Document Type: Research Paper
Authors
^{1} M.S. Student, Department of Chemical Engineering, University of Isfahan, Isfahan, Iran
^{2} University of Isfahan
Abstract
Keywords
One of the severe problems in oil industry is wax crystal formation and deposition phenomenon. Crude oil with a wax content more than 5 wt.% is named waxy crude oil and it has a great potential of wax crystal formation (Zhang and Liu, 2008). Paraffins or waxes are hydrocarbons in the range of C_{10} to C_{70 }(Kané et al., 2003). The solubility of heavy components is high at very high temperature and pressure in reservoirs; however, after oil production and during transportation, the temperature and pressure drop significantly and a sharp decline in their solubility occurs; therefore, the solid phase, i.e. waxy crystal, appears. In the first place, the formation of this solid phase changes the rheological properties of waxy crude oil and converts its rheological nature from a Newtonian fluid to a nonNewtonian fluid. Since the design of pipelines and oil transmission equipment requires detailed rheological information (Adeyanju and Oyekunle, 2012), the rheological behavior of the fluid needs to be studied in different operational conditions. Sometimes emergency shutdowns happen for technical reasons such as maintenance and periodic inspections. In these situations, waxy crude oil is cooled statically, and then solid crystals interlock and build a waxy gel network. This viscoelastic gel has a yield stress which should be supplied adequately to restart the operation (Chang et al., 1999; Visintin et al., 2008). The calculation of the necessary pressure to break down the resistance of the gel network is a challenge that must be solved using the results of rheometry tests. The waxy gel behavior against the applied stress during the restart operation is divided into three regions. First region is the elastic deformation, the second region is creep phase and the third one is destruction of the waxy network. The second region i.e. the creep phase starts after a limitelastic yield stress and continues to a static yield stress. The rheological behavior of waxy crude oil in the first and third region can be described with Hooke's law and rheological models like power law, Bingham plastic, HerschelBulkley and Casson models, respectively (Chang et al., 1998; Kané et al., 2004). The behavior description of the second zone (creep phase), which is dependent on time, requires the use of viscoelastic models.
The studies of the rheological behavior of waxy crude oil have started since about three decades ago. In all the shearrotational tests studies, the nonNewtonian behavior of waxy crude oil samples has been reported at a temperature less than wax appearance temperatures (WAT). The viscosity and yield stress of these nonNewtonian fluids increase by decreasing the temperature and shear rate and increasing the wax content of oil samples (Agarwal et al., 1989; Alboudwarej et al., 2006; Chandaa et al., 1998; ElGamal, 1998; ElGamal and Gad, 1998; Lorge et al. , 1997). Several studies were performed to evaluate the effect of factors like temperature and wax content on the thixotropic behavior of waxy crude oil. The results showed that a decrease in temperature and an increase in wax content lead to an increases in hysteresis loop area of flow curve, which represents the increase in the thixotropic behavior of the waxy crude oil under these conditions (Guo et al., 2015; JapperJaafar et al., 2014; Van Der Geest et al., 2015). A few creep studies were conducted to achieve a better understanding of the creep phenomenon of waxy crude oil (Ghannam, 2004; Rønningsen, 1992). Ho and Zhang reported that an increase in the temperature of waxy crude oil leads to an increase in strain in an applied constant stress condition. The reduction in the internal resistance of crude oil and softening of gel structure at high temperatures were discussed as the reasons (Hou and Zhang, 2010). Many researchers have tried to propose some models to predict the rheological behavior of waxy crude oil in recent years. The studies in this area can be divided into four categories, including the studies in order to provide experimental and quasiexperimental models, the implementation of classical fluid mechanics models, the combined classical and theorybased models on the basis of thermodynamics and kinetics laws, and viscoelastic models. In some studies, the experimental rheological models such as models in (AlFariss et al., 1993; AlZahrani and AlFariss, 1998) were proposed to predict the rheological properties such as viscosity of waxy crude oil. The constant values of these models were determined through fitting experimental results (Iktisanov and Sakhabutdinov, 1999). The ability of the classical nonNewtonian fluid mechanical models like power law, Bingham plastic, and Casson models in predicting the rheological properties of waxy crude oil was compared (Hasan et al., 2010) and several models were combined to achieve a more comprehensive model (Rønningsen, 1992). Several classical rheological models were modified to increase the ability of the classical rheological models by introducing some parameters related to suspension rheology and considering the wax crystal formation phenomenon to the thermodynamic relationships (Adeyanju and Oyekunle, 2012; Li and Zhang, 2003; Matveenkov et al., 1995; Palermo and Tournis, 2015; Pederens and Ronningsen, 2000). Several studies have focused on the waxy gel formation phenomenon inside the pipelines in static conditions. The prediction of the rheological behavior of waxy gel is the important task for the determination of the necessary pressure to restart the flow. In (Li et al., 2015) a model was proposed to predict the yield stress. This model is based on the difference between the desired temperature and the gel point and density of the oil sample and agrees reasonably with experimental results (Li et al., 2015). To describe the complex behavior of a waxy gel, one requires a model which consists of the elastic, viscoplastic, and thixotropic properties. Such models were developed to predict the minimum restarting pressure (Luthi, 2013; Van Der Geest et al., 2015). Some models were developed to describe the viscoelastic behavior of waxy crude oil. Dante et al. have proposed a model for predicting the behavior of waxy crude oil in terms of time and shear rate by combining AlZahrani empirical and Maxwell viscoelastic models (Dante et al., 2006). Moreover, some viscoelastic models based on suspension rheology theory was proposed for predicting the dynamic modulus of waxy crude oil (Dante et al., 2007).
