Study on the Parametric Regression of a Multiparameter Thixotropic

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Study on the Parametric Regression of a Multiparameter Thixotropic Model for Waxy Crude Oil Qing Yuan, Hongfei Liu, Jingfa Li, Bo Yu, and Changchun Wu Energy Fuels, Just Accepted Manuscript • DOI: 10.1021/acs.energyfuels.8b00626 • Publication Date (Web): 02 Apr 2018 Downloaded from http://pubs.acs.org on April 2, 2018

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Energy & Fuels

Study on the Parametric Regression of a Multi-parameter Thixotropic Model for Waxy Crude Oil Qing Yuan1, Hongfei Liu1, Jingfa Li2, Bo Yu2*, Changchun Wu1 1. National Engineering Laboratory for Pipeline Safety, Beijing Key Laboratory of Urban Oil and Gas Distribution Technology, China University of Petroleum, Beijing 102249, China; 2. School of Mechanical Engineering, Beijing Key Laboratory of Pipeline Critical Technology and Equipment for Deepwater Oil & Gas Development, Beijing Institute of Petrochemical Technology, Beijing 102617, China (*Corresponding author: Tel: +86 10 81292805, E-mail: [email protected])

Abstract The waxy crude oil exhibits shear-dependent and time-dependent thixotropic behavior below the threshold temperature. Various models have been proposed to describe this type of behavior in previous literatures. Due to the complexity of thixotropic behavior, the models with good-description ability generally contain a large number of parameters, which results in the difficulty of parametric regression. The double-structural-parameter model shows common features of complex thixotropic models, i.e. containing a great many parameters and possessing good fitting effect, and it is a representative model for the study on parametric regression. In this study, the double-structural-parameter model is introduced, and three regression strategies of this model are proposed. To obtain excellent fitting effect as much as possible and make estimation of the obtained parameters easy, an advanced self-adaptive differential evolution algorithm is adopted, and then it is improved to accommodate parametric regression. Through the implement of parametric regression, it is found that the obtained partial parameters show unstable characteristic for three regression strategies of double-structural-parameter model. By the in-depth analysis of results, the unstable phenomenon is explained, and it can be avoided by using a new thixotropic experimental method. The final regressive results indicate that the regression strategy III can compromise good fitting effect and physical interpretation, and it is recommended in this study. In addition, it is pointed out that the fitting effect, stability of parameters and physical interpretation should be considered simultaneously for the parametric regression. Although this study is conducted on the basis of double-structural-parameter model, it can provide beneficial guidance to parametric regression of other complex thixotropic models, or even other multi-parameter models in other research fields.

Keywords: waxy crude oil; thixotropic behavior; double-structural-parameter model; parametric regression; self-adaptive differential evolution algorithm; new thixotropic experimental method

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1. Introduction The petroleum is a primary and essential energy source around the world [1]. The waxy crude oil comprises about 20% of the world petroleum reserves and is being produced owing to scarcity of more conventional crude oil [2]. The shutdown and restart of the pipeline transporting waxy crude oil are important consideration for the flow assurance [3]. During the shutdown of pipeline in low ambient temperature, the molecules of paraffin start to precipitate and form solid wax crystals in the crude oil when the oil temperature decreases to the wax appearance temperature (WAT). As the temperature further drops, the precipitated wax crystals become more and more, and they interlock and form a three-dimensional sponge-like network when the temperature falls down to the threshold temperature (or gelation temperature). The strength of interlocking network gradually increases with the continuing decrease of temperature, which brings a threat to the restart of pipeline and may cause the congelation of pipeline [4]. In order to ensure the restart safety of pipeline, it is very necessary to investigate the rheological behaviors of gelled crude oil, such as thixotropy and yield stress [5]. The thixotropic behavior of waxy crude oil is complex, because it is influenced by lots of factors including shear history, thermal history, wax content, wax type and so on. Therefore, it is challenging to predict the thixotropic behavior accurately. In spite of this reason, some in-depth studies have been conducted through unremitting efforts of researchers, and various models have been proposed to describe thixotropic behavior. The numerous thixotropic models are based on three different methods: a continuum mechanics method, a microstructural one and a structural kinetic one [6, 7]. Among these three methods, the structural kinetic method might be more suited for a generally thixotropic model than other two methods, and so far the structural kinetic model is commonly adopted in the literature [6]. This type of model generally utilizes a structural parameter λ (there is more than one structural parameter in some models) to describe the state of structure [8]. And as for the structural parameter λ, values of zero and one respectively represent the completely broken down and the entirely built up structures. The structural kinetic model usually comprises a rheological constitutive equation and a kinetic equation (there is more than one kinetic equation in some models). In general, the first one describes the constitutive relation between shear stress and shear rate for a given λ, and the second one expresses the change of λ as a function of the flow conditions [9]. So far, there are some representative structural kinetic models in the literature, such as Houska model [10], Zhao model (also named double-structural-parameter model) [11-14], Dullaert model [15], Mendes model [16, 17], Teng model [1, 18] and Geest model [19]. Due to the complexity of thixotropic behavior, thixotropic models with good fitting effect generally contain a large number of parameters, which need to be determined. In some reports [1, 9, 13-20], it is pointed out that these parameters can be obtained by the least square method on the basis of experimental data. However, for the parametric regression of a thixotropic model, the process of implementation is not specific and not given in detail. The implementation of parametric regression may be various, such as an iterative method based on the partial derivatives of objective function to determined parameters equaling zero [21, 22], steepest descent algorithm, Newton iterative algorithm, conjugate gradient algorithm and so on [23, 24]. These traditional iterative methods or algorithms are closely related to the iterative initial-values of the parameters, i.e., in which the obtained parameters may be different with the change of the initial-values of the parameters. The reason is that the traditional methods or algorithms only 2


