Effect of pre-shear on structural behavior and pipeline restart of gelled waxy crude oil

Abstract

Gelled waxy crude oil is a soft material with very complex rheological behavior that plays a critical role in one of the most concerning problems in the flow assurance of deep-water petroleum development—i.e., pipeline restart. Because the rheological behavior of gelled waxy crude oil is very dependent on shear history, it may hopefully reduce the pipeline blocking risk if a transitory startup of the pump (known as transitory restart) is implemented during the pipeline shutdown period and causes the nearly gelled oil to flow. The essence of transitory restart is the influence of pre-shear on the structural behavior of gelled oil. In the present work, transitory restart was simulated by a rheometer to study the structural behavior of crude oil after being pre-sheared with and without subsequent cooling. Numerical simulations were carried out to investigate the effect of structural behavior on pipeline restart. The results show that the gelled crude oil structure is incompletely recoverable once broken in the isothermal condition. As a benefit from the incompletely recoverable behavior, pre-shear can reduce the structural strength of gelled oil and cause the structure to break down more rapidly. Based on this mechanism, transitory restart can reduce the minimum pressure difference required for successful restart and save restart time. In addition, it is found that the propagation velocity of the yield cross-section gradually decreases during the restart process rather than always being equal to the speed of sound in crude oil. The attenuation of the propagation velocity of the yield cross-section decreases with increasing structure strength of crude oil and increasing restart pressure.

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1 Introduction

Wax, one of the principal components of crude oil, can exist in different forms, which makes waxy crude oils exhibit various rheological behaviors. When wax dissolves in oil in its molecular form, which is the case at high temperatures, the whole oil system behaves as a Newtonian fluid. When the crude oil cools to a temperature called the wax appearance temperature (WAT), the dissolved wax begins to precipitate from the oil. With further decrease in temperature, the precipitated wax turns the oil system from a Newtonian to non-Newtonian liquid. Once the mass percentage of the precipitated wax reaches approximately 2% (wt) of the total oil system, the wax crystals interlock with each other and form a sponge-like network entrapping the liquid oil¹ and the whole oil system turns from a sol to a gel.

Gelled waxy crude oil features the complex rheological behavior of structured fluids, such as yield stress, creep and recoil,^2–4 hysteresis loop,^5–7 structural breakdown and recovery,^2,8,9 etc. Among such behavior, the recoverability of the broken structure, also called the structure reversibility, of the gelled waxy crude oil deserves special attention. Wardhaugh et al.¹⁰ and Chang et al.¹¹ observed that the broken-down structure of a waxy crude oil could be only poorly recovered, so they suggested that it would be more appropriate to call the gelled waxy crude oil a shear-degrading material rather than a thixotropic material. While Rønningsen et al.,² Kané et al.⁴ and Visintin et al.¹² reported that the recoverability of the broken-down gel structure was related to the shear rate imposed to break the gelled oil structure. On the other hand, El-Gamal,¹³ Ding et al.,⁹ Kané et al.,⁴ Singh et al.,¹⁴ Hénaut et al.¹⁵ and Visintin et al.¹² all observed that the formation of the gelled structure could be delayed and the yield stress could be reduced by shearing the waxy crude oil during the development of the wax crystal structure—i.e., in the cooling process. Indeed, this structural strength weakening phenomenon, also called the structure behavior effect of pre-shear, is an expression of the partial recoverability of the structure.

Restartability of a pipeline is among the major concerns in the flow assurance of waxy crude transportation. The rheological behavior of gelled crude oil intrinsically determines the restartability of a pipeline. Taking advantage of the partial recoverability of the gelled waxy crude structure, if possible, could be quite helpful to improve the restartability of a pipeline. Specifically, this involves implementing a transitory startup after a certain period of shutdown and before the final startup of the pipeline. The role of this transitory flow is to break the building-up but weak gel structure and thus help the eventual restart of the pipeline. This is the objective of the present study.

The pressure required to initiate the flow of a pipeline filled with a structured fluid such as gelled waxy crude might be simply predicted by the following equation:¹⁶

ΔP = 4Lτ_y/D (1)

where τ_y is the yield stress, and L and D are the length and diameter of the pipeline, respectively. This equation is derived by simply assuming that the gelled crude oil within the whole pipeline begins to flow at the same moment upon application of a driving force—i.e., based on a force balance assumption. However, the result of this equation is too conservative because the gelled oil at different axial positions of a long pipeline does not yield—i.e., flow initiation does not take place—at the same moment because of the time lag for pressure propagation.^15,17,18