The main aim of this study is to investigate the flow curve and the compliance function trends of the waxy crude oil samples, to propose a modified Casson model for predicting the flow curve, and to suggest a modified Burger model to describe the behavior of the compliance function of each types of waxy crude oil in terms of the amount of wax content and temperature.
The oil samples with different wax contents are provided by adding the desired amount of a raffinate sample provided from Sepahan Oil Company to the Isfahan refinery crude oil as the base oil fluid. The density, wax, and asphaltene content of the crude oil are 895.8 kg/m^{3}, 11.68 and 0.6 wt.% respectively. The pour point and kinematic viscosity of raffinate are 51 °C and 37.71 mm^{2}/s at 60 °C respectively. The wax contents of the oil samples are measured according to the modified UOP 4664 method (Elsharkawya et al., 2000). Eight shearrotational tests and eight creep and recovery tests are conducted to determine the flow curve and compliance function by means of a MCR 300 AntonPaar rheometer with a cone and plate geometry. The diameters of plate and gap size are 25 mm and 0.05 mm respectively. The temperature control is provided by Peltier elements.
The flow curve is a plot of shear stress against applied shear rate. This curve is the result of shearrotational tests and can be used to evaluate the rheological properties such as viscosity and yield stress of materials. In this study, eight flow curves are determined by adopting a rheometer with the ability to control shear rate at different temperatures and wax contents. The shear rate is of 100 s^{1}1000 s^{1} range. The capability of the power law, Bingham plastic, and Casson models of describing the flow curves are assessed. The Casson model is modified in order to match better with experimental results
Compliance functions are measured as a function of time by conducting creep and recovery tests. The creep and recovery test is a subset of unsteady shear tests. A constant shear stress is applied to the sample and the resulted strain is calculated as a function of time. After achieving a maximum strain, the stress is suddenly removed and the resulted strain gradually decreases to achieve equilibrium constant. Elastic materials recover quickly after the removal of stress and viscous materials will never recover, while viscoelastic materials recover gradually. The linear viscoelastic region (LVR) had to be determined in order to conduct the creep and recovery test. The linear viscoelastic region (LVR) is a region in which dynamic modules do not depend on applied stress (Steffe, 1996). In fact, the behavior of the viscoelastic materials in the linear viscoelastic region is only a function of time and temperature and does not depend on the amount of applied stress or strain (Fecarotti et al., 2012). In order to measure the linear viscoelastic properties of the waxy crude oil, the creep and recovery test is conducted with small stress within the LVR. The stress sweep test is conducted to measure the LVR at a frequency of 1 Hz. The creep phase results are fitted with the fourelement mechanical viscoelastic Burger model.
Eight shearrotational tests are performed in accordance with Table 1. The flow curves of the oil sample with a wax content of 22 wt.% are shown in Figure 1(a) at temperatures of 5, 15, 25, and 35 °C. The results show that with a decrease in the temperature, the shear stress values and viscosity of the oil samples increase (Figure 1(b)). It means that a decline in temperature strengthens the waxy network. The solubility of waxes drops by decreasing the temperature, and it causes more waxy crystals to form (Vignati et al., 2005).
Figure 1
a) Flow curve and b) viscosityshear rate curve at different temperatures.
The trend of viscosity reduction by increasing the shear rate represents a nonNewtonian rheological behavior of the waxy crude oil at temperatures of 5, 15, and 25 °C; however, the crude oil viscosity values do not change at the temperature of 35 °C, which means the fluid is Newtonian at this temperature. An increase in the shear rate causes a decrease in the slope of viscosityshear rate curve at high shear rates. It can be concluded that forces on the waxy network increase, break down the bonds between crystals, and reduce the network strength; therefore, the rheological behavior of the crude oil is described as a Newtonian fluid in this condition (ElGamal and Gad, 1998).
Figure 2
a) Flow curve and b) viscosityshear rate curve for the samples with different wax contents.
Four flow curves are shown in Figure 2(a) for the oil samples with wax contents of 12, 17, 22, and 27 wt.% at 25 °C. The changing in the wax contents of the oil samples has a significant influence on the flow curves. An increase in the wax content causes the flow curves to move toward higher stress values.
The wax content increases the possibility of the formation of waxy crystals. The presence of a high amount of waxy crystals leads to forming a stronger waxy network. According to the above results, the shear stress and viscosity of the crude oil increase in this case. The increasing trend of the viscosity values with increasing wax content is clear as depicted in Figure 2(b). An increase in wax content causes the slope of viscosityshear rate curve to increase, which represents a nonNewtonian behavior of the oil samples containing higher amounts of wax.
Waxy crude oil has been introduced as a nonNewtonian timeindependent fluid (Matveenkov et al., 1995; Rønningsen, 1992). The shearrotational test results are fitted with the power law, Binghamplastic, and Casson models, and the results of the model parameters are reported in Table 1. According to this table, it is clear that all three models are successful in fitting the experimental results; however, the power law model is weaker than the Binghamplastic and Casson models. The reason is that the power law model is for pseudoplastic fluids and cannot describe the rheological behavior of a waxy crude oil under different situations. The Bingham plastic model is a linear model and is usually applied to describing the behavior of Binghamplastic fluids, while the Casson model has an ability to describe the curvature of the flow curve, and it is suitable to predict the rheological behavior of a viscoplastic fluid (Chhabra and Richardson, 2008).
Table 1
Comparing the rheological models.
Test number 
Temperature (°C) 
Wax content (wt.%) 
Power Law 
Binghamplastic 
Casson 