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obtain a local minimum of objective function but does not ensure to obtain a global minimum [24]. Thus the problem appears: a thixotropic model possessing good fitting effect may display unfavorable performance if the iterative initial-values are not carefully chosen. In other words, an excellent thixotropic model may be misjudged as a poor one due to the unreasonability of parametric regression. To match the fitting effect with the thixotropic model, the good regression strategy and regression algorithm are required, which can make the study or comparison of thixotropic models convincing. Based on this reason, the study on parametric regression of a multi-parameter thixotropic model is conducted in this work. The double-structural-parameter model is representative for the study on parametric regression, because it contains twelve parameters and is complex [11-14]. A great many parameters generally mean that it is difficult to perform the parametric regression. On the other hand, many previous works reported that this thixotropic model possessed good fitting effect [11-14]. Based on the consideration of complexity and fitting effect, we choose the double-structural-parameter model in this study. The features, i.e., containing a great many parameters and possessing good fitting effect, are common for representative thixotropic models nowadays. This study is very meaningful and it can provide the reference to the parametric regression of some complex thixotropic models. Besides, this study can also make the evaluation and comparison of thixotropic models more convincing in the future work.

2. A multi-parameter thixotropic model The double-structural-parameter model was put forward by Xiaodong Zhao [11], which is based on Houska model [10] and is inspired by the Cheng’s hypothesis [25] (there are reversible and irreversible structures in waxy crude oil). This model consists of a rheological constitutive equation and two kinetic equations, as shown in Eqs. (1) - (3). The first kinetic equation contains a structure buildup term and a structure breakdown term, in which λ1 represents the reversible structure in waxy crude oil. The second one only contains a structure breakdown term, in which λ2 represents the irreversible structure. This thixotropic model contains twelve parameters, which are τy0, τy1, τy2, K, ΔK1, ΔK2, a1, b1, b2, m1, m2 and n.

   y0   y11   y22    K  K11  K22   n

(1)

d1  a1 1  1   b1 m1 1 dt

(2)

d2  b2 m2 2 dt

(3)

where τ is the shear stress, Pa. τy0 is the permanent part of static yield stress, Pa. τy1 and τy2 are the thixotropic parts of static yield stress corresponding to recoverable and unrecoverable structures respectively, Pa. λ1 and λ2 are the structure parameters of recoverable and unrecoverable structures respectively, which range from 0 to 1. K is the permanent part of consistency index, Pa·sn. ΔK1 and ΔK2 is the thixotropic parts of consistency index corresponding to recoverable and unrecoverable structures respectively, Pa·sn.  is the shear rate, s-1. n is the power-law exponent. a1 is the buildup rate constant of recoverable structure, s−1. b1 and b2 are the breakdown rate constants of recoverable and unrecoverable structures respectively, s m1 1 , s m2 1 . m1 and m2 are model parameters controlling the degree of the shear-dependency of recoverable and unrecoverable structures respectively. 3


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3. Regression strategies In contrast to various thixotropic models, regression strategies are much less, and they can be divided into two categories: holistic regression [9, 13, 14, 18, 20] (named strategy I) and stepwise regression [1, 6, 16, 17, 19] (named strategy II). In holistic regression strategy, it only takes one time to obtain all parameters by parametric regression based on thixotropic experimental data while two or more times are needed by utilizing thixotropy and other tests in stepwise regression strategy. These two strategies will be introduced in detail on the basis of double-structural-parameter model in this section. Besides, the third strategy is proposed and depicted in this study, which is a combination (named strategy III) of holistic regression and stepwise regression. 3.1 Strategy I Through the thixotropy test such as hysteresis loop or stepwise changes in shear rate, the experimental data can be obtained. According to the least square method, the regression model can be written, as shown in Eq. (4). In the Eq. (4), twelve parameters need to be determined, and they can be obtained by using regression algorithm. The detail regression algorithm is not given in this section, and it will be introduced in Section 4. f  XI  



1 N   e k   c  X I  k N k 1



2

(4)

11 12 where f represents the objective function. X I   x1I , xI2 , xI3 , xI4 , xI5 , xI6 , xI7 , xI8 , xI9 , x10 , and represents (τy0, τy1, I , xI , xI 

τy2, K, ΔK1, ΔK2, a1, b1, b2, m1, m2, n). N is the total number of experimental data in thixotropy test. k is the serial number of experimental data in thixotropy test. τe is the shear stress which is obtained from the thixotropy test, Pa. τc is the shear stress which is obtained from the thixotropic model, Pa. 3.2 Strategy II The strategy I is simple and it is easy to understand. But the obtained parameters generally lack physical interpretation [21]. To settle this problem, the stepwise regression, namely strategy II, is proposed, and it is introduced in the following text. If recoverable and unrecoverable structures are not broken down (λ1=λ2=1) and the flow does not but will occur (  =0), the Eq. (1) can be simplified, as shown in Eq. (5).

   y0   y1   y2

(5)

The above description is analogous to the static yield stress test, and thus the static yield stress satisfies the below relation with some parameters of double-structural-parameter model.

 y   y0   y1   y2

(6)

where τy is the static yield stress of waxy crude oil, Pa. For the constant shear rate, the integration calculation can be carried out for Eq. (2) and Eq. (3), and Eq. (7) and Eq. (8) can be obtained respectively.