Accurate prediction of the restart process of a gelled waxy crude pipeline requires numerical computation. For this purpose, it is generally assumed that the restart process deals with an isothermal and compressible flow through a pipeline with a constant diameter, and the rheological behavior of the gelled oil varies in the radial and axial directions during the restart process, indeed reflecting the time-dependent structure breakdown behavior of the gelled waxy crude oil.^7,19–25 The mathematical model consists of the continuity equation, the momentum equation and the constitutive equation. According to the dimensionality of the continuity equation and momentum equation, the mathematical models could be divided into a 2D model,²⁰ 1.5D model^22,24,25 and 1D model.^19,21,23 Compared with the 1D model, the 2D model could provide more information on the restart process, such as the radial profiles of flow velocity and rheological properties, but at the cost of longer computing time. Different constitutive equations were adopted to describe the structural breakdown behavior of gelled waxy crude oil, such as the time-dependent Bingham equation,^19,26 the Houska model^22,25,27 and some elasto-viscoplastic thixotropic models.^7,23,24,28 However, most of the studies on numerical computation were focused on the numerical algorithm, with few of them integrating both the real rheological behavior of the crude oil and the numerical algorithm.

In the present work, the effect of the pre-shear on the structural behavior of the gelled waxy crude oil was first studied experimentally to investigate the isothermal recovery of the gelled oil, and microscopic observations were simultaneously made to probe into the mechanism of the macroscopic rheological behavior. The structural breakdown behavior was then quantitatively characterized. Finally, numerical computations of pipeline restart were performed to study the effect of the pre-shear at various shear rates and temperatures on the restart behavior of the gelled waxy crude oil pipeline.

2 Experimental section

2.1 Material

A typical waxy crude oil was used in the present work. The physical properties of the crude oil are listed in Table 1.

Table 1 Physical properties of the studied waxy crude oil

Parameter	Value	Test method
Density at 20 °C (kg m^−3)	863.6	ISO 3675-1998
WAT (°C)	48.6	Differential scanning calorimetry^29–31
Pour point (°C)	37.0	ASTM D5853-11
Gelation point (°C)	36.1	Dynamic rheological measurement^4,32,33
Wax content (wt%)	24.2	Differential scanning calorimetry^31,34
Resins (wt%)	7.9	ASTM D4124-09
Asphaltenes (wt%)	2.9	ASTM D4124-09

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2.2 Sample pretreatment

The rheological behavior of the waxy crude oil is very sensitive to shear and thermal history. For better repeatability of the experimental results, the thermal and shear history was unified by the following procedures: (1) filling the oil sample into several 100 mL sealed reagent bottles; (2) heating the bottles to 80 °C and keeping them isothermally for 2 h so that the oil could adjust to a uniform state; and (3) leaving the oil to cool quiescently and keeping it at room temperature for at least 48 h before being used.

2.3 Rheological experiments

All rheological measurements were performed with a HAAKE MARS III rheometer. The yield stress was measured with an immersion sensor system (vane, FL40), whereas other experiments were performed with a coaxial cylindrical sensor system (rough surface, Z38 Ti P). The temperature was controlled by a HAAKE AC 200 water bath with a control accuracy of 0.01 °C.

The thermal and shear history of a waxy crude oil before pipeline restart was simulated via the procedures shown in Table 2.

Table 2 Procedures for the simulation of the thermal and shear history

Step	Contents	Engineering background
Step 1	Preheat the sample and measuring system to 50 °C, then load the sample into the measuring system and keep isothermally for 10 min	Heat the oil before it is pumped into the pipeline
Step 2	Cool the sample to 37 °C at a rate of 0.1 °C min^−1 while sheared at 30 s^−1	Normal operation of the pipeline
Step 3	Statically cool the sample to pre-shear temperature Tp-s	Shutdown of the pipeline
Step 4	Shear the sample at a shear rate of [small gamma, Greek, dot above]p-s for 30 min (i.e., pre-shear)	Transitory restart
Step 5	Statically cool the sample to 34 °C at a rate of 0.1 °C min^−1	Shutdown of the pipeline
Step 6	Isothermally hold for th, SAOS (strain amplitude 0.05% and frequency 1 Hz) to track the structure recovery process of the sample	—

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Yield stress measurements. After experiencing the thermal and shear history shown in Table 2, the yield stress of the sample was measured via the vane method.^35,36

Structural breakdown behavior measurements. Shear rate was stepwise increased to study the structural breakdown behavior. The shear rate was set to 1 s⁻¹, 2 s⁻¹, 4 s⁻¹, 8 s⁻¹, 16 s⁻¹ and 32 s⁻¹ successively with each step lasting for 30 min except for the first step, which lasted for 1 h.