m (Pa.s^{n}) 
n 
R^{2} 
(Pa) 
(Pa.s) 
R^{2} 
(Pa) 
(Pa.s) 
R^{2} 

1 
5 
22 
63.02 
0.2453 
0.9526 
189.8 
0.1762 
0.9912 
145.6 
0.04565 
0.9955 

2 
15 
22 
33.53 
0.2533 
0.9491 
104.1 
0.1024 
0.9784 
77.80 
0.02838 
0.9854 

3 
25 
22 
10.46 
0.2942 
0.9739 
35.29 
0.05259 
0.9752 
25.38 
0.01721 
0.9943 

4 
35 
22 
0.02407 
0.9909 
0.9976 
0.3837 
0.01273 
0.9986 
0.1799 
0.009211 
0.9755 

5 
25 
12 
0.02063 
0.9706 
0.9993 
0.1894 
0.01669 
0.9995 
0.1099 
0.01290 
0.9871 

6 
25 
17 
0.1210 
0.7497 
0.9900 
1.58 
0.02116 
0.9969 
0.6168 
0.01491 
0.9975 

7 
25 
22 
10.46 
0.2942 
0.9739 
35.29 
0.05259 
0.9752 
25.38 
0.01821 
0.9943 

8 
25 
27 
9.323 
0.5029 
0.9742 
55.80 
0.2841 
0.9925 
34.30 
0.1415 
0.9960 

By comparing the R^{2} values for Bingham and Casson models in each test, it can be concluded that Binghamplastic model is better than Casson model for the experiments No. 4 and 5, while Casson model matches better with the experimental results for the rest. The test No. 4 is performed at a temperature of 35 °C and the oil sample in test No. 5 has a low wax content of 12 wt.%; therefore, it is clear that the behavior of those oil samples are similar to a Binghamplastic fluid under this condition. A decrease in temperature or an increase in wax content causes the waxy crude oil to become a viscoplastic fluid. The change of temperature and wax content changes the rheological nature of the waxy crude oil. At high temperatures or low wax content, there is a difficult condition to form waxy crystals and the waxy crystal formation rate is very low. Moreover, the rheological behavior of the crude oil has a linear relationship with the shear rate, while a strong waxy network exists within the oil at low temperatures or high wax content. This network changes the rheological behavior of the waxy crude oil. In this case, the dependency of the rheological behavior of the waxy crude oil to the shear stress is more complicated. In another similar study, which was performed to provide a model for the prediction of the flow curve of waxy crude oil, it was shown that the waxy crude oil had different rheological behaviors and its rheological nature was changed from a Newtonian to a viscoplastic fluid followed by a viscoelastic behavior (Iktisanov and Sakhabutdinov, 1999).
In this study, the Casson model is selected as the base model for providing a modified model to predict the flow curve of the oil samples with various wax amounts at different temperatures. In the following, the Casson model parameters changing trends are interpreted.
: This parameter is the Casson yield stress, which increases with a decrease in temperature and an increase in wax content. This parameter is a measure of yield strength of material, and by definition, yield stress is a minimum stress required to flow the fluid. The change of this parameter can be related to the changes in the internal structure of the materials. The solubility of waxes in the crude oil decreases with decreasing temperature and it leads to produce waxy crystals. The waxy crystals are linked together and form waxy agglomerates and network at low temperatures. As temperature is lowered, the links become stronger and the resulted network has a higher yield stress (Aiyejina et al., 2011). As the oil wax content increases, the waxy network grows and the Casson yield stress increases.
: In the Casson rheological model, this parameter is a measure of the apparent viscosity of fluid and is named as Casson viscosity. According to Table 1, the decreases inversely with temperature but directly with wax content. The solubility of waxy crystals in the crude oil increases by temperature but drops by wax content, which in turn causes the waxy network to become a weaker structure. The weakening of the structure of the network is followed by Casson viscosity reduction.
The Casson rheological model is able to match the experimental results successfully. The dependency of the model parameters on temperature and wax content are correlated by applying four different combinations of power and exponential functions through Equations 14. The fitting results and the statistical parameters are reported in Table 2. It should be noted that the experimental results of the test which is conducted for the oil sample with wax content of 22 wt.% at 5 °C is not included in the regression and it is used to validate the proposed modified Casson model prediction.
(1) 

