- Email: [email protected]
Validation of wax deposition models with recent laboratory scale flow loop experimental data
Author’s Accepted Manuscript Validation of Wax Deposition Models with Recent Laboratory Scale Flow Loop Experimental Data Auzan A. Soedarmo, Nagu Daraboina, Cem Sarica
www.elsevier.com/locate/petrol
PII: DOI: Reference:
S0920-4105(16)30613-1 http://dx.doi.org/10.1016/j.petrol.2016.10.017 PETROL3677
To appear in: Journal of Petroleum Science and Engineering Received date: 12 May 2016 Revised date: 3 October 2016 Accepted date: 6 October 2016 Cite this article as: Auzan A. Soedarmo, Nagu Daraboina and Cem Sarica, Validation of Wax Deposition Models with Recent Laboratory Scale Flow Loop Experimental Data, Journal of Petroleum Science and Engineering, http://dx.doi.org/10.1016/j.petrol.2016.10.017 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Validation of Wax Deposition Models with Recent Laboratory Scale Flow Loop Experimental Data Auzan A. Soedarmo, Nagu Daraboina, Cem Sarica McDougall School of Petroleum Engineering, The University of Tulsa Abstract Experimental data at various conditions are important for improving wax deposition prediction and upscaling efforts. This study consolidates several single-phase wax deposition data acquired in recent years and validates available wax deposition models for a comprehensive range of experimental conditions. The film mass transfer (FMT), equilibrium (EM), Matzain (1999), and Venkatesan (2004) models were tested against 70 experimental data points (19 sets of initial operating conditions), obtained with four different oils and flow loop facilities. Uncertainty propagation analysis was performed to estimate reasonable ranges of error in the prediction that may be caused by fluid properties or standard correlations instead of the model formulation itself. The FMT model regularly over-predicts the deposition rate, which is consistent with previous theoretical, experimental, and visualization studies. It is observed that EM does not necessarily serves as the lower bound of deposition rate, suggesting that some form of flux reduction effects may exist. Sherwood number analysis was performed to validate conclusions from FMT and EM model assessment. A fix set of shear coefficient values in Venketasan’s model is not sufficient to achieve acceptable accuracy in general, indicating that the shear coefficients are likely to be oil and flow conditions dependent. Matzain’s model delivers relatively superior performance on deposit wax mass and flux predictions among the four models, although it is still unable to predict aging mechanistically and the deposition trends as a whole. In conclusion, more mechanistically rigorous wax deposition models and experimental data are still needed. Keywords: wax deposition models, flow loop experiments, model verification 1.
Introduction Lack of mechanistic understanding has been highlighted as one of the primary shortcomings in
available wax deposition models (Dwivedi et al., 2013; Sarica and Panacharoensawad, 2012; Venkatesan and Creek, 2007). Flow loop testing serves as the primary mean for model benchmarking prior to field application (Huang et al., 2015) and, to some extent, explain the deposition physics. It is important to organize available data and validate existing models with more comprehensive range of experimental conditions. Accurate prediction of single-phase wax deposition is the key enabler to develop predictive tools for more complex multiphase cases. Due to complex nature of wax deposition experiments, models are usually validated against limited data set. Recently, wax deposition flow loop data have been generated extensively, which involve different testing fluids, expanded experimental conditions range, and improved measurement techniques in general. Therefore, it is necessary to continue the models validation efforts with these newer sets of data. Moreover, with time constraints in collecting statistically meaningful data set, an effort is needed to consolidate available experimental data and enhance their
1
accessibility to the best extent possible. A consolidated data also enables more comprehensive data analyses and identification of current knowledge gaps to direct future studies. The wax deposition models need to perform two main tasks in series: quantification of the total wax mass flux coming onto the cold interface, and distribution of this quantity into deposit aging and growth. Error from the first task will conceivably be propagate to the second one. Therefore, it is imperative to prioritize the wax mass flux prediction accuracy as the first step of model improvement/ validation efforts. The second task itself is dependent on the crystal aspect ratio (Hernandez et al., 2003; Sarica and Panacharoensawad, 2012; Soedarmo et al., 2016a), for which a proper closure relationship is still not available. To systematically assess the models, this study focuses on model-to-experiment comparison of deposit wax mass per deposition area (σ), which is more directly related to the performance of models for flux prediction compared to separated thickness or wax content data. The flux quantification depends on the physical assumptions inherent in the models regarding solid-liquid equilibrium state and flow effects to the diffusion process. This manuscript covered the film mass transfer (FMT), equilibrium (EM), Matzain (1999), and Venkatesan (2004) models. By evaluating the capability of these for predicting σ, one could assess the applicability of the physical assumptions made during their formulation. The FMT model assumes independent heat and mass transfer processes which lead to maximum super-saturation and wax concentration gradient near the interface. On the other hand, EM assumes thermodynamic equilibrium condition, which leads to much smaller concentration gradient than FMT. Matzain’s model modifies the EM diffusion equation with empirical correlation to take account of shear stripping and trapped oil in the deposit. The Venkatesan’s model uses an empirical relationship with two coefficients to quantify the shear effect in wax mass flux reduction. With the exception of Matzain’s model, other models use Singh et al. (2000) model to mechanistically predict the increase of wax content in deposit over time (aging). Matzain’s model is the only self-sufficient model among the four and does not need any arbitrary fitting coefficients. However, it is important to note that the empirical constants inherent in this model are determined solely based on South Pelto crude experimental data, hence the model’s performance for other oils at different operating conditions remains to be seen. For convenience, the mathematical formulations of these models are described briefly in the appendix. 2.