1 =

m  a1  b1 1 t a1 b1 m1  e a1  b1 m1 a1  b1 m1

2 = eb  2

m

2t

4


(7) (8)

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The buildup rate of structure is generally very small, and is much less than the breakdown rate of structure [5], which means that the a1 is much less than the b1 m1 in the condition where the shear rate is greater than or equal to one. If the flow occurs and the loading time of the shear is long enough, it can be known from Eq. (7) and Eq. (8) that the λ1 and λ2 are close to zero. Thus the Eq. (9) can be written by simplifying Eq. (1).

   y0  K n

(9)

It is known that the Eq. (9) is the expression of Herschel-Bulkley model [26]. And the parameters τy0, K and n can be obtained from the equilibrium flow curve (or steady state curve). The remaining parameters τy1, τy2, ΔK1, ΔK2, a1, b1, b2, m1 and m2 can be determined via fitting to thixotropic experimental data. In conclusion, the relation of the parameters τy0, τy1, τy2 and the values of τy0, K, n can resolved by static yield stress and equilibrium flow curve respectively in this study, and the values of the parameters τy1, τy2, ΔK1, ΔK2, a1, b1, b2, m1 and m2 can be obtained from the thixotropy test. In strategy II, the regression model can be written, as shown in Eq. (10) and Eq. (11).



1 N   e k   c  X II  k N k 1

f  X II  



2

x1II  xII2   y   y0

(10)

(11)

where X II   x1II , xII2 , xII3 , xII4 , xII5 , xII6 , xII7 , xII8 , xII9  , and represents (τy1, τy2, ΔK1, ΔK2, a1, b1, b2, m1, m2). 3.3 Strategy III There is an experimental error in the experiment because of the sample, equipment and operation, which is the inherent feature of experiment. When two or more different experiments correlate with each other, the tiny error of one experiment may significantly influence another experiment or other experiments [22]. Considering this fact, in strategy II, the errors of static yield stress and equilibrium flow curve, attributing to experiment, may result in poorly fitting to thixotropic experimental data. Based on this consideration, the strategy II is improved and thus the strategy III is proposed. In strategy III, the static yield stress and equilibrium flow curve are amended slightly to obtain better fitting effect, and the regression model can be written, as shown in Eqs. (12) - (14).



1 N   e k   c  X III  k N k 1

f  X III  

x

1 III

x

1 III



2

2 3  xIII  xIII   y   y



4  xIII  xIII   y0'  K ' n '     y0'  K ' n '  12

(12)

(13) (14)

3 4 5 6 7 8 9 11 12 where X III   x1III , xIII2 , xIII , and represents (τy0, τy1, τy2, K, ΔK1, ΔK2, a1, b1, b2, m1, , xIII , xIII , xIII , xIII , xIII , xIII , x10 III , xIII , xIII 

m2, n). δ is a ratio of acceptable deviation for the static yield stress and equilibrium flow curve.  y0' , K ' and n ' are the parameters of Herschel-Bulkley model, which can be obtained by utilizing equilibrium flow curve. According to the static yield stress, equilibrium flow curve and acceptable deviation, the ranges of x1, x2, x3, x4 and x12 are restricted in strategy III. It can be easily known that the strategy III degrades to the strategy II when 5


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the δ equals zero and the strategy III degrades to the strategy I when the δ equals positive infinity. Thus the strategy III can represent other two strategies, and the strategies I and II are only special cases of strategy III. In fact, the value of δ is crucial for the parametric regression. If the value is too small, the static yield stress and equilibrium flow curve, which are obtained from the experiment, will excessively influence parametric regression of thixotropic model. On the contrary, if the value is too large, the static yield stress and equilibrium flow curve are meaningless and the obtained parameters may lack physical interpretation. Thus the moderate value of δ may can settle this conflict. The 10% deviation of static yield stress and equilibrium flow curve is generally acceptable, and thus it is adopted in this study. Other reasonable deviations (for example, 5%~25%) can also be adopted, which depends on the required precision for the static yield stress and equilibrium flow curve. The larger δ generally results in larger deviation of obtained parameters from the measured results from the static yield stress and equilibrium flow curve test, but induces better fitting to thixotropic experimental data due to less strict constraints.

4. Regression algorithm In Section 1, some algorithms have been introduced and their shortcomings also have been pointed out. To overcome initial-value problem of parametric regression as much as possible, an advanced differential evolution algorithm with self-adaptive control parameters setting [27] (or called self-adaptive differential evolution algorithm [27, 28]) is adopted, which is proposed by Brest et al. This algorithm is an effective global optimization algorithm, and it has been verified by using 21 representative benchmark optimization problems. In the process of optimization implement, the initial-values of parameters can be chosen randomly, which hardly affects the final results of optimization. Thus the adoption of this algorithm contributes to the study on the parametric regression of thixotropic model. According to the reference [27], this algorithm completely consists of initialization, mutation, crossover, selection and update, and it can be summarized as follows. (1) Initialization: G  0  j j j j  xi ,G   xu    xl    rand  0,1   xl    Fi ,G  0.5  Ci ,G  0.9

(15)

where G is the generation. i=1, 2, … , NP (NP is the population size), and represents the individual in the population. j=1, 2, … , D (D is the dimension of variable vector, which equals 12 in strategies I or III and 9 in strategy II), and represents the serial number of variable (i.e. parameter). xl and xu are lower and upper bounds of variable respectively. rand(0, 1) represents a uniformly distributed random variable within the range [0, 1]. F is the amplification factor of difference vector. C is the crossover control parameter. (2) Mutation:  Fa  rand  0,1  Fb , if rand  0,1  1  Fi ,G 1   otherwise   Fi ,G ,



yij,G 1  xr1j ,G  Fi ,G 1 xrj2 ,G  xr3j ,G 6




(16)

(17)