(2) 

(3) 

(4) 
The statistical parameters such as the coefficient of determination (R^{2}), mean percent error (MPE), mean bias error (MBE), and standard error (RMSE) are evaluated to judge the mentioned functions. The above statistical parameters are calculated as the following equations (Toǧrul and Arslan, 2004).
(5) 

(6) 

(7) 

(8) 
where, is the experimental result; is the result predicted by the model; is the average of the experimental results, and N is the number of experimental results; k stands for the number of independent variables in the model. In multiple regression, a greater R^{2} value and lower absolute values of MPE, MBE, and standard error for each model mean that the model fits the experimental results reasonably (Toǧrul and Arslan, 2004).
Table 2
The effect of temperature and wax content on the Casson model parameters.

Parameters of the proposed model 
Statistical parameters 

a 
b 
c 
R^{2} (%) 
MPE 
MBE 
RMSE 

19.96 
0.0926 
0.1126 
95.96 
17.42 
2.231 
9.466 

86.324 
1.103 
0.1056 
89.96 
24.77 
2.336 
14.61 

0.09 
0.091 
2.545 
96.45 
14.83 
2.142 
8.88 

0.5349 
1.086 
2.386 
90.57 
22.02 
2.146 
14.46 

0.000007 
0.04665 
0.4118 
97.62 
0.1362 
0.0025 
0.000638 

0.000014 
0.5876 
0.4106 
97.31 
0.1519 
0.0026 
0.00679 

3*10^{14} 
0.0367 
8.783 
18.35 
0.6882 
0.0245 
0.0388 

1.6*10^{14} 
0.515 
9.21 
22.15 
0.6806 
0.0241 
0.0365 
The best model for describing and parameters are Equations 3 and 1 respectively. These relations have the highest R^{2} and the lowest absolute values of MPE, MBE, and standard error, which reflects the high ability of these equations to match with the experimental results. The final form of the modified Casson model is reported in Table 3.
Table 3
Modified Casson model.
R^{2} (%) 