Experimental Data Collection and Interpretation To conduct analyses based on σ, experiments producing both raw deposit mass (including the
trapped oil) and wax content data are more reliable. These experiments obtain deposit mass data by pigging the whole pipe or a removable spool piece section upon completion of each test. Mass of the collected deposit is then converted to deposit thickness, knowing the deposit density and test section 2
dimension. Panacharoensawad and Sarica (2013), Singh (2013), Rittirong et al. (2014), Chi (2015), and Agarwal (2016) have performed a total of 19 single-phase experiments (70 data points) with this method using four oils and four different experimental flow loop facilities. In these studies, wax content data from high temperature gas chromatography (HTGC) are available. The operating range for these experiments and the specifications of the four facilities are described in Table 1. The C17+ carbon number distributions and fluid properties of the oils used in these experiments are shown in Figs. 1 and 2, respectively. These oils are the model oil (MO-14), Garden Banks condensate (GB), South Pelto oil (SP), and Caspian Sea condensate (CS). Table 1. Operating conditions of the experiments covered in this study, * denote the length of the removable spool piece only instead of the whole test section.
Panacharoensawad (2012)
Singh (2013)
Rittirong et al. (2014) Chi (2015)
Agarwal (2016)
Fluid Test #ID SP-01 SP-02 SP SP-02R SP-03 GB-01 GB-02 GB GB-03 GB-04 GB-05 GB GB-06 CS-01 CS CS-02 MO-01 MO-02 MO-03 MO-04 MO-05 MO-06 MO-07 MO-08 MO-14
Study
ID [in] 0.651 0.651 0.651 0.651 0.651 0.651 0.651 0.651 2.067 2.067 1.05 1.05 0.622 1.61 0.622 1.049 1.61 0.622 1.049 1.61
L [m] 0.84 0.84 0.84 0.84 0.84 0.84 0.84 0.84 1.52* 1.52* 1,00 1.00 2.44 2.44 2.44 2.44 2.44 2.44 2.44 2.44
Tb [°F] 105 105.4 105.4 105.4 77.6 78.5 78.3 79.2 76.6 77.2 95 95 95 95 95 95 95 95 95 95
3
Ti,ini[°F] 85 85 85 85 61 61 60 61 62 63 77 78 91 91 91 91 91 91 91 91
NRE [-] 2732 4232 4232 5210 7423 11282 15000 18989 3788 11448 1064 5471 12907 12643 5031 5059 4986 8532 8431 8451
τw [Pa] 7.2 15.1 15.1 21.6 13.5 28.2 47.8 72.2 0.4 2.7 0.4 6.9 25.5 3.4 4.7 1.6 0.7 12.0 3.9 1.6
Time points [h] 4, 16, 32, 48 4, 16, 32, 48 4, 16, 32, 48 4, 16, 32, 48 2.67, 8, 16, 24 2.67, 8, 16, 24 2.67, 8, 16, 24 2.67, 8, 16, 24 4, 12, 24 4, 12, 24 2, 8, 20, 24, 48, 72 2, 8, 20, 24, 72 2, 4, 12, 24 2, 4, 12, 18, 24 2, 4, 12, 24 2, 4, 12, 24 2, 4, 12, 24 2, 4, 12, 24 2, 4, 12, 24 2, 4, 12, 24
Figure 1. C17+ carbon number distribution of GB (Rittirong, 2014; Singh, 2013), SP (Panacharoensawad, 2012), CS (Chi, 2015), and MO-14 (Agarwal, 2016) oils.
