6.09	0.019	1.93
Stainless-Steel Plate	29	7.90	0.13	0.51
	33	6.14	0.81	1.72
	36	6.05	0.59	2.13
	37	6.02	0.63	2.16

Note: 1#, 2#, 3# and 4# denote four different crude oil samples.

.

For two solid surfaces interacting across a water medium, the total XDLVO work of adhesion was calculated as the sum of the Lifshitz–van der Waals and Lewis acid–base contributions, following the van Oss–Chaudhury–Good formulation [53,54]:

$$\begin{array}{cc}\hfill W_{OWS}& ={(\sqrt{{\gamma }_O^{LW}}-\sqrt{{\gamma }_W^{LW}})}^2+{(\sqrt{{\gamma }_S^{LW}}-\sqrt{{\gamma }_W^{LW}})}^2+{(\sqrt{{\gamma }_O^{LW}}-\sqrt{{\gamma }_S^{LW}})}^2\hfill \\ \hfill & -2\left[\sqrt{{\gamma }_W^{+}}\left(\sqrt{{\gamma }_O^{-}}+\sqrt{{\gamma }_S^{-}}-\sqrt{{\gamma }_W^{-}}\right)+\sqrt{{\gamma }_W^{-}}\left(\sqrt{{\gamma }_O^{+}}+\sqrt{{\gamma }_S^{+}}-\sqrt{{\gamma }_W^{+}}\right)\right]\hfill \\ & -2\sqrt{{\gamma }_O^{+}{\gamma }_S^{-}}-2\sqrt{{\gamma }_O^{-}{\gamma }_S^{+}}\hfill \end{array}$$

(8)

In Equation (8), the subscripts “O”, “S”, and “W” denote the gelled oil layer, stainless-steel surface, and water medium, respectively.

The calculated work of adhesion values is summarized in

Table 7. Work of adhesion and adhesion forces between gelled oil layer and stainless-steel plate.

Oil Sample	Test Temperature/°C	Work of Adhesion/mJ/m2	Adhesion Force/N
#1	33	66.95	0.0015
#2	29	79.36	0.0013
#3	36	68.82	0.0012
#4	37	66.76	0.0010

Note: 1#, 2#, 3# and 4# denote four different crude oil samples.

. According to the XDLVO theory, the acid–base (AB) interaction is the dominant contribution to the adhesion between the gelled oil and the pipe wall.

The adhesion force characterizes the ability of gelled crude oil particles to adhere to the gelled oil layer on the pipe wall. A larger adhesion force indicates stronger adhesion, making the particle less likely to slip during shutdown. Conversely, particles with a smaller adhesion force are more prone to slip and accumulate at the high point of the pipeline.

3.1.2. Axial Force Analysis

In Figure 5, taking the upward direction along the pipe wall as the positive x-axis, the force in this direction is called the axial force; taking the direction perpendicular to the pipe wall upward as the positive y-axis, the force in this direction is called the normal force. When the axial force F_x > 0, the gelled oil particle slips upward along the pipe wall. By resolving the net buoyant force along the inclined wall and substituting Equations (1)–(3), the axial resultant force was derived in the present study as

$$F_x=Fgsin\alpha -f={\displaystyle \frac{\pi }{6}}\left({\rho }_w-{\rho }_o\right)d^{3}gsin\alpha -\mu F_n$$

(9)

When F_x > 0, the gelled oil particle initiates slip along the pipe wall. However, the theoretically calculated critical condition does not perfectly correspond to the experimentally measured critical slip temperature due to the finite sensitivity of macroscopic displacement detection. The experimentally observed critical slip temperature, determined by the onset of visible displacement of the gelled oil, is therefore slightly higher than the theoretical critical value. Using Equation (9), the axial forces for oil samples 1#–4# at their respective critical slip temperatures were calculated for inclination angles of 15°, 30°, 45°, and 60° (see Figure 7).

The calculated results show that for all oil samples, the axial force, F_x, increases with increasing inclination angle. For example, for oil sample 1#, F_x rises from 0.0010 N to 0.0032 N, an absolute increase of 0.0022 N. For the same oil sample, with a constant particle diameter and density, the tangential component of the net buoyant force increases with angle, resulting in a larger axial driving force. The error bars increase slightly with inclination angle, indicating that the propagated variability in the experimental input parameters becomes more pronounced as F_x increases; however, their relatively limited magnitude does not alter the overall monotonic increasing trend.

Force analysis indicates that the pipeline inclination angle, θ, affects both the driving force (proportional to sinθ) and the normal force (proportional to cosθ), which in turn influences the static friction. As θ increases, sinθ increases while cosθ decreases. The combined effect results in a nonlinear increase in F_x with θ. In the lower inclination range, the increase in driving force dominates, leading to a rapid rise in F_x. At higher inclinations, the reduction in normal force and associated friction partially offsets the increase in driving force, resulting in a slower rate of increase in F_x.