96.45 


97.62 

The predictions of the proposed model are validated through the experimental results of the Casson model parameters for all the eight tests and the flow curve of the sample with wax content of 22 wt.% at a temperature of 5 °C. These comparisons are shown in Figures 3 and 4.
a) 
Figure 3
Comparing the experimental and predicted results for a) Casson yield stress and b) Casson viscosity.
Figure 4
Comparing the experimental and predicted flow curve of the oil sample with wax content of 22 wt.% at 5 °C.
There is reasonable agreement between the experimental results and the predictions of the model. This model matches with the experimental results with R^{2 }= 97.33% as shown in Figure 4.
In this study, four different waxy crude oil samples with different wax amounts are prepared and used to perform creep and recovery tests at four different temperatures. The linear viscoelastic region should be valid for all the samples within the test temperature range; therefore, the LVR should be measured under a condition that the crude oil has the weakest waxy structure. Therefore, the sample with the lowest wax content (12 wt.%) is selected to perform the stress sweep test at the highest temperature (35 °C). The stress sweep test is performed at a frequency of 1 Hz and 0.01 and in stress range of 50 Pa; the results show that G" and G' moduli are independent of the stress up to stress values of 35 Pa and 8 Pa respectively. To ensure that the selected stress is within the linear viscoelastic region, the stress should be less than 8 Pa; herein, the stress of 0.5 Pa is considered.
In the creep phase, a constant stress of 0.5 Pa is applied to the oil samples for 319 seconds and then it is suddenly removed. Next, the recovery phase continues for 319 seconds. The compliance functions in terms of time are shown in Figures 5 and 6.
a) 
Figure 5
Compliance function of the oil sample with wax content of 22 wt.% at a) 5 °C, b) 15 °C, c) 25 °C, and d) 35 °C.
The general trend of the results in Figure 5 represents a viscoelastic nature and timedependent rheological behavior of the waxy crude oil. The compliance function value at 35 °C is much higher than the others because of the high solubility of waxes at high temperatures. The internal structure is weak at high temperatures due to the lake of waxy crystals to form a strong waxy network. This weak structure is deformed quickly, and its deformation is large. As the recovery phase starts, the compliance function drops to an equilibrium value, and, at the end of the recovery phase, a full recovery is not observed at all four temperatures. Since viscoelastic materials have a partial recovery and reach their full recovery at a long time, it can be concluded that the waxy crude oil samples have viscoelastic behavior (Steffe, 1996). After the removal of the stress, as shown in Figure 5, at 5 °C the waxy crude oil achieves the 99% of its final recovery suddenly; however, the compliance functions decline gradually at the other temperatures. It represents that the rheological behavior of the waxy crude oil is similar to an ideal elastic material at 5 °C. A very little recovery can be observed at 35 °C because the absence of the waxy crystal network has caused the crude oil to behave like a viscous fluid.
Figure 6
Compliance functions of the oil samples with wax content of a) 12 wt.%, b) 17 wt.%, c) 22 wt.%, and d) 27 wt.% at 25 °C.
The compliance function for the oil sample with wax content of 12 wt.% has a higher value compared to the other samples as shown in Figure 6. Increasing the wax content, the compliance decreases since the compliance function depends on the strain and the strain depends on the strength of the internal structure of the material directly. As the wax content of the oil samples increases, according to the results of this research, a crystal network is formed; therefore, the internal strength of the crude oil increases.
Moreover, the wax content has a significant influence on the recovery phase. The sample with a wax content of 12 wt.% does not recover, which represents a viscous fluid. An increase in wax content leads to higher recovery levels of the waxy crude oil, so its behavior tends to a solid viscoelastic solid.
In order to evaluate the performance of the viscoelastic Burger model, the creep test experimental results are fitted with this model. The results are presented in Table 4. The Burger model has a good accuracy in predicting the compliance function according to the reported R^{2} values. However, this model has the lowest accuracy in runs No. 4 and 5, where temperature is at the lowest level and wax content is in the highest level respectively. It can be explained that a favorable condition is provided for the formation of waxy crystals at either low temperatures or high wax content. In this situations, many waxy crystals form and a waxy network propagates in the whole crude oil medium. It changes the rheological behavior of the crude oil from a viscoelastic material to a Hookean solid. It is not surprising that the viscoelastic Burger model cannot predict the behavior of a Hookean solid very well.
Table 4
The Burger model parameters.
Temperature (°C) 
Wax content (wt.%) 
J_{0 }(Pa^{1}) 
J_{1 }(Pa^{1}) 
𝜆 (s) 
𝜇_{0 }(Pa.s) 
R^{2} 
5 
22 
0.0003108 
0.001479 
12.37 
254653 
0.9854 
15 
22 
0.0003196 
0.002799 
39.18 
131268 
0.9988 
25 
22 
0.001317 
0.004679 
58.28 
54678 
0.9981 
35 
22 
0.008769 
0.1462 
68.95 
2272 
0.9998 
25 
12 
0.05895 
1.463 
61.23 
232.5 
0.9999 
25 
17 
0.004547 
0.1336 
59.81 
2840 
0.9999 
25 
22 
0.001317 
0.004679 
58.28 
54678 
0.9981 
25 
27 
0.0004321 
0.001163 
5.049 
795213 
0.9720 
The Burger model parameters have particular physical interpretations. The functionality of these four parameters with temperature and wax content are studied.
The J_{0} is instantaneous compliance, and J_{1} represents retardation compliance; λ and μ_{0} are retardation time and zero shear viscosity respectively. J_{0 }is the reverse of the Young’s modulus of Maxwell spring (G_{0}), and J_{1} is the reverse of the Young’s modulus of Kellvin portion (G_{1}) in Burger model (Steffe, 1996). The Young’s modulus is the slope of shear stressstrain curve. This modulus is a measure of the hardness of the internal structure of materials (Morrison, 2001). The λ is a measure of the response delay of materials to stress. In other words, as the value of this parameter is low, the system responds to the applied force quickly and deforms very fast. The value of this parameter relates to the viscoelastic nature of the material. Ideal elastic materials have a low λ, whereas ideal viscous fluids have a high λ (Steffe, 1996). The μ_{0} is the viscosity of Maxwell dashpot in Burger model and represents the viscous properties of materials in a long time. It is the inverse of creep phase curve slope at the end of this phase. It is infinite for the ideal elastic solids and is very low for ideal viscous fluids (Steffe, 1996). Therefore, the values of J_{0,} J_{1, }λ, and μ_{0} relate to the strength of waxy network and the viscoelastic nature of waxy crude oil in different conditions. According to Table 4, a decrease in temperature or an increase in wax content causes J_{0}, J_{1}, and λ to decrease but μ_{0} to increase. It shows that a strong waxy network is formed under these conditions and the rheological behavior of the crude oil approaches viscoelastic solids with decreasing temperature or increasing wax content.
c. Modified viscoelastic Burger model
In order to determine the best model for Burger model parameters predictions, four different combinations of exponential and power functions (Equations 14) are applied. The results are reported in Table 5. It should be noted that the experimental results of a test which is conducted by the oil sample with wax content of 12 wt.% at 25 °C is not included in the regression and it is used to evaluate the validity of the proposed Burger model.
Table 5
The effect of temperature and wax content on the Burger model parameters.
Proposed models 
Parameters of proposed model 
Statistical parameters 