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where y is the mutant variable. r1, r2, r3 are random indexes within the range [1, NP], which should be mutually different and also different from index i. Fa, Fb, η1 are control parameters of F, which are set as 0.1, 0.9 and 0.1 respectively [27]. It is noteworthy that the mutant variable may goes off the range [xl, xu] in the mutation. In this case, the mutant variable will be set to random value within the range [xl, xu] in this study. (3) Crossover:  rand  0,1 , if rand  0,1  2 Ci ,G 1   otherwise  Ci ,G ,

(18)

j   yi ,G 1 , if rand  0,1  Ci ,G 1 or j  jrand zij,G 1   j   xi ,G , otherwise

(19)

where z is the trial variable. jrand is a randomly chosen index within the range [1, D], which ensures that at least one variable is from mutation in every individual. η2 is a control parameter of C, which is set as 0.1 [27]. (4) Selection:  zi1,G 1 , zi2,G 1 ,…, ziD,G 1  , if g  zi1,G 1 , zi2,G 1 , …, ziD,G 1   g  xi1,G , xi2,G , …, xiD,G  1 2 D x , x , … , x   i,G1 i,G1 i,G1   1 2 D otherwise   xi ,G , xi ,G ,…, xi ,G  ,

(20)

where g is the fitness function, whose expression is complex in this study and will be given in Section 6. (5) Update: The variables for all individuals in Gth generation are updated by those in (G+1)th generation. Then the mutation, crossover and selection are repeated until the prescribed maximum generation is satisfied. The implementation process of self-adaptive differential evolution algorithm is shown above. It can be found that the ranges [(xl)j, (xu)j ] (j=1, 2, …, D) of parameters (i.e., variables) need to be given in the implementation process. However, for parametric regression, the ranges of parameters generally are unknown and need to be set artificially, which is different from ordinary optimization. In order to obtain optimal values of parameters as much as possible, the ranges are often set as large enough on the basis of the physical interpretation of parameters. This method is feasible for the regression of most parameters. However, the exception sometimes appears, such as the regression of parameters a1, b1 and b2, which is found by numerical experiments. The explanation of this problem is given in the below. The real values of parameters a1, b1 and b2 are small and are close to zero. However, it cannot be ascertained how small these three parameters are, such as 10-1 or 10-3, even smaller value. To contain enough small-value belonging to different orders of magnitude, the ranges are set as large enough, for example, [0, 1], which contains small-values 10-1, 10-2, 10-3 and so on. Although these small-values are contained in the range [0, 1], it can be known that the values within the range [10-3, 1] easily appear and those within the range [0, 10-3] barely appear in the implement process of algorithm according to the principle of generating random numbers. This means that a great many small-values are neglected, which may lead to the failure to obtain the optimal parameters a1, b1 and b2. In order to solve this problem, the new initialization and mutation are proposed for the parameters a1, b1 and b2, as shown in Eq. (21) and Eq. (22) respectively. 7


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

j' j' j' xij,0'  10 ^ lg  xu   lg  xl    rand  0,1  lg  xl   







yij,G' 1  10 ^ lg xr1j ,' G  Fi ,G 1 lg xrj2',G  lg xr3j ',G   

(21) (22)

where x j ' only represents a1, b1 or b2. In Eq. (21) and Eq. (22), the zero cannot be used as xl, but a small-value can be employed, such as 10-6 or 10-7 which is small enough. It is noteworthy that the Eq. (21) and Eq. (22) increase the possibility of the appearance of small-value, but each order of magnitude can only contain a fairly small number of individuals in the initialization if the population size (NP) is small. The recommended NP is between 5*D and 10*D [29], which may be not applicable due to less individuals at each order of magnitude. According to previous recommended range of NP and taking two orders of magnitude as the benchmark, the NP is amplified to obtain more individuals at each order of magnitude in this study, as shown in Eq. (23). And the value of NP is calculated by using Eq. (23). Three control parameters of differential evolution algorithm (i.e., F, C and NP) are determined by Eq. (16), Eq. (18) and Eq. (23), and the self-adaptive differential evolution algorithm can applied to the parametric regression of thixotropic model. lg  xu   lg  xl  NP  lg102 j'

j'

 5* D ~ 10* D 

(23)

It is worth noting that this regression algorithm is also applicable to other thixotropic models because its implementation does not vary with the change of the expression of thixotropic model. As for different thixotropic models, only the number of parameters (or variables) and the selections of initialization & mutation are different. The number of parameters is 12 (or 9) for the double-structural-parameter model in this study while it may be another value for other thixotropic models. In the initialization and mutation, the Eq. (21) and Eq. (22) should be selected for the parameters close to zero while the second expression of Eq. (15) and Eq. (17) should be selected for the large parameters.

5. Experiments As is introduced in Section 2, the thixotropy test is only needed for strategy I and the thixotropy, equilibrium flow curve and static yield stress tests for strategies II and III. And these three tests will be introduced in the following text. All measurements including thixotropy, equilibrium flow curve and static yield stress tests are performed by using the rheometer HAAKE RS150H configured with the coaxial cylinder sensor system Z41Ti, and the temperature of crude oil is controlled by a programmable water bath HAAKE AC200 with the accuracy of 0.01 ℃. Firstly, the prepared fresh waxy crude oil sample (whose WAT and gelation temperature are 45 ℃ and 39 ℃ respectively) is heated to 50 ℃ and kept at this temperature for 20 min. Subsequently, the sample is loaded into the measuring cylinder which is preheated to 50 ℃, and then it is kept at 50 ℃ for 10 min. Thereafter, the sample is statically cooled to the test temperature at a cooling rate of 0.5 ℃·min-1, and then it is kept isothermally for 90 min to make the wax crystal fully developed. Finally, the thixotropy, equilibrium flow curve or static yield stress test is performed isothermally at the required temperature. 8