Figure 2. Viscosity (a) and solubility (b) curves of MO-14 (Agarwal, 2016), CS (Chi, 2015), GB (Rittirong et al., 2014; Singh, 2013), and SP (Panacharoensawad and Sarica, 2013) oils. It can be seen from Table 1 and Fig. 2 that all of the tests were performed under sub-cooled conditions, hence the thermal restriction effect can be neglected (Venkatesan, 2004). The uncertainty of wax content measurement using HTGC is reported to be 1.5%-wt. (Panacharoensawad and Sarica, 2013), while the relative uncertainty of direct mass measurement is expected to be less than 1% (Panacharoensawad, 2012). Note that since the wax content uncertainty is given in absolute value, the uncertainty ranges for data with lower wax content would be greater. The uncertainties caused by procedural error is assumed minimal owing to good experimental practice. The uncertainties caused by 4
errors due to operating conditions settings (flow rates and temperatures) will be covered in the uncertainty propagation analysis for the model predictions instead of the experimental data. The experimental data used for this analysis is summarized in Table 2. Errors in deposition prediction accumulate over time. Given this consideration, a model analysis solely based on σ parameter might become vague at later time although it will still serve as a valuable performance indicator in a hindsight. Therefore, the experimental wax mass flux ( ̇
) values were
extracted from the deposit mass data to exclude the time effects. To further normalize the driving force discrepancies, the actual mass transfer coefficient ( ⁄
(
|
) and interfacial concentration gradient
) are back-calculated from experimental data. These values indirectly represent the actual
wax concentration gradient at the interface for FMT and EM models, respectively.
With these
procedures, one could assess each model performance in predicting mass transfer rate more thoroughly. The procedures to extract the ̇
,
, and
⁄
|
values from experimental data are shown in
Eqs. 1 and 2. One should be aware that these values may have greater uncertainties than σ due to the inherent uncertainties in derivative operation of experimental data and wax precipitation curves. (
(
̅ )
̅ )
( ) (
)
(
)
( )
̇
( ̇
|
( ))
Eq. 1
|
Eq. 2
PVT simulations was used to generate the wax precipitation curves (as shown in Fig. 2b). To illustrate the magnitude of uncertainties in the wax precipitation curves, Fig. 3 provides the MO-14 precipitation curves generated with various methods (Agarwal, 2016). At the experimental conditions (Tb = 95° F and Ti, ≈ 91 to 94.8° F), the uncertainties of
and
⁄
| values are estimated to be 29.3
– 33.1% and 33 – 39%, respectively (assuming no error in temperature measurements/ calculations). Due to unavailability of data, it is not possible to conduct similar analysis for other oils. Therefore, the uncertainty values for MO-14 are assumed applicable for other oils discussed in this study.
Table 2. Summary of the experimental data (Agarwal, 2016; Chi, 2015; Panacharoensawad and Sarica, 2013; Rittirong et al., 2014; Singh, 2013). ̅
σ [g/m2]
m [g]
δ [mm]
25.6
0.37
11.0
23.3
29.3
0.43
12.8
31.0
12
57.1
0.87
14.6
68.9
24
61.2
0.93
20.4
103.0
Test t [h] 2 4 12 18 24 MO-02
MO-01
Test t [h] 2 4
5
m [g] δ [mm]
̅
σ [g/m2]
116.0
0.67
5.9
21.7
154.9
0.91 1.37
11.9 11.5
58.8 84.6
1.58
12.4
107.5
1.64
12.3
111.4
230.9
271.0 282.9
79.3
107.2
1.73
10.8
95.8
314.3
1.90
4.6
46.3
356.8
2.16
5.4
60.9
511.3
3.17
5.1
83.4
579.0
3.63
6.2
113.9
110.4
0.98
5.1
27.4
124.1
1.11
8.9
54.3
183.1
1.68
10.7
95.7
186.1
1.71
13.0
118.6
9.7
0.22
6.1
13.6
23.8
0.61
9.1
49.7
36.1
0.96
12.1
100.2
44.0
1.19
12
121.1
11.7
0.28
8.1
21.6
34.7
0.92
12.6
100.1
42.8
1.14
18.3
179.4
38.4
1.03
23
202.2
7.1
0.19
8.0
13.1
11.7
0.32
11.1
29.7
15.4
0.42
11.8
41.6
16.6
0.46
16.3
62.2
7.6
0.21
10.0
17.4
10.6
0.29
16.9
41.0
11.4
0.31
24.6
64.3
12.8
0.35
35.5
104.1
29.1
0.17
6.8
7.8
78.9
0.46
7.1
20.6