Microscopic observations. To study the change in the size and morphology of the wax crystal structure due to the pre-shear, sheared oil specimens were taken from the gap between the cylinders and then placed on the stage of a Nikon OPTIPHOT2-POL polarizing microscope, which had been preheated to the test temperature. To obtain a statistically reliable result, at least 10 micrographs covering different visions were taken for each specimen. Microscopic images were analyzed by the Image J analysis software program. The total count, average area, average perimeter, and boundary box fractal dimension of wax crystals determined based on 10 micrographs were used to characterize the wax crystals.

The history simulation conditions and experimental designs were set as shown in Table 3. The repeatability tests of SAOS and stepwise increases in shear rate are shown in Fig. S1 and S2.†

Table 3 History simulation conditions and experimental designs

Experiment	History simulation condition			Test^a
	Tp-s (°C)	[small gamma, Greek, dot above]p-s (s^−1)	th (min)	Yield stress	Structural breakdown behavior	Micro-examination
a Y denotes that the test was conducted.b Step (5) was not applicable.c Step (4) and step (5) were not applicable for Tp-s = 34 °C.
Isothermal recovery^b	35	0/1/10	1/60/120/240/480	Y	N.A.	Y
Effect of the rate of pre-shear^c	34	0	120	Y	Y	N.A.
	35	1/10	120	Y	Y	N.A.
Effect of the pre-shear temperature	36	10	120	Y	Y	N.A.

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3 Results and discussion

3.1 Effect of pre-shear on the oil without subsequent cooling

The loss angle, defined as δ = arctan(G′′/G′), is usually used to evaluate whether a material is viscously (δ > 45°) or elastically (δ < 45°) dominated. Fig. 1 compares the loss angles of the samples after pre-shearing at 1 s⁻¹, after pre-shearing at 10 s⁻¹, and with no pre-shearing. It can be seen that pre-shear significantly breaks the structure down and turns the oil from an elastically dominated (δ ≈ 17.5°) to a viscously dominated material (δ ≈ 47° and 75°). Obviously, the higher the rate of pre-shear, the greater is the loss of elasticity.

Fig. 1 Loss angle variation due to pre-shear at 35 °C.

The gelation of waxy crude oils is attributed to the formation of the 3D network of wax crystals resulted from the attractive forces between the crystals.^2,12,37 Pre-shear can influence both the formation of the wax crystal network, which is partially irreversible,^38,39 and the geometric characteristics (such as count, size, morphology, and so on) of wax crystals, and the latter in turn changes interactions between the wax crystals and then results in different rheological behavior. Fig. 2 shows the microscopic images of the samples non-sheared and pre-sheared at different rates at 35 °C. According to the upper part of Table 4 (t_h = 1 min), analysis of microscopic images using Image J reveals that (1) pre-shear makes the wax crystal smaller; (2) the higher the rate of pre-shear, the smaller the wax crystals becomes; and (3) the boundary box fractal dimension increases with increasing rate of pre-shear, which means that the morphology of waxy crystals becomes more complex.

Fig. 2 Micrographs of samples non-sheared and pre-sheared at different rates at 35 °C.

Table 4 Characterization parameters of wax crystals after isothermal holding for different rest periods for samples non-sheared and pre-sheared at different rates at 35 °C

[small gamma, Greek, dot above]p-s (s^−1)	th (min)	Total count	Area of wax crystal (μm^2)		Perimeter of wax crystal (μm)		Boundary box fractal dimension
			Average	Variance	Average	Variance
0	1	3106	8.24	25.76	9.70	15.39	1.09
1	1	3335	7.79	23.11	9.24	13.74	1.16
10	1	4106	6.51	20.02	8.17	10.53	1.30
0	480	3022	8.30	24.96	9.79	15.21	1.08
10	480	4210	6.43	19.90	8.08	11.07	1.27

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Fig. 3 shows the isothermal recovery of the broken-down structure represented by the modulus and loss angle, which is compared with the structural build-up of the non-sheared sample. This structural evolution may be well described by the exponential correlation proposed by Visintin et al.¹² as shown in eqn (2):

where G represents the modulus, G′ or G′′, at any isothermal holding time t. In eqn (2), a higher value of t_cr means a longer time required to reach the quasi-steady state, and a higher value of n means a milder recovery process. The values of the parameters in eqn (2) are listed in Table 5 with 95% confidence intervals, with G₀ measured directly and others fitted according to the data of Fig. 3(a).

Fig. 3 Evolution of the modulus and loss angle with isothermal holding time for samples non-sheared and pre-sheared at 10 s⁻¹ at 35 °C (symbol represents measured data, and line represents the fitted result using eqn (2)).