The dominant uncertainty in the measured critical slip temperature arose from the 1 °C temperature interval used near the transition boundary and the finite sensitivity of the visual criterion used to identify the onset of particle displacement. These contributions were more significant than the temperature-control accuracy of the water bath (±0.01 °C). Accordingly, the experimental resolution of the critical slip temperature was approximately ±0.5 °C. Secondary uncertainty sources included irregularities in particle shape, ImageJ spatial calibration and boundary identification, between-droplet variation in the contact-angle measurements, and possible nonuniformity of the pre-coated gelled oil layer [55,56]. Repeated slip tests, six ImageJ determinations for each oil sample, and replicate contact-angle measurements were used to reduce and quantify these random contributions.

The experimental apparatus was designed to isolate the local quasi-static flotation, adhesion, and slip-initiation behavior of a gelled oil particle on an inclined stainless-steel surface, rather than to provide a geometrically scaled simulation of an entire pipeline. In the local force-balance model, particle diameter and wall inclination are included explicitly, whereas pipe diameter is not a direct parameter. Nevertheless, in an actual pipeline, pipe diameter and wall curvature may affect oil–water redistribution, confinement of the gelled oil particles, multiparticle accumulation, and the resulting restart pressure. Therefore, the measured critical slip boundary should be transferred only to systems with comparable crude oil properties, wall materials, inclination angles, and shutdown conditions. Application to pipelines with substantially different diameters or surface conditions requires additional validation.

3.2. Critical Slip Temperature Model

The critical slip temperature is the temperature limit below which the gelled oil begins to exhibit macroscopic slip along the pipe wall. Its value directly affects the shutdown safety assessment of high-water-cut inclined pipelines. Based on the experimental observations and mechanical analysis presented in the previous section, three key factors governing gelled oil slip were identified. The gel point is the characteristic temperature at which the oil loses fluidity, and the critical slip temperature must be lower than the gel point. A larger oil–water density difference results in a larger net buoyant force and stronger tangential driving force, making slip easier. A higher wax content leads to a denser wax crystal network, greater yield stress, and stronger resistance to slip. These three factors were selected as the basic parameters for the prediction model. The model was regressed using the properties of oil samples 1# and 2#, and its accuracy was verified using oil samples 3# and 4#.

To eliminate dimensional effects and reflect physical laws, the following dimensionless quantities are defined:

The dimensionless slip temperature difference, β, is used to characterize the relative difficulty of gelled oil slip, defined as

$$\beta ={\displaystyle \frac{T_s-T_c}{T_S^K}}$$

(10)

where β is the dimensionless slip temperature difference; T_s is the gel point in Celsius, °C; $T_S^K$ = $T_s$ + 273.15, is the gel point in Kelvin, K; and T_c is the critical slip temperature, °C. This ratio represents the relative magnitude of the slip temperature below the gel point. A larger difference between the gel point and the critical slip temperature indicates that the gelled oil can remain stationary over a wider temperature range below the gel point, meaning that the slip temperature is relatively lower. Conversely, a smaller difference indicates that the gelled oil slips just after falling below the gel point, meaning the slip temperature is relatively higher.

The dimensionless density difference reflects the relative magnitude of the buoyancy driving force. Taking the density of water as 1000 kg/m³ as a reference, it is defined as

$$\Delta \rho ={\displaystyle \frac{\Delta {\rho }^{20}}{1000}}$$

(11)

where Δρ is the dimensionless density difference and Δρ²⁰ is the water-oil density difference at 20 °C, kg/m³. From the previous analysis, the tangential component of the net buoyant force is proportional to the density difference. Therefore, Δρ directly reflects the intensity of the driving force for gelled oil floating and slipping. A larger Δρ results in a larger buoyancy force per unit volume of gelled oil and a stronger tangential driving force, making slip easier. Conversely, a smaller Δρ results in a weaker driving force, making slip more difficult.

The dimensionless wax content converts the percentage wax content into a decimal, defined as Equation (12).

$${\varpi }^{\ast }={\displaystyle \frac{\varpi }{100}}$$

(12)

where ${\varpi }^{\ast }$ is the dimensionless wax content and $\varpi$ is the wax content of the oil, wt%. Wax content is a key parameter determining the gel strength of waxy crude oil. During cooling, wax crystals precipitate and form a three-dimensional network structure, giving the gelled oil yield stress and enabling it to resist external forces. The higher the wax content, the denser the wax crystal network, the greater the yield stress, and the stronger the resistance to slip. Therefore, ${\varpi }^{\ast }$ is a dimensionless measure of the structural resistance of the gelled oil. A larger value makes slip more difficult.