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In this study, the chosen thixotropy test is the stepwise changes in shear rate, which is deemed to be a good test for the study on the thixotropic behavior of waxy crude oil [1, 5]. In thixotropy test, the adopted shear rates are 1, 2, 4, 8, 16, 32, 64 and 128 s-1 for each chosen test temperature 35, 37 or 39 ℃, and each shear rate lasts for 5 min. All results of thixotropy test are shown in Fig. 1. 27

140 35℃ 37℃ 39℃ Shear rate

24

18

120 100

15

80

12

60

9

40

-1

Shear stress (Pa)

21

Shear rate(s )

6 20

3 0 0

300

600

900

0 1200 1500 1800 2100 2400 Time (s)

Fig. 1 Stress responses to stepwise changes in shear rate at different temperatures

The equilibrium flow curve test is similar to the thixotropy test in this study, but the former adopts stepwise decreases in shear rate while the latter adopts stepwise increases in shear rate [30, 31]. In this test, the adopted shear rates are 128, 64, 32, 16, 8, 4, 2 and 1 s-1 for each chosen test temperature 35, 37 or 39 ℃, and each step is kept for 30 min so that the steady state (or equilibrium) has been reached. And the obtained equilibrium flow curves are shown in Fig. 2. 15 35℃ 37℃ 39℃

12

Shear stress (Pa)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Energy & Fuels

9 6

3

.0.64 8 5 5 . 2 +0 .0.74 51 2.0 84 2 = . 0 8  .0.80 71+ 1.6 70 1 = . 0 90+ 0.7 =

3 0 0

20

40

60

80

100

120

140

Shear rate (s ) -1

Fig. 2 Equilibrium flow curve at different temperatures

We use stress ramp to determine the static yield stress [32, 33]. The shear stress is applied from 0 to 60 Pa at rate of 6 Pa·min-1. The yield point is considered as the point at which there is a remarkable increase in the strain, and the shear stress at the yield point is determined as the static yield stress. The Fig. 3 shows static yield stress at different temperatures.

9


Energy & Fuels

14 11.6

12 10

Yield stress (Pa)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 10 of 21

8 6.1

6 4

3.2

2 0 34

35

36 37 38 Temperature (℃)

39

40

Fig. 3 Static yield stress at different temperatures

6. Implement of parametric regression The regression models have been introduced in Section 3, but they cannot match regression algorithm completely because the regression models contain constraints while regression algorithm does not. To settle the problem, the penalty function method is applied [22, 23]. In this section, taking the most complex strategy III as an example, the regression model can be rewritten, as shown in Eq. (24). f   X III  



1 N   e k   c  X III  k N k 1





2

1 2 3   min  xIII  xIII  xIII     1 y ,0   min   x1III  xIII2  xIII3     1 y ,0





(24)



12 12 1 4 1 4   min  xIII  xIII  xIII    1  y0'  K ' n '  , 0   min  xIII  xIII  xIII    1  y0'  K ' n '  ,0    

where σ is a positive scalar parameter, and it is set as 100 in this study. In Fig.1, the experimental data shows that the beginning stage of structure breakdown contains very few data points, which is general feature of stepwise shear rate test. These few data points may be flooded by other dense data points, because more data points better matching thixotropic model can make the total deviation between experimental and regressive data smaller. However, the beginning stage of structure breakdown is highly important for the thixotropy behavior of waxy crude oil. Considering the important significance of the beginning stage of structure breakdown, a constraint should be added, as shown in Eq. (25). Due to the existence of deviation for experimental data, an acceptable deviation is needed in Eq. (25).

 e 1   c  X III 1     e 1

(25)

where the subscript 1 represents the first data point in the beginning stage of structure breakdown. δ′ is a ratio of acceptable deviation for the first data point and is set as 10% in this study, whose value equals value of δ. Based on the consideration of the beginning stage of structure breakdown, the regression model is further amended, as shown in Eq. (26). f   X III  



1 N   e k   c  X III  k N k 1





2

1 2 3   min  xIII  xIII  xIII     1 y ,0   min   xIII1  xIII2  xIII3     1 y ,0







12 12 1 4 4   min  xIII  xIII  xIII    1  y0'  K ' n '  ,0    min   x1III  xIII  xIII    1  y0'  K ' n '  , 0    







(26)



  min  c  X III  1   '  1  e 1 ,0   min   c  X III  1   '  1  e 1 , 0

In this study, the f′′ is used as the fitness function, i.e., g=f′′. Then according to the regression algorithm introduced in Section 4, the parametric regression of double-structural-parameter model can be implemented. 10


Page 11 of 21

7. Results and discussion 7.1 The parametric regression for the stepwise shear rate In this study, the population size and maximum generation are set as 360 and 3000 respectively. Average fitness of all individuals in each generation is shown in Fig. 4. The Fig. 4 indicates that the average fitness drops gradually with the increase of generation and then becomes stable. The stable average fitness implies that the fitness of each individual is the same (see Fig. 5) for each regression strategy and that the parametric regression is completed. Besides, it can be found from Fig. 4 and Fig. 5 that the final fitness corresponding to strategy I is minimal and the final fitness corresponding to strategy II is maximal. The finding can be explained easily. From the viewpoint of optimization, the stricter constraints of regression model cause higher fitness due to the decrease of freedom degree of variables (or parameters). 2

Average fitness

1

Strategy I Strategy II Strategy III

35℃

0 2 1

37℃

0 2 1 39℃ 0 0

500

1000

1500 2000 Generation

2500

3000

Fig. 4 Average fitness of all individuals in each generation 0.6 0.3

Average fitness

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Energy & Fuels


35℃

0.0 0.6 37℃ 0.3 0.0 0.6 0.3

39℃

0.0 0

60

120

180 Individual

240

300

360

Fig. 5 Fitness of each individual in 3000th generation

In 3000th generation, the fitting effect corresponding to each individual is the same for each regression strategy, and the fitting effects corresponding to three regression strategies are shown in Fig. 6. It can be known from Fig. 6 that three regression strategies obtain good fitting effects when the shear rate is small, but the fitting effects corresponding to strategies II and III gradually become poor with the increase of shear rate. And the strategy III obtains better fitting effect than strategy II, but worse fitting effect than strategy I. It is easily understood that the good fitting effect is consistent with the small fitness.