Table 5 Fitted parameters of eqn (2) for Fig. 3(a)

[small gamma, Greek, dot above]p-s (s^−1)	G′					G′′
	G′0 (Pa)	G′∞ (Pa)	t′cr (min)	n′ (—)	AADs (%)	G′′0 (Pa)	G′′∞ (Pa)	t′′cr (min)	n′′ (—)	AADs (%)
0	99.24	690.79 ± 8.81	133.41 ± 6.59	0.5869 ± 0.0122	1.50	41.05	142.78 ± 1.80	114.42 ± 7.33	0.5387 ± 0.0143	1.26
10	0.41	104.41 ± 0.84	254.97 ± 3.04	1.3977 ± 0.0138	7.33	1.75	51.50 ± 0.39	206.34 ± 2.89	1.1922 ± 0.0142	2.97

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It is obvious that although the broken-down structure recovers somewhat during the isothermal holding period, it cannot reach the state of the non-sheared oil. From Table 5, it can be seen that the asymptotic values of G′_∞ and G′′_∞ of the sample pre-sheared at 10 s⁻¹ are only 15.1% and 36.1% of the corresponding values of the non-sheared sample. These recovery rates are basically equal to those observed by Ronningsen et al.,² Kané et al.⁴ and Visintin et al.¹²

In addition to the modulus, the yield stress, as shown in Fig. 4, also indicates a weak recovery of the broken-down structure. After isothermal holding for 480 min, the yield stress of the sample pre-sheared at 10 s⁻¹ is 39.6% of the non-sheared sample. Therefore, it is important to note that the recovery rate is not a unique value, and its value depends on the parameter used to evaluate the recovery behavior, such as storage modulus, loss modulus, loss angle and yield stress. However, no matter which parameter is used, all results show that the broken-down structure of a waxy crude oil could be only partially recovered.

Fig. 4 Yield stress versus time elapsing at 35 °C.

For the samples non-sheared and pre-sheared at 10 s⁻¹, the micrographs of wax crystals were taken after isothermal holding 480 min at 35 °C, and the results are shown in Fig. 5. By comparing the statistical analysis results of Fig. 5 with those of Fig. 2, i.e. Table 4 and Fig. 6, it may be found that the count, average size and size distribution of wax crystals change little after isothermal holding for 480 min at 35 °C. This illustrates that the partial recovery of the broken-down structure during the isothermal holding period results from the evolution of the interlocks of particles rather than the changes of the geometric characteristics of particles.

Fig. 5 Morphology of wax crystals after isothermal holding for 480 min for samples non-sheared and pre-sheared at 10 s⁻¹ at 35 °C.

Fig. 6 Size distribution of wax crystals after isothermal holding for 480 min for samples non-sheared and pre-sheared at 10 s⁻¹ at 35 °C.

3.2 Effect of pre-shear on the oil with subsequent cooling

Fig. 7 shows the aging processes of samples non-sheared and pre-sheared at different rates at 35 °C/36 °C and subsequent cooling to 34 °C. The structural evolution during the aging processes is quantitatively characterized by eqn (2), and the values of the parameters are listed in Table 6 with 95% confidence intervals. Owing to the partial recoverability of the waxy crude oil structure as verified in the previous section, the structural strength of samples pre-sheared with subsequent cooling becomes weaker than the non-sheared sample, the aging process becomes slower, and the elapsed time reaching the quasi-steady state becomes longer. A higher rate of pre-shear makes the relevant changes more significant, whereas a larger temperature drop after pre-shearing weakens the relevant changes.

Fig. 7 Evolution of the modulus and loss angle with isothermal holding time for samples non-sheared and pre-sheared at different rates at 35 °C/36 °C and subsequent cooling to 34 °C (symbol represents measured results, and line represents fitted result by eqn (2)).

Table 6 Fitted parameters of eqn (2) for Fig. 7(a) and (b)

Tp-s (°C)	[small gamma, Greek, dot above]p-s (s^−1)	G′					G′′
		G′0 (Pa)	G′∞ (Pa)	t′cr (min)	n′ (—)	AADs (%)	G′′0 (Pa)	G′′∞ (Pa)	t′′cr (min)	n′′ (—)	AADs (%)
34	0	289.14	1211.25 ± 9.92	30.04 ± 1.01	0.6511 ± 0.0122	1.62	94.22	227.79 ± 1.01	18.85 ± 0.51	0.5986 ± 0.0117	1.02
36	10	45.70	760.90 ± 11.08	37.16 ± 1.36	0.9224 ± 0.0271	1.56	27.71	199.67 ± 3.26	27.92 ± 1.42	0.9120 ± 0.0483	1.6
35	1	28.93	528.23 ± 3.59	55.63 ± 0.77	1.0393 ± 0.0089	3.39	19.76	160.25 ± 0.99	41.30 ± 0.64	0.9819 ± 0.0122	3.06
35	10	10.72	248.05 ± 1.44	59.67 ± 0.65	1.0792 ± 0.0070	3.46	9.09	87.50 ± 0.35	46.17 ± 0.43	0.9619 ± 0.0060	1.69