From the above analysis, the characteristic stress under critical slip conditions is proportional to $\Delta \rho$. Considering that the density difference variation range is limited, this paper assumes a linear relationship between the characteristic stress and $\Delta \rho$, while expressing the nonlinear effect of wax content in a power-law form, as shown in Equation (13).

$$\beta =A△\rho {({\varpi }^{\ast })}^{\delta }$$

(13)

where A and δ are constants. Taking natural logarithms on both sides and rearranging gives Equation (14).

$$\ln \beta -\ln \Delta \rho =\ln A+\delta \ln {\varpi }^{\ast }$$

(14)

For linear parameter estimation, the auxiliary variable Z was defined in the present study as

$$Z=\ln \beta -\ln \Delta \rho$$

(15)

Using data from oils 1# and 2# at the critical slip temperature, the dimensionless parameters and Z values were calculated and are presented in

Table 8. Calculation results of dimensionless parameters and Z values.

Oil Sample	Δ ρ	ϖ*	ln Δρ	ln ϖ*	ln β	Z
1#	0.134	0.1733	−2.010	−1.753	−5.037	−3.027
2#	0.132	0.1735	−2.025	−1.751	−5.023	−2.998

.

Linear regression of Z against $\ln {\varpi }^{\ast }$ yields δ = −1.256, lnA = −5.508, and A = 0.00407.

Substituting these values into Equation (13) and rearranging gives the final critical slip temperature prediction model:

$$T_c=T_s-0.00407T_s^k({\displaystyle \frac{\Delta {\rho }^{20}}{1000}}){({\displaystyle \frac{\varpi }{100}})}^{-1.256}$$

(16)

Substituting the physical property parameters of oil samples 3# and 4# into the model, the predicted critical slip temperatures were calculated and compared with the experimental values. The predicted values for oil samples 1# and 2# are also listed to assess the model’s fit to the calibration data, as shown in

Table 9. Comparison of predicted and experimental critical slip temperatures.

Oil Sample	Ts/°C	Δρ20/kg/m3	$\mathit{\varpi }$/wt%	Tsk/K	Predicted Tc/°C	Experimental Tc/°C	Absolute Error/°C
1#	35	134	17.33	308.15	33.5	33	0.5
2#	31	132	17.35	304.15	29.5	29	0.5
3#	37	131	16.39	310.15	35.4	36	−0.6
4#	38	131	21.68	311.15	36.9	37	−0.1

.

For oil samples 1# and 2#, which were used for parameter calibration, the absolute prediction errors were both 0.5 °C. For the two validation oils, the absolute errors were 0.6 °C for sample 3# and 0.1 °C for sample 4#. Across all four oils, the absolute errors ranged from 0.1 to 0.6 °C, with a mean absolute error of approximately 0.4 °C.

Although the agreement between the predicted and experimental values is encouraging, the present database is too limited to establish the general applicability of the correlation. In particular, only two crude oils were used to determine the two model coefficients, leaving insufficient statistical information to evaluate coefficient uncertainty, goodness of fit, or model robustness. Moreover, the two validation oils represent only a narrow range of crude oil properties and experimental conditions.

The proposed equation should therefore be regarded as a preliminary empirical correlation applicable only to crude oils with physicochemical properties and experimental boundary conditions close to those investigated in this study. The current samples cover gel points of 31–38 °C, densities of 866–869 kg/m³ at 20 °C, and wax contents of 16.39–21.68 wt%. The correlation has not been validated for crude oils with substantially different compositions, rheological characteristics, cooling or shear histories, particle-size distributions, wall materials, surface roughnesses, aqueous-phase properties, or inclination angles. Additional crude oil samples and independent experimental conditions are required before broader application of the model.

3.3. Restart-Pressure Calculation Results and Verification for High-Water-Cut Inclined Pipelines

After completing the design and development of the restart-pressure calculation software, verification was conducted to ensure the correctness of its theoretical model, the reliability of its algorithm implementation, and its applicability in actual engineering scenarios. The winter, spring–autumn, and summer operating conditions of Pipeline A in a certain oilfield were selected for verification, along with the relevant oil parameters.

The oil sample in Pipeline A has a gel point of 38 °C, a density of 869 kg/m³, a wax peak temperature of 26.92 °C, a total wax content of 21.66 wt%, and a specific heat capacity of 2010 J/(kg·K). The pipeline has a length of 4500 m, an inner diameter of 0.089 m, an inclination angle of 31°, and a water cut of 87%. The operating condition parameters are listed in

Table 10. Operating condition parameters for Pipeline A.