11


Energy & Fuels

27

27 Experimental data Regressive data

24

Experimental data Regressive data

24

21

21

18

18

Shear stress (Pa)

Shear stress (Pa)

15 12 9

15 12 9

6

6

3

3

0

0 0

300

600

900

1200 1500 1800 2100 2400 Time (s)

0

300

(a) Strategy I

600

900

1200 1500 1800 2100 2400 Time (s)

(b) Strategy II 27 Experimental data Regressive data

24 21

Shear stress (Pa)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 12 of 21

18 15 12 9 6 3 0 0

300

600

900

1200 1500 1800 2100 2400 Time (s)

(c) Strategy III Fig. 6 Fitting effects for three regression strategies in 3000th generation

From the above results, it cannot be concluded that the strategy I is optimal for the parametric regression. The fitting effect is important for a thixotropic model, but the obtained parameters are equally important and should be concerned specially. The previous studies generally focused on the former and rarely concerned the latter, which may lack a suitable regression algorithm. In this study, based on the obtained parameters at 35 ℃, the importance of the latter will be introduced in detail below. According to regression models, regression algorithm and experimental data, the parameters of double-structural-parameter model are obtained, and they are displayed in two kinds of ordinates to make results clearer, as shown in Figs. 7-9 (all zero values are not shown in Fig. 7(b), Fig. 8(b) and Fig. 9(b)). The regressive results are puzzling. The obtained parameters τy2, ΔK2, m2 show unstable characteristic but other parameters do not in Fig. 7, while the parameter m2 shows unstable characteristic but other parameters do not in Fig. 8 and Fig. 9, although the final fitness of each individual and the fitting effect corresponding to each individual are the same for each strategy. The detail explanation is required for this phenomenon and it is given as follows.

12


10

y0 y1 y2 K K1 K2 a1 b1 b2 m1 m2 n

Obtained value

8 6 4 2

10

y0 y1 y2 K K1 K2 a1 b1 b2 m1 m2 n

1 0.1 0.01 1E-3 1E-4

0 0

60

120

180 240 Individual

300

0

360

60

(a) Ordinary ordinate

120

180 240 Individual

300

360

(b) Logarithmic ordinate

Fig. 7 Obtained parameters for the strategy I in 3000th generation 10

10

Obtained value

6 4

y1 y2 K1 K2 a1 b1 b2 m1 m2

1

Obtained value

y1 y2 K1 K2 a1 b1 b2 m1 m2

8

0.1 0.01 1E-3

2

1E-4

0 0

40

80

120

160 200 240 280 Individual

320

360

0

60


120

180 240 Individual

300

360


Fig. 8 Obtained parameters for the strategy II in 3000th generation y0 y1 y2 K K1 K2 a1 b1 b2 m1 m2 n

8 6 4 2

10

y0 y1 y2 K K1 K2 a1 b1 b2 m1 m2 n

1

Obtained value

10

Obtained value

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Energy & Fuels

Obtained value

Page 13 of 21

0.1 0.01 1E-3 1E-4

0 0

60

120

180 240 Individual

300

360

0

60


120

180 240 Individual

300

360


Fig. 9 Obtained parameters for the strategy III in 3000th generation

It is known that all unstable parameters τy2, ΔK2, m2 are related to the structure parameter λ2, as shown in Eq. (1) and Eq. (3), and thus the analysis begins with this structure parameter. Although the obtained parameter m2 is unstable, the structure parameter λ2 can be computed by using Eq. (3) in the first stage of stepwise shear rate owing to the shear rate equaling one. And the changing trend of λ2 in the first stage is obtained, as shown in Fig. 10. It can be seen that the λ2 drops quickly and then becomes zero which is the limit of λ2. The emergence of zero value implied that the unrecoverable structure no longer breaks down and thus the λ2 is also zero in other stages of 13


Energy & Fuels

stepwise shear rate. This indicates that it is only in the first stage that the valid value (nonzero value) of λ2 can work. Therefore, the double-structural-parameter model can be rewritten, as shown in Eqs. (27) - (29). The constraints of three regression strategies are different: the strategy I does not contain the constraints of static yield stress and equilibrium flow curve while the strategies II and III do. Because the  s1 equals to one, it can be known easily from Eq. (29) that the difference of m2 does not affect the change of λ2 for three regression strategies. Besides, it is found from Fig.7 that the sum of τy2 and ΔK2 equals to the constant 6.03 although they both are unstable. According to the Eq. (27), this phenomenon is understood easily, and the reason is that the sum of τy2 and ΔK2 is optimal but the τy2 and ΔK2 can be free to combine for the obtaining of minimum fitness. However, due to the constraint of static yield stress on τy2 for strategies II and III, the τy2 cannot be free, which also induces the fixation of ΔK2. And thus the τy2 and ΔK2 are stable for strategies II and III. 1.2 Strategy I Strategy II Strategy III