a 
b 
c 
R^{2} (%) 
MPE 
MBE 
RMSE 

0.01051 
0.3085 
0.4990 
99.96 
52.31 
0.000317 
0.000512 

0.00003 
3.5692 
0.3943 
0.3943 
45.05 
74.71 
0.00635 

0.014 
7078.77 
7.1512 
99.92 
58.84 
0.000397 
0.000535 

1.308 
4.1428 
6.7258 
91.33 
51.28 
0.002818 
0.00556 

0.8759 
0.2521 
0.2424 
99.94 
16.22 
0.00105 
0.003734 

0.001 
3.583 
0.39 
84.80 
33.62 
0.0266 
0.1846 

44.732 
0.1983 
7.0718 
99.74 
110.74 
0.0031 
0.00845 

0.3631 
5.8916 
7.071 
99.73 
99.64 
0.00283 
0.00848 

19.144 
0.057 
0.0197 
62.17 
72.55 
0.3034 
13.9 

7.717 
0.845 
0.0419 
61.68 
73.11 
0.712 
14.02 

149.56 
0.043 
0.742 
57.71 
78.87 
0.3307 
14.72 

23.96 
0.8456 
0.6652 
57.02 
79.46 
0.56 
14.84 

2.786 
0.0783 
0.538 
99.86 
1.267 
1632.87 
9291.35 

11.501 
0.9345 
0.242 
99.39 
2.28 
1927.75 
19767.9 

7.8*10^{12} 
0.934 
12.8 
99.39 
2.04 
1662.37 
19672.6 
According to the results of Table 5 and comparing the statistical parameter values, it can be concluded that the best model for describing the behavior of J_{0}, J_{1}, and μ_{0} is Equation 1; however, none of the equations are appropriate to predict λ. For this reason, more terms need to be included to this equation in order to comply with the experimental results of λ. A complex term as a function of wax content is included to Equation 1. The best relations with reasonable predictions are reported in Table 6.
Table 6
Modified Burger model.
R^{2} (%) 

99.96 


99.94 


87.9 


99.86 

In these equations, T and W are temperature (°C) and wax content (wt.%) respectively.
d. Model validation
The proposed model is validated by the comparison of the experimental and predicted results of Burger parameters and the compliance function of oil sample with wax content of 12 wt.% at temperature 25 °C. The comparison results are shown in Figures 7 and 8.
Figure 7
Comparing the experimental and predicted results of a) instantaneous compliance, b) retardation compliance, c) retardation time, and d) zero shear viscosity of the Burger model.
There is reasonable agreement between the experimental results and the predictions of the model. The model matches with the experimental results with R^{2 }= 99.7% as shown in Figure 8.
Figure 8
Comparing the experimental and predicted compliance of the oil sample with 12 wt.% wax content at 25 °C.
4. Conclusions
In this study, several shearrotational tests are conducted to investigate the timeindependent rheological behavior of an Iranian waxy crude oil sample. The flow and the viscosityshear rate curves show that a decrease in temperature or an increase in wax content raises the viscosity. The power law and Binghamplastic models cannot predict the flow curves in all the wax amounts and temperature ranges since the rheological behavior of the waxy crude oil changes from a Newtonian to a viscoplastic fluid when temperature decreases or wax content increases. The Casson model is the best model for predicting the rheological behavior of the waxy crude oil. This model has two parameters and , which are increased as temperature decreases or wax content increases. It is concluded that a decrease in temperature or an increase in wax content causes the internal structure of the waxy crude oil to strengthen, and as a result, viscosity and yield stress increase. A modified model, based on Casson model, is developed to predict the flow curves of the waxy crude oil samples. The viscoelastic behavior of the waxy crude oil is studied using creep and recovery test. The Burger model can fit with compliance function results adequately. The functionality of the four parameters of this model is determined, and it is shown that a decrease in temperature or an increase in wax content leads to a decrease in the J_{0}, J_{1}, and λ but an increase in the 𝜇_{0}. It means that a strong waxy network forms in the crude oil, and the viscoelastic nature of the fluid tends to a viscoelastic solid in these conditions. A modified model, based on Burger model, is proposed, which reasonably predicts the compliance function of the waxy crude oil in a vast range of temperatures and wax content.
Nomenclature
J 
: Compliance function 
J_{0} 
: Instantaneous compliance 
J_{1} 
: Retardation compliance 
LVR 
: Linear viscoelastic region 
t 
: Time 
UOP 
: Universal oil products (standards) 
WAT 
: Wax appearance temperature 
Greek letters 