138.4
0.81
7.2
36.4
68.0
0.86
9.6
78.2
175.4
2.35
9.7
203.4
250.1
3.53
10.8
321.3
283.5
4.10
10.9
370.2
389.1
6.20
11.5
532.7
433.3
7.15
11.5
595.6
MO-04
56.9
9.0
MO-06
8.0
1.71
MO-08
1.37
SP-02
85.8 106.2
SP-03
39.0
GB-02
7.1
GB-04
1.06
GB-06
66.5
CS-02
MO-03 MO-05 MO-07 SP-01 SP-02R GB-01 GB-03 GB-05 CS-01
2 4 12 24 2 4 12 24 2 4 12 24 4 16 32 48 4 16 32 48 2.67 8 16 24 2.67 8 16 24 4 12 24 2 8 20 24 48 72
6
2 4 12 24 2 4 12 24 2 4 12 24 4 16 32 48 4 16 32 48 2.67 8 16 24 2.67 8 16 24 4 12 24 2 8 20
147.9
1.37
5.4
39.4
171.1
1.59
5.7
47.4
255.6
2.44
7.4
92.3
282.2
2.71
8.9
122.9
32.7
0.49
9.0
24.4
51.6
0.79
9.9
42.3
76.2
1.19
13.8
87.0
85.2
1.34
15.7
110.2
214.5
1.24
5.9
40.1
231.2
1.34
11.9
87.8
338.0
1.99
11.5
123.9
396.0
2.36
12.4
157.0
11.7
0.28
8.1
21.6
37.5
1.00
14.1
121.0
42.3
1.16
18.1
175.5
42.6
1.15
22
214.9
15.9
0.39
10.8
39.4
25.8
0.66
19.8
117.1
28.5
0.74
23.2
151.6
30.6
0.80
28.6
200.6
7.2
0.20
9.2
15.3
10.8
0.29
15.7
38.8
11.9
0.33
21.1
57.5
14.0
0.38
25.2
80.8
7.3
0.20
11.5
19.2
8.8
0.24
23.2
46.8
10.8
0.29
30.0
74.3
11.1
0.30
35.7
90.7
29.5
0.17
16.0
18.4
48.1
0.29
21.9
41.9
67.6
0.39
24.7
66.4
45.0
0.55
12.66
68.0
66.5
0.82
13.92
110.6
94.4
1.18
14.86
167.3
24
97.4
1.22
16.22
188.5
72
186.0
2.48
18.53
411.4
Figure 3. MO-14 wax precipitation curves based on PVT simulations and DSC measurements (Agarwal, 2016). It is generally accepted that direct numerical differentiation amplifies the noises in experimental data and may lead to unreliable results (Lubansky et al., 2006). Singh (2013) performed the flux analysis with first central difference method suppressing the noise, however several computations still yielded peculiar trends, which show increasing flux with increasing time. These trends are not realistic since the deposition rate is expected to decrease over time due to self-insulation effects or diminishing driving force (Huang et al., 2015; Lu et al., 2012; Venkatesan, 2004). The power law curve fitting to correlate the ̅ data with respect to time is used in this study to suppress the data noise. The wax mass flux for a
given test condition is then calculated based on the derivative of the power law equation. However, due to the plateauing nature of wax deposition process (the power law coefficient is less than unity), the analytical derivative is not doable at t = 0. Therefore, the initial fluxes are excluded from the analyses performed in this manuscript. That being said, the application of power law curve fitting equation have improved the continuity of wax deposition rate trends compared to central time differentiation by suppressing data noises as illustrated in Fig. 4.
7
Figure 4. Examples of wax deposition rate quantification using Central Time Difference and Power Law Curve-Fit methods: (a) CS-01 test wax mass as a function of time, (b) CS-01 test wax deposition rate as a function of time; (c) GB-02 test wax mass as a function of time; (d) GB-02 test wax deposition rate as a function of time. 3.
Uncertainty Propagation and Model Sensitivity Analyses To objectively assess the individual model performance, it is necessary to separate the
uncertainties originated from the model inputs (fluid properties, measurements) and from the model formulation. Uncertainty propagation analyses (Dieck, 2007), provided in the appendix, are performed to serve this purpose. The assumptions of variable uncertainties used in this study are shown in Table 3, while the results are shown in Fig. 5. Based on these results, the tolerance ranges for FMT and EM are set at 60%. From Fig. 5, one could see that the sensitivity of EM in general is slightly higher than FMT. Sensitivity analyses were performed to identify the most influential input parameters in wax deposition model. Fig. 6 shows the results of sensitivity analyses, which conclude that prediction results (particularly EM model) are affected the most by the accuracy of temperature measurement.