Season	Ground Temp/°C	Starting Oil Temp/°C	Heat Transfer Coeff/W/m2·k	Mass Flow Rate/kg/s	Blockage Length/m
Winter	5	48	0.799	1.59	451
Spring–Autumn	15	47	0.890	1.53	451
Summer	25	46	1.072	1.54	/

, and the calculation results are presented in Figure 8.

The laboratory conditions reproduce the principal local features of the field shutdown condition, including a water-continuous environment, gelled oil particles, an inclined stainless-steel wall, and quasi-static thermal equilibration. The field pipeline considered in the engineering comparison had an inner diameter of 0.089 m, an inclination angle of 31°, and a water cut of 87%. Therefore, the laboratory experiments are representative of the local adhesion and slip-initiation mechanism in high-water-cut inclined gathering pipelines. However, the apparatus does not reproduce all full-scale effects. The field comparison should therefore be regarded as preliminary engineering support rather than a complete full-scale validation [57].

For the winter condition, the minimum temperature along the pipeline is 36.4 °C, which is 0.5 °C below the critical slip temperature of 36.9 °C. According to the criterion established in Section 3.1, the gelled oil should remain stably adhered under this condition. However, given the small margin and potential temperature fluctuations during long-term shutdown, a certain risk of pipe blockage cannot be completely ruled out. At the minimum temperature of 36.4 °C, the static yield stress is 37.70 Pa, and the calculated restart pressure is 0.84 MPa.

For the spring–autumn condition, the minimum temperature along the pipeline is 37.2 °C, and the critical slip temperature is 36.9 °C. Since the minimum temperature exceeds the critical slip temperature, this condition is classified as a risk condition. At 37.2 °C, the static yield stress is 29.71 Pa, and the calculated restart pressure is 0.66 MPa.

For the summer condition, the minimum temperature along the pipeline is 38.6 °C, which is above the gel point of 38 °C. Since the crude oil does not gel under this condition, there is no risk of gelled oil slip or blockage. This condition is therefore classified as no-risk, which is consistent with actual field observations where no pipe blockage occurred during summer operations.

The field test data were compared with the software predictions, and the results are summarized in

Table 11. Error analysis between predicted and actual restart pressures.

Season	Minimum Pipeline Temperature/°C	Static Yield Value/Pa	Actual Restart Pressure/MPa	Predicted Value/MPa	Relative Error
Winter	36.4	37.70	0.78	0.84	7.7%
Spring–Autumn	37.2	29.71	0.62	0.66	6.45%
Summer	38.6	/	/	/	/

.

For the two field restart cases, the relative errors of the predicted restart pressures were 6.45% and 7.70%, with a maximum absolute error of 0.06 MPa. The remaining discrepancy is mainly associated with uncertainty in the blockage length and yield stress estimation, together with the simplified assumptions of one-dimensional steady heat transfer and simultaneous yielding of the gelled oil plug. These two cases provide preliminary field-scale support for the calculation framework under comparable pipeline conditions. Additional field validation covering different pipe diameters, inclination angles, crude oil properties, and shutdown histories is required before broader engineering application [58].

4. Conclusions

This study systematically investigated the slip behavior of gelled crude oil in high-water-cut inclined pipelines and the corresponding restart pressure through visualized experiments, mechanical analysis and numerical calculation. The main conclusions and prospective research directions are summarized as follows:

(1)

The critical slip temperatures of the four tested crude oils all fall within the range of 1–3 °C below their gel points. After floating to the upper pipe wall, the gelled oil presents three typical behaviors: stable adhesion, adhesion–slip transition and bulk floating aggregation, among which only the adhesion state corresponds to a safe shutdown condition. Force analysis indicates that static friction and interfacial adhesion force act as slip resistance, while the tangential component of net buoyancy provides the driving force for upward slip. In future work, machine learning-based image recognition can be introduced to realize intelligent identification of slip states and automatic risk grading of pipeline blockage.

(2)

A prediction model for critical slip temperature was constructed with gel point, oil–water density difference and wax content as core parameters. The model yields a maximum absolute error of 0.6 °C and an average absolute error of 0.4 °C, which can meet the requirements of engineering estimation. Subsequent research can expand the crude oil property database and optimize model parameters through machine learning regression algorithms so as to further improve the universality of the model for different oil types.

(3)

The restart-pressure calculation software developed by coupling the Sukhov temperature-drop formula and mechanical equilibrium equation was verified by field data, with a maximum relative error of 7.7% and a maximum absolute error of 0.06 MPa. It can provide a theoretical basis and engineering tool for safe shutdown judgment and restart-pressure evaluation of high-water-cut inclined pipelines. Follow-up studies can further incorporate factors such as pre-shear history and pipe-wall surface characteristics and build a data-driven prediction framework combined with field operation data to enhance prediction accuracy under complex working conditions.