The waxy crude oil needs to be heated usually when transported in pipelines. However, these pipelines may inevitably be shutdown due to transport plans or accidents. The temperature of heated oil is usually higher than the environment temperature and drops after the shutdown of pipelines. Waxy crystals will dissipate and build up gelled structure which increase the oil viscosity and impair the restart of pipe flow. The rheological properties of gelled crude oil are quite crucial for calculating the pipeline restart properties. Elastic-limit stress, static stress and dynamic yield stress were proposed by Cheng Chang to describe the yielding property of gelled crude oil during the pipeline restart [1]. The Houska thixotropic model was selected to represent the flow property of gelled crude oil in the 1.5D start-up model proposed by Wachs, A. [2]. When investigating the pressure wave velocity during the initial startup of gelled crude oil pipeline, a pure thixotropic model was chosen for calculating, and the initial shear stress was regarded as the yield stress by Zhang, G. [3]. The shear stress growth process before yielding was not considered when investigating the pipeline restart in the above literature. But Negr*ã*o, C.O.R. [4] regarded that the initial stress growth before yielding should be considered when exploring the restart properties of drilling fluid pipeline numerically. Similar to the drilling fluid, gelled crude oils always have a significant shear stress growth before yielding [5]. Based on the experimental data in literature [3], we can find that the time period needed for the start-up wave passing through the whole pipe is quite short. During such a short time period, the thixotropic process may be insignificant but the stress response process before yielding plays a leading role. So, when investigating the restart of gelled crude oil pipeline, the initial mechanical response and the yielding properties of gelled crude oil should not be ignored [6].

For the yielding properties of gelled crude oil, scholars have carried out many research studies. Among the properties, the yield stress is an important parameter to describe the yielding process. However, the yield stress of gelled crude oil is correlated significantly with the shear conditions, such as the loading rate of shear stress [7]. Through large amount of experiments, Li, C.X. [8] and Hou, L. [9] found out that the critical yield strain of gelled crude oil under different loading conditions remains unchanged and the shear strain was chosen as the index to measure the yielding behavior of gelled crude oil. But, just relying on the yield strain of gelled crude oil cannot figure out the restart process of pipeline. Thus, scholars pay more attention to the whole process of the shear stress or shear strain changing with time during the shear deformation of gelled crude oil.

When studying the rheological properties of gelled crude oil, loading conditions of constant shear stress and constant shear rate are mostly used. For the experiments under constant shear stress condition, scholars have built models to describe the relationship between the shear strain and time. But the deformation of gelled crude oil is often accompanied with structure damage. Classical models, such as the Maxwell model, Kelvin model and Burgers model, cannot describe the rheological behavior accurately. Wang, Z. [10] established a quasi-Burgers rheological model by combining the Maxwell model and the quasi-Kelvin model with fractional order derivatives to describe the creep behavior of gelled crude oil. Li, C.X. [11] established a nonlinear creep model which forms a creeping equation combining the nonlinear intenerating Hooken body and nonlinear hardening Newtonian fluid body by introducing intenerating and hardening variables. Liu, G. [12] introduced a damage variable which is relevant to micro-parameters to the Burgers model and built a mathematical model which can describe the viscoelastic properties of crude oil. But the fitted parameters in above models vary when the loading shear stress changes. Furthermore, calculation of the relationship between the shear strain and time under a time-dependent shear stress must be carried out through complex nonlinear superposition based on the fitted parameters obtained from constant shear stress conditions. The loading-dependent parameters and the complex nonlinear superposition limit the actual application of this kind of model in the calculation of pipeline start-up pressure.

As shown in literature [2,3,4], scholars often adopt the rheological model with shear rate controlled when studying the start-up of gelled crude oil pipeline. Initially, research studies mainly focused on the thixotropic process of material under constant shear rate. Thixotropic models which describe the shear stress decline of materials are divided into two categories: one is a direct expression between the shear stress and time, such as the thixotropic models proposed by literature [13,14]. This kind of model only applies to fitting the stress decrease under constant shear rate condition. Another model regards the structural parameter *λ* as a intermediate quantity to indirectly show the changes of shear stress with time, such as the thixotropic models proposed by Cheng, D.C.H. [15], Houska, M. [16], etc. Theoretically, this kind of model can be applied to the dynamic shear rate situation. However, the two kinds of models above have not considered the shear stress increasing stage before yielding.