EM
prediction results may be heavily affected by temperature measurement errors especially if the oil has a steep precipitation curve slope at the experimental conditions. Temperature measurement uncertainty of ± 2°F, which is approximately ± 2-4% of the typical temperature range in the experiments, may results in wax prediction error up to 50%. The effect of precipitation curve uncertainty is uniform for all models and oils assuming that no significant uncertainty arises from the derivative operation of the precipitation 8
curve, which in reality might be too optimistic. This assumption however, was reasonable for MO-14 experiments as described earlier. Note that while this analysis may not provide an exact uncertainty value, it still provide a useful insight on the effect of input values uncertainties to the wax deposition prediction results. Referring to Niesen (2002), it can be inferred that 100% conservatism in wax deposition prediction may impose up to $13M yearly revenue deferral due to over-pigging (in 2002 fiscal year condition), depending on the wax deposition case severity. Unfortunately, for wax deposition case, 100% uncertainty of wax deposition prediction is still somewhat common. The uncertainty propagation analyses show that the expected uncertainty due to fluid properties and measurement may reach 60%, even in a relatively controlled laboratory environment. This uncertainty range may expand for field applications, where subsea temperature and heat transfer coefficient are usually based on rule of thumb at best. Seasonal temperature changes may complicate the deposition prediction further for long period cases. The diffusivity of wax in oil to date is still estimated based on correlation developed for binary mixture. However, efforts to develop necessary correlations for crude oil may not be easily justified due requirement of complex and extensive testing. The best chance to reduce the range of model uncertainty induced by fluid properties inputs might be through better measurement of wax solubility/ precipitation curve and viscosity. Table 3. Assumptions on model input uncertainties. Fluid Properties ε-ρ ± 5% ε-µ ± 20% ε-CP ± 5% ε-k ± 5% ε-Vw ± 5% ε-Cws(T)a ± 33% ε-hcoolant ± 10%
Measurement ε-velocity ± 2% U-Tb ± 0.5°F ± 1.5%-wt. U- ̅ ε-m ± 1% c ε-δwax ± 10%
Correlation ε-Hayduk-Minhasd ε-Maxwell ε-H/M transport
20% 10% 5%
a) Based on MO-14 precipitation curves information b) Additional tolerance is added on top of reported 0.2° F value (Panacharoensawad, 2012) c) Panacharoensawad (2012) reported 5-6% uncertainty for deposit mass less than 50g d) Additional tolerance is added to the reported 3.4% value (Hayduk and Minhas, 1982) to take account of non-binary system effects Note: unless noted, the values in this table are simply arbitrary estimation by the authors and should not be taken as facts, better values from experiments might be available in the future and should be used in-lieu of these values.
9
Figure 5. Estimated FMT and EM models uncertainties caused by fluid properties or measurement errors.
Figure 6. FMT and EM model sensitivity to uncertainties in temperature (a), viscosity (b), wax precipitation curve (c), and Hayduk-Minhas correlation (d). 10
4.
Model Performance Assessment The assessment was performed using TUWAX software developed by Tulsa University Paraffin
Deposition Projects (TUPDP). This software is equipped with four models used in this study. Following assumptions were made for this study.
The “end-effects” are negligible (the flow is assumed to be fully developed thermally and hydrodynamically throughout the test section).
Manabe (2001) model is used for heat transfer equation
For EM, FMT, and Venkatesan’s models, the value of Kα (aspect ratio proportionality constant) is assumed to be 5. It is assumed that the change in aspect ratio has negligible impact to the total incoming flux quantity.
For Venkatesan’s model, the values for shear stripping coefficients m and n are set at 0.8E-12 and 1.9, respectively. These are the default TUWAX values suggested by Singh et al. (2011). The mass transfer coefficient is calculated based on FMT. The FMT model performance for σ and ̇ predictions are summarized in Fig. 7. This figure
shows that FMT consistently over-predicts the wax deposit mass and deposition rate. This observation is consistent with flow loop experiments (Venkatesan, 2004), heat/ mass transfer theory (Lee, 2008; Venkatesan and Fogler, 2004) and microscopic observation in the boundary layer (Soedarmo et al., 2016b). This suggests that the assumption of complete super-saturation (independent heat-mass transfer) inherent in FMT model is not realistic most of the times. The CS data sets show a distinctive difference between laminar and turbulent flow. The FMT prediction for CS-01 (laminar, denoted with letter “L”) is observed to be closer to experimental values compared to CS-02 (turbulent), which is consistent with literature (Singh et al., 2000; Singh et al., 2001). Fig. 8 shows the comparison between the mass transfer coefficient back-calculated from experimental data (
) and the ones determined based on FMT
theory. Both Gnielinski (1976) equation (
) and back-calculation from TUWAX FMT
simulation (
) results are used to determine the FMT mass transfer coefficients. These two
values would differ greatly at later time due to accumulation of errors over time on TUWAX. The mass transfer coefficient is proportional to the concentration gradient at the interface (inversely proportional to the boundary layer thickness). Consistent with the conclusion from Fig. 7, Fig. 8 shows that FMT (both calculation methods) over-predict the mass transfer coefficient.
11
Figure 7. FMT model performance in predicting: (a) the normalized wax mass in deposit (σ) and (b) wax mass flux ( )̇ .