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To study the structural breakdown behavior of samples experiencing different history conditions, the stepwise increases in shear rate were imposed, and the results are shown in Fig. 8. The viscoplastic thixotropic model of waxy crude oil proposed by Teng et al.⁶ was used to quantitatively characterize these structural breakdown processes. Teng's model is

Fig. 8 Shear stress responses to stepwise increases in shear rate at 34 °C for samples experiencing different pre-shear conditions.

Teng's model is a viscoplastic thixotropic model, so only the experimental data after stress overshoot were used to determine the model parameters by least square fitting. The contrast between measured results and fitted results is shown in Fig. S3.† The relevant parameters of Teng's model are listed in Table 7 in which τ_y is the yield stress measured by the van method and τ_y1 is the yield stress fitted by eqn (2). There are slight differences between the values of τ_y and τ_y1.

Table 7 Fitting parameters of Teng's model for samples experiencing different pre-shear conditions

Tp-s (°C)	[small gamma, Greek, dot above]p-s (s^−1)	τy (Pa)	τy1 (Pa)	k (Pa s^n1)	Δk (Pa s^n1)	n1 (—)	n2 (—)	a (—)	b (—)	m (—)	AADs (%)
34	0	25.33	26.09 ± 0.11	0.79 ± 0.01	14.92 ± 0.07	0.7583 ± 0.0014	0.8406 ± 0.0006	0.00339 ± 0.0001	0.0181 ± 0.0002	0.9916 ± 0.0026	1.1
36	10	16.01	16.89 ± 0.07	0.61 ± 0.01	8.19 ± 0.04	0.7873 ± 0.0014	0.8353 ± 0.0090	0.0074 ± 0.0020	0.0231 ± 0.0030	0.9906 ± 0.0038	0.8
35	1	13.29	12.67 ± 0.05	0.65 ± 0.01	6.05 ± 0.02	0.7756 ± 0.0011	0.8324 ± 0.0012	0.0063 ± 0.0003	0.0185 ± 0.0002	1.0134 ± 0.0053	1.5
35	10	8.81	7.51 ± 0.03	0.58 ± 0.01	5.05 ± 0.02	0.8025 ± 0.0011	0.8350 ± 0.0011	0.0246 ± 0.0004	0.0567 ± 0.0007	1.0373 ± 0.0060	1.2

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As shown in Table 7, at the same pre-shear temperature T_p-s = 35 °C, a higher rate of pre-shear increases the reductions of yield stress, structure-independent consistency and structure-dependent consistency of the sample at 34 °C; i.e., the stress overshoot during the structural breakdown process is reduced from 41.6 Pa to 17.6 Pa and 12.7 Pa by pre-shearing at 1 s⁻¹ and 10 s⁻¹, respectively. At the same rate of pre-shear [small gamma, Greek, dot above]_p-s = 10 s⁻¹, a larger temperature drop after pre-shearing reduces the reductions of yield stress, structure-independent consistency and structure-dependent consistency of the sample at 34 °C; i.e., the stress overshoot during the structural breakdown process is reduced from 41.6 Pa to 24.0 Pa and 12.7 Pa by pre-shearing at 10 s⁻¹ at 36 °C and 35 °C, respectively. The increase in n₁ with increasing rate of pre-shear indicates that the oil structure becomes less non-Newtonian. In addition, the structural breakdown rate b of the pre-sheared sample significantly increases, which means that the structure of the sample becomes more easily broken down.

Although both the rate of pre-shear and pre-shear temperature are different for the pre-shear conditions of T_p-s = 36 °C and [small gamma, Greek, dot above]_p-s = 10 s⁻¹ vs. T_p-s = 35 °C and [small gamma, Greek, dot above]_p-s = 1 s⁻¹, there is an interesting result regarding the stress responses to stepwise increases in shear rate for these two cases. As shown in Fig. 8, the stress overshoot of the case with T_p-s = 36 °C and [small gamma, Greek, dot above]_p-s = 10 s⁻¹ is higher than that of the case with T_p-s = 35 °C and [small gamma, Greek, dot above]_p-s = 1 s⁻¹, which means that the initial structure of the case with T_p-s = 36 °C and [small gamma, Greek, dot above]_p-s = 10 s⁻¹ is stronger. However, the strengths of both structures become comparable once the structure is broken, and the structural strength of the case with T_p-s = 36 °C and [small gamma, Greek, dot above]_p-s = 10 s⁻¹ becomes even weaker than the case with T_p-s = 35 °C and [small gamma, Greek, dot above]_p-s = 1 s⁻¹ in the higher shear rate steps resulting from the different structural breakdown behavior. This phenomenon illustrates the importance of structural breakdown behavior; in other words, a stronger initial structure does not mean a stronger broken structure.