: Shear rate 

: Retardation time 

: Viscosity 

: Shear stress 

Superscripts 

B 
: Bingham 
C 
: Casson 
References
Adeyanju, O. and Oyekunle, L., An Experimental Study of Rheological Properties of Nigerian Waxy Crude Oil, Petroleum Science and Technology, Vol. 30, No. 11, p. 11021111, 2012.
Agarwal, K. M., Purohit, R. C., Surianarayanan, M., Joshi, G. C., and Krishna, R., Influence of Waxes on the Flow Properties of Bombay High Crude, Fuel, Vol. 68, p. 937939, 1989.
Aiyejina, A., Chakrabarti, D. P., Pilgrim, A., and Sastry, M. K. S., Wax Formation in Oil Pipelines: A Critical Review, International Journal of Multiphase Flow, Vol. 37, No. 7, p. 671694, 2011.
AlFariss, T. F., Jang, L. K., Ozbelge, H. O., and Ghasem, N. M. A New Correlation for the Viscosity of Waxy Oils, Journal of Petroleum Science and Engineering, Vol. 9, No. 2, p. 139144, 1993.
AlZahrani, S. M. and AlFariss, T. F. A, General Model for the Viscosity of Waxy Oils, Chemical Engineering and Processing, Vol. 37, p. 433–437, 1998.
Alboudwarej, H., Kempton, E., and Huo, Z., Flowassurance Aspects of Subsea Systems Design for Production of Waxy Crude Oils Paper Presented at the SPE Annual Technical Conference, San Antonio,Texas, U.S.A.
Chandaa, D., Sarmaha, A., Borthakura, R A., Raoa, K. V., Subrahmanyama, B., and Das, H. C., Combined Effect of Asphaltenes and Flow Improvers on the Rheological Behavior of Indian Waxy Crude Oil, Fuel, Vol. 77, No. 11, p. 11631167, 1998.
Chang, C., Boger, D. V., and Nguyen, Q. D., The Yielding of Waxy Crude Oils, Industrial and Engineering Chemistry Research, Vol. 37, No. 4, p. 15511559, 1998.
Chang, C., Nguyen, Q. D., and Rønningsen, H. P., Isothermal Startup of Pipeline Transporting Waxy Crude Oil, Journal of NonNewtonian Fluid Mechanics, Vol. 87, No. 2–3, p. 127154, 1999.
Chhabra, R. P. and Richardson, J. F., Chapter 1NonNewtonian Fluid Behavior, In R. P. C. F. Richardson (Ed.), NonNewtonian Flow and Applied Rheology, Second Edition, Oxford: ButterworthHeinemann, p. 155,2008.
Dante, R. C., GeffroyAguilar, E., and Chávez, A. E., Viscoelastic Models for Mexican Heavy Crude Oil and Comparison with a Mixture of Heptadecane and Eicosane, Part I. Fuel, Vol. 85, No. 4, p. 559568, 2006.
Dante, R. C., Geffroy, E., and Chávez, A. E., Viscoelastic Models for Mexican Heavy Crude Oil and Comparison with a Mixture of Heptadecane and Eicosane, Part II., Fuel, Vol. 86, No. 15, p. 24032409, 2007.
ElGamal, I. M., Combined Effects of Shear and Flow Improvers: The Optimum Solution for Handling Waxy Crudes Below Pour Point. Colloids and Surfaces A, Physicochemical and Engineering Aspects, Vol. 135, No. 1–3, p. 283291, 1998.
ElGamal, I. M. and Gad, E. A. M., Low Temperature Rheological Behavior of Umbarka Waxy Crude and Influence of Flow Improver, Colloids and Surfaces A: Physicochemical and Engineering Aspects, Vol. 131, No. 1–3, p. 181191, 1998.
Elsharkawya, A. M., AlSahhafb, T. A., and Fahimb, M. A., Wax Deposition from Middle East Crudes, Fuel, Vol. 79, p. 1047–1055, 2000.
Fecarotti, C., Celauro, C., and Pirrotta, A., Linear Viscoelastic (LVE) Behavior of Pure Bitumen Via Fractional Model, Procediasocial and Behavioral Sciences, Vol. 53, p. 450461, 2012.
Ghannam, M. T., Creep–recovery Experimental Investigation of Crude Oil–polymer Emulsions. Journal of Applied Polymer Science, Vol. 92, No. 1, p. 226237, 2004.
Guo, L., Wang, Y., Shi, S., Yu, X., and Chen, X., Study on Thixotropic Properties of Waxy Crude Oil Based on Hysteresis Loop Area, Engineering, Vol. 7, No. 07, p. 469, 2015.