Figure 8. FMT model(s) performance in predicting the convective mass transfer coefficient value; (a) forward calculation using Gnielinski equation and (b) back-calculation from TUWAX simulations. The EM model performance for σ and ̇ predictions are summarized in Fig. 9. Assuming the thermodynamic equilibrium (instantaneous precipitation) in the boundary layer is expected to result in lower limit of deposition rate (Huang et al., 2011; Lee, 2008). However, Fig.9 shows EM may over12
predict the deposition rate as well. It is important to note that EM over-predicts MO-14 and CS data only at relatively high shear stress conditions (τw,ini > 25 Pa for MO-14 and τw,ini > 7 Pa for CS), while it overpredicts all of GB data. Hence, it can be inferred that the onset of EM over-prediction appears to be oil dependent. Note that EM is able to predict all SP data accurately. By comparing Fig. 9 and Fig.7, one could infer that although EM produces more scattered behavior, it is considerably more accurate than FMT.
Figure 9. EM model performance in predicting: (a) the normalized wax mass in deposit (σ) and (b) wax mass flux ( )̇ . The interfacial wax precipitation curve derivatives ( ⁄
experimental data. Fig. 10 shows the values of ⁄
calculated from TUWAX EM simulation ( information (
⁄
|
). The
⁄
|
|
|
⁄
|
) are back-calculated from the
, compared to the same parameter back) and calculated from the fluid properties
values serve as direct indication of wax super-saturation
near the interface (in the boundary layer). If the
⁄
|
is greater than
⁄
|
, it indicates that
super-saturation exists near the interface, leading to greater concentration gradient than thermodynamic equilibrium prediction. Theoretically, the opposite behavior of the abovementioned statement indicates that some form of flux reduction mechanisms such as: shear reduction (Matzain, 1999; Venkatesan, 2004) or shifts in solid-liquid equilibrium condition at the deposit interface may exist. Figs. 9 and 10 show that although EM under-predict the interfacial concentration gradient and σ for majority of the data, it still over-predicts all GB data and some of high shear stresses CS and MO-14 data. Other possibilities for the
13
cause of EM over-prediction are excessively poor performance of the diffusivity correlation (Sarica and Panacharoensawad, 2012) and/ or inaccurate precipitation curves beyond the expected uncertainty range.
Figure 10. EM model(s) performance in predicting the interfacial wax precipitation curve gradient value; (a) back-calculation from TUWAX simulation and (b) based on fluid properties data. Matzain’s model uses EM principle thermodynamically, with empirical coefficients to take account of shear stripping and thickness enhancement due to trapped oil in deposit. Fig. 11 shows the performance of Matzain’s model in predicting the normalized wax mass in deposit (σ) and wax mass flux ( ̇). Fig. 11 shows that, despite being empirically developed for South Pelto crude, Matzain’s model performs satisfactory for MO-14. Matzain’s model prediction for GB data is somewhat similar to EM with more scattered behavior. On the other hand, the prediction for CS data appears to be less scattered. Fig. 11(b) implies that the Matzain model tends to over-predict the early time fluxes (the larger values in each data series) and under-predict the late time ones (the smaller values in each data series), this eventually evens out resulting in generally reasonable σ prediction. This phenomena will be discussed later when all models are compared to each other.
14
Figure 11. Performance of Matzain’s model in predicting: (a) the normalized wax mass in deposit (σ) and (b) wax mass flux ( )̇ .
15
The performance of Venkatesan’s model in predicting σ and wax mass flux is shown in Fig. 12. This figure shows the model’s performance with constant shear coefficients is rather unpredictable although most of the σ wax mass flux data are still over-predicted. This is conceivable as the model only takes account of shear stress in determining the flux reduction; hence, it is less influential in low shear stress data. Comparing Fig. 12 with Fig. 7, for example, the correction effects on GB-01 to GB-04 data (indicated by yellow diamonds) which were generated under high shear stresses (τw > 13 Pa) are more pronounced than GB-05 and GB-06 (indicated by red diamonds), which were generated under less than 2.7 Pa shear stresses. These results implies that shear stress may not be only determining factor in quantifying the flux reduction from FMT theory, and/ or that shear reduction coefficients are indeed not constant (changing with different oils and flow conditions).
Figure 12. Venkatesan’s model performance in predicting: (a) the normalized wax mass in deposit (σ) and (b) wax mass flux ( )̇ . The summary of models performance in predicting the wax mass flux is shown in Table 4, where the highlighted cells indicate the error values within expected uncertainty range caused by fluid properties and measurement errors. Table 4 coarsely shows that Matzain’s model delivers relatively better performance compared to the other models. All models over-predict GB data in considerably different magnitudes compared to their prediction for other oils. In addition to model formulation deficiencies, this behavior may be influenced by inaccurate fluid properties data or mass transport correlations as well. With the exception of GB, performances of EM and Matzain’s models are less sensitive to the type of oil compared to those of FMT and Venkatesan’s models.