4 Case studies of gelled waxy crude oil pipeline restart

Based on the rheological experiment results in the previous section, we will study the effect of structural behavior on the pipeline restart.

4.1 Physical model and governing equations

The physical model of pipeline restart is shown in Fig. 9. The pipeline is initially filled with gelled waxy crude oil. A suitable constant pressure difference is then applied to the pipeline inlet, and the structure of gelled oil is broken gradually from upstream to downstream until the steady state is reached.

Fig. 9 Physical model of gelled waxy crude oil pipeline restart.

To simplify the problem, the flow is assumed to be isothermal and in a horizontal pipeline with a uniform cross-section. The gelled waxy crude oil is assumed to be weakly compressible.

The conservation equations of mass and momentum are given, respectively, as²⁴

where V is the average velocity across the sectional area, and

The weak compressibility of gelled waxy crude oil is characterized by a constant isothermal compressibility number:

Substituting eqn (6) into (4), the continuity equation becomes

The flow is assumed to be fully developed, so the shear stress along the radial direction is linear as expressed in eqn (8):

Substituting eqn (8) into Teng's model as shown in eqn (3), the thixotropic model becomes

where

Before the restart, gelled waxy crude oil in the pipeline is assumed to be uniform with the velocity profile v(r, z, t = 0) = 0, pressure field P(z, t = 0) = 0, and structural parameter λ(r, z, t = 0) = 1. Once the restart is initiated, the inlet pressure instantaneously increases to P(z = 0, t) = P_in. The outlet pressure is assumed to be P(z = L, t) = 0 at all times. The no-slip condition is set on the velocity at the pipeline wall—i.e., v(r = R, z, t) = 0.

The relevant parameters of eqn (5), (7) and (9) are nondimensionalized as follows:

where the reference viscosity

The dimensionless governing equations are given by

where

The dimensionless numbers appearing in the dimensionless analysis are shown in Table 8. The algorithm proposed by Negrao et al.²⁴ is adopted to solve eqn (11)–(13) numerically. See ref. 24 for details about the mesh generation, solution of governing equations and algorithm realization. The present work focuses on the effect of the structural behavior of gelled waxy crude oil on pipeline restart, so the dimensionless parameters are set as shown in Table 9 according to the rheological parameters shown in Table 7.

Table 8 Dimensionless numbers appearing in the dimensionless governing equations

Definition	Symbol	Name
Reynolds number	Re	ρ0cd/μr
Aspect ratio	δ	D/L
Yield number	Γ	τy1α
Viscosity ratio	S	Δk/k
Build-up number	Bu	aL/c
Break down number	Bd	bρ0^mL^1−m/c^1−3m

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Table 9 Dimensionless parameters for case studies of pipeline restart

Tp-s (°C)	[small gamma, Greek, dot above]p-s (s^−1)	Re	δ	Γ	S	Bu	Bd	n1	n2	m
34	0	52 563.53	5.00 × 10^−4	1.30 × 10^−6	18.89	0.0222	3.1932 × 10^5	0.7583	0.8406	0.9916
36	10	71 883.33		8.45 × 10^−7	13.43	0.0484	4.0149 × 10^5	0.7873	0.8353	0.9906
35	1	65 994.02		6.34 × 10^−7	9.31	0.0412	4.5197 × 10^5	0.7756	0.8324	1.0134
35	10	77 790.04		3.76 × 10^−7	8.71	0.1608	1.9794 × 10^6	0.8025	0.8350	1.0373

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To ensure the mesh independence of the simulation results, the effect of mesh size on the simulation results is investigated (see Fig. S4 and S5† for more details). The radial grid number N_r = 50 and the axial grid number N_z = 500 are confirmed.