1.0 0.8

λ2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 14 of 21

0.6 0.4 0.2 0.0 0

50

100

150 Time (s)

200

250

300

Fig. 10 Changing trend of structure parameter λ2 in the first stage of stepwise shear rate for three regression strategies

   y0   y11    K  K11   sn ,s ,   y2  K2 sn  2 1

2

(27)

1

d1  a1 1  1   b1 sm1 1,s2 ,1 dt d2  b2 sm1 2 2 dt

(28) (29)

where s1, s2, …, represent different stages of stepwise shear rate. According to the analysis above, it may be doubtful that whether or not the unstable phenomenon results from the shear rate equaling one in this first stage of stepwise shear rate. Therefore the further analysis is needed. If the  s1 does not equal to one, based on the similar analysis in the above text, it can be inferred easily that the product of b2 and  sm is optimal but the b2 and  sm can be free to combine for the obtaining of minimum fitness. 2

1

2

1

This inference means that b2 and m2 may be unstable for three regression strategies when the  s1 does not equal to one. Another inference can also be obtained, i.e., the τy2 and ΔK2 are unstable for strategy I but they are stable for strategies II and III on the basis of similar analysis. Therefore, it can be concluded that the unstable phenomenon is existing for three regression strategies no matter whether the  s1 equals one. However, the unstable phenomenon is unacceptable for numerically simulating the restart of waxy crude oil 14


Page 15 of 21

pipeline. In the process of restart, the shear rate is always changing and does not always equal to the constant  s , 1 which means that different values of parameters can get different shear stresses. Based on different shear stresses, it can be known from the governing equations (30) and (31) [34] of restart that the obtained p and u are different, which indicates these results are unreliable. Thus the stable regressive results of parameters are needed for the numerical simulation of restart.  p   u p     u  0  t  

(30)

 u   u u   p     t  

(31)

 



where  is the isothermal compressibility and can be computed by using   1/   /p  , Pa-1. p is the pressure, Pa. u is the velocity tensor, m·s-1. ρ is the density of crude oil, kg·m-3. τ is the stress tensor, Pa. Considering the importance of the stable regressive results of parameters, an effective method needs to be proposed to avoid the emergence of unstable phenomenon. The above analysis is very useful for the settlement of this problem. Because it is only in the first stage of stepwise shear rate that the nonzero value of λ2 can work, partial parameters can be free to combine (or change) and the unstable phenomenon appears. Thus it is inferred easily that the unstable phenomenon will disappear if the nonzero value of λ2 can work in more than one stage of stepwise shear rate. According to this idea, a new experimental method is put forward to avoid the emergence of unstable phenomenon as follows. 7.2 The parametric regression for the new experimental method The new experimental test contains two initial breakdown processes, which can be seen as a combination of constant shear rate and stepwise shear rate, as shown in Fig. 11. In this experimental test, it needs to be pointed out that two fresh waxy crude oil samples were employed at each test temperature, and the first sample was sheared in 1 s-1 while the second one was sheared in 2, 4, 8, 16, 32, 64 and 128 s-1. It is noteworthy that the structural parameters λ1 and λ2 begin to fall from 1 when the fresh waxy crude oil is sheared. 32

140 35℃ 37℃ 39℃ Shear rate

28

100

20 80 16 60

12

40

8

-1

Shear stress (Pa)

24

120

Shear rate(s )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Energy & Fuels

20

4 0 0

300

600

900

0 1200 1500 1800 2100 2400 Time (s)

Fig. 11 Stress responses to shear rate using a new experimental method

In order to confirm the above inference, the obtained parameters are given for three regression strategies, as shown in Figs. 12-14. It can be found from Figs. 12-14 that the all obtained parameters are stable, which indicates that the new experimental method is effective to avoid the emergence of unstable phenomenon. 15


Energy & Fuels

y0 y1 y2 K K1 K2 a1 b1 b2 m1 m2 n

Obtained value

8 6 4 2

10

y0 y1 y2 K K1 K2 a1 b1 b2 m1 m2 n

1 0.1

Obtained value

10

0.01 1E-3 1E-4 1E-5 1E-6 1E-7

0 0

60

120

180 240 Individual

300

360

0

60


120

180 240 Individual

300

360


Fig. 12 Obtained parameters for the strategy I in 3000th generation under a new experimental test 10

10

Obtained value

6 4

y1 y2 K1 K2 a1 b1 b2 m1 m2

1 0.1

Obtained value

y1 y2 K1 K2 a1 b1 b2 m1 m2

8

0.01 1E-3 1E-4 1E-5

2

1E-6 0

1E-7 0

40

80

120

160 200 240 280 Individual

320

360

0

60


120

180 240 Individual

300

360


Fig. 13 Obtained parameters for the strategy II in 3000th generation under a new experimental test y0 y1 y2 K K1 K2 a1 b1 b2 m1 m2 n

8 6 4 2 0

10

y0 y1 y2 K K1 K2 a1 b1 b2 m1 m2 n

1 0.1

Obtained value

10

Obtained value

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 21

0.01 1E-3 1E-4 1E-5 1E-6 1E-7

0

60

120

180 240 Individual

300

360

0

60


120

180 240 Individual

300

360


Fig. 14 Obtained parameters for the strategy III in 3000th generation under a new experimental test

The fitness of each individual and the fitting effects in 3000th generation are shown in Fig. 15 and Fig. 16 respectively. The Fig. 15 and Fig. 16 show that the fitness corresponding to strategy I is minimal and the strategy I obtains better fitting effect than strategies II and III, which is similar to Fig. 5 and Fig. 6. However, it still cannot be concluded that the strategy I is optimal for the parametric regression, because the obtained parameters need to be evaluated further. 16