16
Table 4 also shows that the performance of Matzain’s model is relatively insensitive to changes in initial NRE and τw while FMT and EM uncertainties are expanding with increase in both parameters. The behavior of FMT model is consistent with available literature (Singh et al., 2000; Soedarmo et al., 2016b; Venkatesan, 2004), but further study is needed to explain the behavior exhibited by EM. This table also confirms that Matzain’s model tends to over-predict early time fluxes and under-predict the late time fluxes. The time effects on EM and FMT models behavior appears to be minimal based on Table 4. The behavior of Venkatesan’s model with changes in initial NRE and τw does not exhibit any noticeable pattern. This model exhibits slight reduction in the flux over-prediction at later time, which might be caused by increasing magnitude of shear reduction term over time as the effective pipe diameter is reduced by the deposition. Table 4. Overall model performance summary. Wax Mass Flux Prediction Errors
Time
Initial NRE
Initial τw
Oil
FMT MO-14 SP GB CS < 5 Pa 5-10 Pa 10-20 Pa 20-30 Pa > 30 Pa < 5000 5000 – 10000 10000-15000 > 15000 0-8 h 8-20 h > 20 h
Avg. |ε| 295% 673% 2453% 384% 371% 807% 1019% 1372% 2634% 387% 839% 1246% 2634% 925% 992% 978%
EM Avg. ε 295% 673% 2453% 384% 371% 807% 1019% 1372% 2634% 387% 839% 1246% 2634% 925% 992% 978%
Avg. |ε| 62% 18% 339% 59% 62% 75% 128% 206% 368% 36% 119% 190% 394% 133% 126% 120%
Matzain Avg. ε 34% 17% 339% 56% 36% 75% 127% 206% 368% 18% 105% 187% 394% 107% 123% 120%
Avg. |ε| 20% 40% 107% 37% 61% 38% 51% 34% 31% 64% 39% 47% 33% 55% 42% 17%
Avg. ε -9% -25% 96% -19% 34% 5% -3% -5% 2% 25% -1% 31% 33% 44% 9% -18%
Venkatesan Avg. |ε| 105% 65% 337% 86% 227% 117% 82% 112% 95% 164% 95% 286% 160% 184% 165% 86%
Avg. ε 104% 29% 337% 86% 226% 117% 58% 95% 95% 151% 87% 286% 160% 178% 156% 108%
With the data sets presented in this manuscript, decoupling of time and ΔT effects from NRE and τw is not doable yet. However, oil specific analyses are still feasible to some extent by exploring the fact that temperature difference between bulk oil and cold interface (∆T) diminishes as the time progresses. This phenomena theoretically leads to reduction of equivalent cooling rate and super-saturation in the boundary layer over time. Consequently, it is conceivable that FMT over-prediction expands and EM under-prediction decreases over time. This theory has been supported by visual observation on MO-14 oil (Soedarmo et al., 2016b). Fig.13 shows the trends of
(mass transfer coefficient ratio) over the
deposition time. Fig.13 (a) supports the referred visual observation as it consistently shows that the mass transfer coefficient ratio values for MO-14 decrease with time, indicating an increase in FMT over17
prediction magnitude. Fig.13 (b) shows that only GB’s and several SP data exhibit the similar tendency as MO-14 (with less magnitude), while CS data shows relatively constant mass transfer coefficient ratio over time. Conversely, Fig. 14 shows the change in (
⁄
|
) (
⁄
|
) values (precipitation
curve derivative ratio) over time. Fig 14 (a) shows that for MO-14, most of precipitation curve derivative ratio values are greater than one (except the high shear stress case), indicating that the EM under-predicts the deposition rate. The precipitation curve derivative ratio values tend to be closer to one as time progresses, indicating agreement with visual observation. However, Fig. 14(b) shows that other oils do not necessarily follow the same pattern. These analyses indicate that crude oils may demonstrate different behavior/ sensitivity with respect to change in ∆T compared to model oils.
Figure 13. Mass transfer coefficient ratio value with respect to deposition time.
Figure 14. Solubility curve derivative ratio value with respect to deposition time. 18
The early time over-prediction of Matzain’s model is also demonstrated in examples in Fig. 15. It is likely due to the fact that Matzain’s model does not model aging mechanistically. Instead, it empirically assigns a value for trapped oil fraction in deposit for a given NRE condition. Consequently, the wax fraction predictions increase sharply at early time and tend to be relatively flat afterward, as the NRE changes throughout a single experiment are relatively modest. EM model on other hand, typically predicts gradual increase over time such that it predicts the early time data better compared to Matzain. However, Matzain’s model actually predicts the end of test data slightly better than EM in general (except for several GB cases).