Hasan, S. W., Ghannam, M. T., and Esmail, N., Heavy Crude Oil Viscosity Reduction and Rheology for Pipeline Transportation, Fuel, Vol. 89, No. 5, p. 10951100, 2010.
Hou, L. and Zhang, J., A Study on Creep Behavior of Gelled Daqing Crude Oil, Petroleum Science and Technology, Vol. 28, No. 7, p. 690699, 2010.
Iktisanov, V. A. and Sakhabutdinov, K. G., Rheological Studies of Paraffinbased Oil at Different Temperatures, Colloid Journal, Vol. 61, No. 6, p. 719722, 1999.
JapperJaafar, A., Bhaskoro, P., Sariman, M., and Rozlee, R., Rheological Investigation on the Effect of Shear and Time Dependent Behavior of Waxy Crude Oil, Paper Presented at the MATEC Web of Conferences, 2014.
Kané, M., Djabourov, M., and Volle, J. L., Rheology and Structure of Waxy Crude Oils in Quiescent and Under Shearing Conditions, Fuel, Vol. 83, No. 11–12, p. 15911605, 2004.
Kané, M., Djabourov, M., Volle, J. L., Lechaire, J. P., and Frebourg, G., Morphology of Paraffin Crystals in Waxy Crude Oils Cooled in Quiescent Conditions and under Flow, Fuel, Vol. 82, No. 2, p. 127135, 2003.
Li, H. and Zhang, J., A Generalized Model for Predicting NonNewtonian Viscosity of Waxy Crudes as a Function of Temperature and Precipitated Wax. Fuel, Vol. 82, No. 11, p. 13871397, 2003.
Li, H., Zhang, J., Song, C., and Sun, G., The Influence of the Heating Temperature on the Yield Stress and Pour Point of Waxy Crude Oils, Journal of Petroleum Science and Engineering, Vol. 135, No., p. 476483, 2015.
Lorge, O., Djabourov, M., and Brucy, F., Crystallization and Gelation of Waxy Crude Oils under Flowing Conditions, Oil and Gas Science and Technology, Vol. 52, No. 2, p. 235239, 1997.
Luthi, I. F., Waxy Crude Oil Characterization and Experimental Study of the Restart of a Line Blocked with Gelled Waxy Crude, Paper Presented at the SPE Annual Technical Conference and Exhibition, 2013.
Matveenkov, V. N., Kirsanov, E. A., and Remizov, S. V., Rheologiy of Highly Paraffinaceous Crude Oil, Colloid and Surfaces, Vol. 101, p. 17, 1995.
Morrison, f. A., Understanding Rheology, Michigan State University, 2001.
Palermo, T. and Tournis, E., Viscosity Prediction of Waxy Oils: Suspension of Fractal Aggregates (SoFA) Model, Industrial and Engineering Chemistry Research, Vol., No., p., 2015.
Pederens, K. S., and Ronningsen, H. P. Effect of Precipitated Wax on Viscosity –A Model for Predicting NonNewtonian Viscosity of Crude Oils, Energy and Fuels, Vol. 14, p. 4351, 2000.
Rønningsen, P. H. Rheological Behaviour of Gelled, Waxy North Sea Crude Oils, Journal of Petroleum Science and Engineering, Vol. 7, No. 3–4, p. 177213, 1992.
Steffe, J. F., Rheological Methods in Food Process Engineering, Freeman Press, 1996.
Toǧrul, H. and Arslan, N., Mathematical Model for Prediction of Apparent Viscosity of Molasses. Journal of Food Engineering, Vol. 62, No. 3, p. 281289, 2004.
Van Der Geest, C., Guersoni, V. C. B., MerinoGarcia, D., and Bannwart, A. C., A Modified ElastoViscoplastic Thixotropic Model for Two Commercial Gelled Waxy Crude Oils, Rheologica Acta, Vol. 54, No. 6, p. 545561, 2015.
Vignati, E., Piazza, R., Visintin, R. F., Lapasin, R., Paolo, D., and Lockhart, T. P., Wax Crystallization and Aggregation in a Model Crude Oil, Journal of Physics: Condensed Matter, Vol. 17, No. 45, p. 36513660, 2005.
Visintin, R. F. G., Lockhart, T. P., Lapasin, R., and D’Antona, P., Structure of Waxy Crude Oil Emulsion Gels, Journal of NonNewtonian Fluid Mechanics, Vol. 149, No. 1–3, p. 3439, 2008.
Zhang, J. J. and Liu, X., Some Advances in Crude Oil Rheology and Its Application, Journal of Central South University of Technology, Vol. 15, p. 288292, 2008.