In fact, for most of the thixotropic materials, the shear stress increases with shear strain before yielding when being applied a constant shear rate. Therefore, scholars gradually began to pay attention to the stress increase process. A kind of thixotropic model proposed by Mujumdar, A. [17], Dullaert, K. [18] assumes that the shear stress *σ* consists of the elastic stress *σ*_{e} and the viscous stress *σ*_{v}, where *σ*_{e} is associated with structural parameter *λ* and strain, and *σ*_{v} is a structure and shear rate dependent parameter. This kind of model can express the shear stress rising phenomenon during the initial shear deformation stage. Different from Mujumdar, A. [17] and Dullaert, K. [18], de Souza Mendes, P.R. regards the shear stress *σ* as the sum of the viscous shear stress *σ*_{v} and the viscoelastic shear stress *σ*_{M} of a special Maxwell mechanical analog [19,20]. Elastic modulus *G*_{M} and viscosity *η*_{M} in the Maxwell analog are associated with structural parameter *λ*. When investigating the thixotropic characteristics of waxy crude oil, Teng, H. [21,22] compared the above models and believed that the gelled crude oil with weak structure is more suitable for the kind of model proposed by Mujumdar, A. [17] and Dullaert, K. [18] instead of the one proposed by de Souza Mendes, P.R. [19,20]. Through modifying the structure breakdown rate equation, he built a model of gelled crude oil which can accurately describe the thixotropic characteristics under dynamic shear rate conditions.

Here we regard the models proposed by Mujumdar, A. [17], Dullaert, K. [18] and Teng, H. [21,22] as the Type-I model, and the model proposed by de Souza Mendes, P.R. [19,20] as the Type-II model. The total stress of the Type-I model *σ* is the sum of the elastic shear stress *σ*_{e} and the viscous shear stress *σ*_{v}. Therefore, it can be approximately regarded as the quasi-Voigt-Kelvin mechanical analog, as shown in Figure 1, but the elastic modulus *G* and the viscosity *η* are related to the structural parameter. The mechanical analog of the Type-II model is shown in Figure 1, where *G*_{M} and *η*_{M} in the Maxwell component are structure-dependent parameters while *η*_{∞} is usually constant.

Figure 1 Mechanical analog of different models (a) Type-I model (b) Type- II model

Similar to other viscoelastic materials, gelled crude oil shows significant creep properties under constant shear stress condition and remarkable stress relaxation properties under constant shear strain condition. Actually, the constant shear rate is a shear strain linearly increasing with time. Based on the Boltzmann superposition principle, we can find that the stress relaxation still plays an important role when calculating the shear stress under a linearly increasing shear strain. However, the Voigt-Kelvin model exhibits a strain creep but no stress relaxation [23].

In the Type-I model, the elastic stress *σ*_{e} is usually determined by the elastic strain *ε*_{e}, initial shear modulus, the structural parameter, or the damping function *h*(*ε*_{e}) associated with the shear strain considered in literature [21]. Within the range of initial deformation, the structural parameter almost remains unchanged and the elastic strain *ε*_{e} is approximately equal to the total strain *ε* of material [21]. Therefore, the elastic stress *σ*_{e} is mainly determined by the shear modulus and total strain *ε*, less relevant to the shear rate *ε̇*.

According to literature [17,21], we have *σ*_{v} → 0 when *ε* → 0 or *ε̇* → 0. There will be *σ* ≈ *σ*_{e}, which means that the material shows a significant elastic property and the total shear stress *σ* is almost independent of shear rate *ε̇* when *ε* → 0 or *ε̇* → 0. Based on the Type-I model, we can presume that during the initial deformation stage, the stress-strain curves should be similar when the loading shear rate changes under ultralow levels. An associated problem is that the total shear stress will remain large even if the loading shear rate has been reduced. The shear stresses under ultralow levels of shear rate have been forecasted based on the Type-I model in literature [21]. Result showed that there is little difference between the shear stresses under 0.0001s^{−1} and 0.1s^{−1}. And the one under 0.0001s^{−1} is even higher than the shear stresses under 0.001s^{−1} and 0.01s^{−1}. Changing rule of the predicted shear stresses in literature [21] is significantly different from the changing rule of the measured data shown in Figure 3 and incompatible with the rules in literature [4,5]. For pipeline restart of thixotropic drilling fluid [4] and waxy crude oil [24], velocity is almost constant in the central area of tube. The size of the low-shearing or non-shearing region is remarkable. So a robust model which is also applicable to ultralow shear rates is requried.

Theoretically, the Type-II model built by de Souza Mendes, P.R. [19] is more rigorous when compared with the Type-I model. The Maxwell mechanical analog can characterize the relaxation properties. When constant shear rate *ε̇* is applied, shear stress of the classical Maxwell analog can be described in Eq. 2. Obviously, the shear stress of Maxwell analog is related to shear rate *ε̇* and shear strain *ε*, so the defect of Type-I model is fixed. But in the model proposed by de Souza Mendes, P.R. [19], the elastic modulus *G*_{M} and the viscosity *η*_{M} in Maxwell analog are structure-dependent parameters which make the model complex and difficult to fit the experimental data.

Describing the initial mechanical response and yielding behavior of gelled crude oil accurately is the pre-condition for calculating the start-up process of gelled crude oil pipeline. There is an urgent need for the model considering relaxation properties and fitting the experimental data easily under both normal and ultralow shear rate conditions. Inspired by the Type-II model built by de Souza Mendes, P.R. [19], we improve the classical Jeffreys mechanical analog and build a mathematical model to describe the initial mechanical response characteristics of gelled crude oil under constant shear rate. Based on the mathematical model, we propose a critical damage softening strain, which is irrelevant to loading shear rates, to describe the yielding behavior of gelled crude oil.

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