4.2 Effect of pre-shear on pipeline restart

Fig. 10 shows the steady-state outlet velocity under different restart pressures for different pre-shear history conditions. It can be seen that pre-shear can effectively reduce the minimum pressure difference required for successful restart. The minimum pressure difference is approximately 0.0035 for the non-shear condition, which decreases to 0.00215 and 0.00115 when the rate of pre-shear increases to 1 s⁻¹ and 10 s⁻¹, respectively, at pre-shear temperature T_p-s = 35 °C. In other words, the minimum pressure difference required for successful restart is reduced to 61.4% and 32.9% of the non-shear condition, respectively. In addition, the minimum pressure differences are almost the same for the case with T_p-s = 36 °C and [small gamma, Greek, dot above]_p-s = 10 s⁻¹ and the case with T_p-s = 35 °C and [small gamma, Greek, dot above]_p-s = 1 s⁻¹. As mentioned above, the initial structure of the case with T_p-s = 36 °C and [small gamma, Greek, dot above]_p-s = 10 s⁻¹ is stronger than that of the case with T_p-s = 35 °C and [small gamma, Greek, dot above]_p-s = 1 s⁻¹, whereas the structural breakdown rate of waxy crude oil of the case with T_p-s = 36 °C and [small gamma, Greek, dot above]_p-s = 10 s⁻¹ is higher. Both the initial structure and the structural breakdown behavior cause the minimum pressure differences for the two cases to be almost the same, which embodies the importance of the structural breakdown behavior of gelled waxy crude oil on pipeline restart.

Fig. 10 Steady-state outlet velocity versus restart pressure for different pre-shear history conditions.

The times at which the oil at different axial positions starts to flow (called “restart times”) are given as shown in Fig. 11. The restart times are almost the same for axial position z* < 0.2, whereas the curves no longer overlap for z* > 0.2. The higher the rate of pre-shear, the longer is the restart time. The differences in restart times illustrate that the propagation velocities of yield cross-sections (equivalent to the “yield front” proposed by Davidson et al.¹⁹) for different pre-shear history conditions are unequal.

Fig. 11 Restart times at different axial positions (restart pressure = 0.003).

The derivation of curves in Fig. 11 is obtained by the Origin software program to obtain the propagation velocity of yield cross-section and the results are shown in Fig. 12. The velocity of the yield cross-section is equal to the sound speed in gelled oil in the initial period, and it then decreases during the propagation process. The stronger the structural strength, the greater is the velocity decline. In addition to the structural strength of gelled oil, the restart pressure also has an effect on the propagation of the yield cross-section. Fig. 13 shows the propagation velocities of the yield cross-section for the case with T_p-s = 35 °C and [small gamma, Greek, dot above]_p-s = 1 s⁻¹ under different restart pressures. It is clear that the velocity of the yield cross-section is equivalent to the sound speed in gelled oil in the initial period regardless of how much pressure is imposed, and then decreases sharply with decreasing restart pressure

Fig. 12 Propagation velocities of yield cross-section when passing different axial positions for different pre-shear history conditions (restart pressure = 0.003).

Fig. 13 Propagation velocities of yield cross-section when passing the different axial positions under different restart pressures (pre-shear temperature T_p-s = 35 °C and rate of pre-shear [small gamma, Greek, dot above]_p-s = 1 s⁻¹).

By tracking the propagation process of the yield cross-section for different pre-shear history conditions and restart pressures the propagation velocities of the yield cross-section when it reaches the outlet can be obtained as shown in Fig. 14. It can be seen that the propagation velocity of the yield cross-section increases with increasing rate of pre-shear and increasing restart pressure but to no more than the sound speed in gelled oil. Fig. 12, 13 and 14 all indicate that the assumption of Davidson et al.,¹⁹ that the propagation velocity of the yield front is identical to the sound speed in gelled oil during the pipeline restart process, is unreasonable.

Fig. 14 Propagation velocities of yield cross-section when reaching the outlet for different pre-shear history conditions and restart pressures.

Actually, in the initial restart period, the total pressure difference is applied on a short upstream distance, so the structure of the gelled oil is broken rapidly, and the propagation velocity of the yield cross-section is almost equal to the sound speed. As the pressure wave propagates downstream, the pressure is dissipated and the applied pipeline length becomes longer, so the structure of the gelled oil is broken at a relatively slow rate; i.e., the propagation velocity of the yield cross-section decreases. Thus, the assumption that the speed of the yield front is identical to the sound speed in gelled oil during the pipeline restart process applies only to the situations in which the structure is weaker or the restart pressure is sufficiently higher.

The total restart time is also a main concern when working with the restart problem. Fig. 15 shows the total restart times for different pre-shear history conditions and restart pressures It can be seen that the total restart time is close to unit, which is equal to the total restart time for weakly compressible Newtonian flow, provided that the restart pressure is sufficiently higher. The pre-shear can effectively reduce the total restart time under a given restart pressure, even by orders of magnitude.

Fig. 15 Total restart times for different pre-shear history conditions and restart pressures.