Page 17 of 21

0.9

Average fitness

0.6


35℃

0.3 0.6 37℃

0.3 0.0 0.6

39℃

0.3 0.0 0

60

120

180 Individual

240

300

360

Fig. 15 Fitness of each individual in 3000th generation under a new experimental test 32


28

Experimental data Regressive data

28 24

Shear stress (Pa)

24

Shear stress (Pa)

20 16 12

20 16 12

8

8

4

4

0

0 0

300

600

900

1200 1500 1800 2100 2400 Time (s)

0

300

(a) Strategy I

600

900

1200 1500 1800 2100 2400 Time (s)

(b) Strategy II


28 24

Shear stress (Pa)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Energy & Fuels

20 16 12 8 4 0 0

300

600

900

1200 1500 1800 2100 2400 Time (s)

(c) Strategy III Fig. 16 Fitting effects for three regression strategies in 3000th generation under a new experimental test

Because the value of τy0+τy1+τy2 and the curve corresponding to    y0  K n are determined by static yield stress and equilibrium flow curve respectively in strategy II, they are consistent with the measured results from experiments and can be interpreted physically. It can be seen from Fig. 17 and Fig. 18 that the value of τy0+τy1+τy2 and the curve corresponding to    y0  K n in strategy II are close to those in strategy III but not those in strategy I. This foundation means that the restart of pipeline successes easily using strategy I because the value of τy0+τy1+τy2 is smaller than static yield stress. Besides, the final restart flow rate is larger using strategy I because 17


Energy & Fuels

the curve corresponding to    y0  K n is lower than actual equilibrium flow curve. This unreasonable phenomenon is attributed to the absence to physical explanation for the obtained parameters. Therefore, although the strategy I possesses the best fitting effect, the obtained parameters are lacked of physical explanation, and their adoption may bring danger to the actual restart of pipeline due to small value of τy0+τy1+τy2 and low curve corresponding to    y0  K n . Comprehensively considering the fitting effect and physical explanation, the strategy III is recommended in this study because it can compromise the fitting effect and physical explanation. 18 Strategy I Strategy II Strategy III

y0+y1+y2 (Pa)

15 12 9 6 3 0 34

35

36 37 38 Temperature (℃)

39

40

Fig. 17 Values of τy0+τy1+τy2 for three regression strategies 18

15 Strategy I Strategy II Strategy III

15


12

Shear stress (Pa)

12 9 6

9 6 3

3 0

0 0

20

40

60

80

100

120

140

0

20

40

Shear rate (s )

60

80

Shear rate (s )

-1

-1

(a) 35℃

(b) 37℃ 15 Strategy I Strategy II Strategy III

12

Shear stress (Pa)

Shear stress (Pa)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 21

9 6 3 0 0

20

40

60

80

100

120

140

Shear rate (s ) -1

(c) 39℃ Fig. 18 Curves corresponding to    y0  K n for three regression strategies

18


100

120

140

Page 19 of 21 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Energy & Fuels

8. Conclusions To settle the parametric regression of the multi-parameter thixotropic model, taking representative double-structural-parameter model as example, the regression strategies and regression algorithm are proposed. Besides, the relevant experimental tests are introduced and conducted, and the parametrical regression of double-structural-parameter model is implemented and then the regressive results are obtained. This study can be concluded as follows. (1) For the parametrical regression of double-structural-parameter model, three regression strategies are proposed and are named strategies I, II and III. The strategy I is a holistic regression, and it only takes one time to obtain all parameters by regression based on thixotropic experimental data. The strategy II is a stepwise regression, and two or more times are needed by utilizing thixotropy, static yield stress and equilibrium flow curve tests. The strategy III is a combination of holistic regression and stepwise regression, and firstly some restrictions are provided by utilizing static yield stress and equilibrium flow curve tests, and then all parameters are obtained by regression based on thixotropic experimental data. (2) To overcome initial-value problem of parametric regression as much as possible, an advanced self-adaptive differential evolution algorithm which is an effective global optimization algorithm, is adopted in this study. In addition, the new initialization and mutation are put forward for the parameters a1, b1 and b2 because their real orders of magnitude cannot be known in advance before the regression. (3) Based on the stepwise shear rate, it is found that the obtained parameters τy2, ΔK2, m2 show unstable characteristic but other parameters do not for the strategy I, while the parameter m2 shows unstable characteristic but other parameters do not for strategies II and III. This unstable phenomenon is unacceptable for numerically simulating the restart of waxy crude oil pipeline. To avoid the emergence of unstable phenomenon, we put forward a new experimental method, which can be viewed as a combination of constant shear rate and stepwise shear rate. (4) Compared with the fitting effect and the obtained parameters for three regression strategies, it is found that the strategy I can obtained best fitting effect but the obtained parameters lack physical interpretation while the strategy II is opposed to strategy I, and that strategy III can compromise the fitting effect and the physical interpretation. Therefore, the strategy III is recommended in this study. Through this study, the self-adaptive differential evolution algorithm is recommended for the parametric regression, because it possesses excellent global search capability and is convenient for parameter analysis. In addition, it is suggested that the fitting effect, stability of parameters and physical interpretation should be considered simultaneously for the parametric regression. Although this study is conducted using double-structural-parameter model, some ideas can extend to the parametric regression of other complex thixotropic models, or even other multi-parameter models in other research fields.

Acknowledgments The study is supported by the Project of Construction of Innovative Teams and Teacher Career Development for Universities and Colleges Under Beijing Municipality (No. IDHT20170507) and the Program of Great Wall 19


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Scholar (No. CIT&TCD20180313).

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Study on the Parametric Regression of a Multiparameter Thixotropic

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