5 Conclusions

With the background of transitory restart, the structural behavior of gelled waxy crude oil pre-sheared with and without subsequent cooling was studied in the present work. On this basis, numerical simulations were performed to study the effect of pre-shear on pipeline restart. The following conclusions can be drawn:

(1) The broken-down structure of a waxy crude oil could be only partially recovered in the isothermal condition. The pre-shear makes the wax crystals smaller and the morphology of wax crystals more complex. The changes in the count and size of wax crystals are unobvious during the isothermal aging period after pre-shearing at 10 s⁻¹, whereas the storage modulus, loss modulus and yield stress all weakly recover until becoming steady. The recovery rates of the storage modulus, loss modulus and yield stress are 15.1%, 36.1% and 39.6%, respectively.

(2) For the conditions in which the oil samples were pre-sheared with subsequent cooling, at the same pre-shear temperature, a higher rate of pre-shear increases the reductions in yield stress, structure-independent consistency and structure-dependent consistency of the sample; i.e., the stress overshoot during the structural breakdown process is reduced from 41.6 Pa to 17.6 Pa and 12.7 Pa by pre-shearing at 1 s⁻¹ and 10 s⁻¹, respectively; and makes the oil structure less non-Newtonian as well as the structural breakdown process faster. At the same rate of pre-shear, a larger temperature drop after pre-shearing reduces the reductions in yield stress, structure-independent consistency and structure-dependent consistency of the sample; i.e., the stress overshoot during the structural breakdown process is reduced from 41.6 Pa to 24.0 Pa and 12.7 Pa by pre-shearing at 10 s⁻¹ at 36 °C and 35 °C, respectively.

(3) The propagation velocity of the yield cross-section during the restart process is not yet identical to the sound speed in gelled oil. Actually, the propagation velocity is equal to the sound speed in the initial restart period and then decreases during the propagation process. The propagation velocity even decreases to 1% of the sound speed in gelled oil. The attenuation of the propagation velocity of the yield cross-section decreases with increasing structural strength of crude oil and increasing restart pressure.

(4) Pre-shear can effectively reduce the minimum pressure difference required for successful restart. By pre-shearing at 1 s⁻¹ and 10 s⁻¹ at 35 °C, the minimum pressure difference required for successful restart is reduced to 61.4% and 32.9% of the non-shear condition, respectively. Pre-shear also can save the total restart time under a given restart pressure, even by orders of magnitude.

Nomenclature

Tp-s	Temperature at which pre-shear is imposed [°C]
th	Isothermal holding time [min]
G′	Storage modulus [Pa]
G′′	Loss modulus [Pa]
ts	Total restart time [s]
k	Structure-independent consistency in Teng's thixotropic model (eqn (3)) [Pa s^n1]
Δk	Structure-dependent consistency in Teng's thixotropic model (eqn (3)) [Pa s^n1]
v	Axial velocity [m s^−1]
V	Average velocity [m s^−1]
Vy	Velocity of the yield cross-section [m s^−1]
L	Pipeline length [m]
D	Pipeline diameter [m]
P	Gauge pressure [Pa]
N	Number of nodes
c	Sound speed in gelled oil [m s^−1],
n1, n2, a, b, m	Parameters in Teng's thixotropic model (eqn (3))
tcr, n	Parameters in Visintin et al.'s exponential correlation (eqn (2))
Δt	Time step [s]
Δz	Space step in the axial direction [m]

Greek symbols

[small gamma, Greek, dot above]	Shear rate [s^−1]
[small gamma, Greek, dot above]p-s	Rate of pre-shear [s^−1]
γ	Shear strain
δ	Loss angle [°]
τ	Shear stress [Pa]
τy	Yield stress [Pa]
τy1	Yield stress in Teng's thixotropic model (eqn (3)) [Pa]
λ	Structural parameter, varying between 0 and 1
ρ	Oil density [kg m^−3]
ρ0	Oil density at the pipeline outlet [kg m^−3]
τw	Shear stress at the pipe wall [Pa]
α	Compressibility of oil [Pa^−1]

Subscripts

ss	Steady state
in	Pipeline inlet
out	Pipeline outlet
z	Axial direction
r	Radial direction

Superscripts

*	Dimensionless variable

Acknowledgements

Support from the National Natural Science Foundation of China (Grant No. 51134006, 51534007) is greatly acknowledged.

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Footnote

† Electronic supplementary information (ESI) available: Repeatability tests of SAOS and stepwise increases in shear rate; comparison between test results and fitted results of shear stress responses to stepwise increases in shear rate; mesh size study. See DOI: 10.1039/c6ra16346g

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