Figure 15. Examples of simulation results for CS (a), SP (b), GB (c), and MO-14 (d) data. 5.
Sherwood Number Analysis This analysis offer a way to validate the TUWAX simulation results for FMT and EM by with
two available Sherwood number calculation methods, which directly related to the prediction of concentration profile near the interface. Huang et al. (2011) presented a model with precipitation kinetics 19
term. In this model, the deposition rate prediction is bounded by FMT (referred as independent heat/ mass transfer or IMHT in the original publication) as upper limit and instantaneous precipitation theory, also referred as solubility method (Venkatesan and Fogler, 2004), as lower limit. The solubility method is essentially similar with EM. These two methods differ in the Sherwood number calculations. For convenience, mathematical formulations of these two methods are provided in the appendix. experimental Sherwood number is back-calculated from
The
obtained from Eq. 1 and compared to the
Sherwood number predicted by these two methods. Gnielinski (1976) equation is used in the Sherwood number calculation for FMT and Nusselt number calculation for solubility method. This comparison is shown in Fig. 16. Fig. 16 shows that the FMT over-predicts the Sherwood number, which is consistent with FMT model performance assessment described earlier. On the other hand, although majority of the experimental Sherwood are under-predicted by solubility method, some of the data are still overpredicted.
This observation agrees with EM model performance assessment described earlier, and
indicates that the thermodynamic equilibrium assumption may not necessarily serves as lower bound of deposition rate prediction.
Figure 16. Comparison between experimental and predicted Sherwood number values. Although neither methods predict the experimental Sherwood number accurately, the average absolute error of solubility method (102%) is considerably lower than FMT (960%). The superior overall performance of solubility method is illustrated further with some examples in Fig. 17. Fig.17 (a) shows two experimental data (MO-01 and MO-02) which were generated under similar NRE (~13,000) but different initial wall shear stresses (25 Pa and 3.4 Pa, respectively). This figure shows that at low shear stress condition, the solubility method under-predicts the experimental Sherwood number. However, at higher shear stress, the experimental Sherwood number is much closer to that of the solubility method 20
prediction, indicating the effect of shear stress in wax thermodynamic equilibrium state near the interface. Fig. 17 (b) shows that solubility method performs well for SP oil, supporting the information provided in Table 4, which states that the EM works best for SP oil data. Fig. 17 (c) supports the EM performance analysis earlier by showing that even the solubility method still over-predicts the Sherwood number value for GB data. Given the consistency of the data sets from two different investigators, it is fair to assume that the data quality is acceptable. Fig. 17 (d) shows the analysis for CS-01 (the only laminar data set) and demonstrates that FMT model is satisfactory for laminar condition. This observation is in agreement with well-established conclusions from previous studies (Singh et al., 2000; Singh et al., 2001). However, for turbulent data (CS-02), the solubility method prediction is considerably more accurate than FMT.
Figure 17. Sherwood number analysis examples for MO-14 (a), SP (b), GB (c), and CS (d) data. 6.
Field Implications
21
This study provides a review of several simple wax deposition models performance to predict recent laboratory/ bench-scale flow loop data. It is observed that the EM and Matzain Model are able to predict majority of the normalized wax mass in deposit (σ) data within the expected uncertainty range, with the exception of GB data. However, using the EM as the base case for flow assurance strategy development increases the risk of being overly optimistic. It is also observed that Matzain’s model predicts the end-of-test (later time) data better than EM, which is conceivably more relevant to field applications. That being said, as Matzain’s model is still unable predict the deposition trends as a whole; it is uncertain whether the model is applicable for longer test durations. Accordingly, it implies that more experimental/ field data and more mechanistically rigorous models are still needed. The σ prediction has not covered the whole wax deposition modeling objectives, as it is still needed to be translated into deposit thickness (δ) and wax content ( ̅ ). Nevertheless, the benefit of reasonably accurate σ predictions could not be undermined. From modeling perspective, reliable aging/ growth prediction is not plausible if the total mass prediction is inaccurate. For practical purposes, if a reasonable range of wax content could be estimated then thickness prediction can be extracted from σ. Fig. 18 shows the range distribution of
̅ data with respect to initial NRE and τw, where
̅ is defined
as the total wax content increase from initial condition (wax content of oil) to the end of test condition. Fig. 18 may not necessarily represent the field cases wax content, as there are other parameters aside from NRE and τw, which differentiate the typical flow loop and field conditions (e.g.: ∆T, duration). However, it illustrates that if more experimental data are available, a plausible range of wax content might be reasonably estimated with greater confidence. Availability of more wax content data is also essential for aging modeling as the next step of comprehensive wax deposition models improvement.
Figure 18. Wax content distribution with respect to initial NRE and τw. MO-02 is one of the data sets which exhibit a resemblance to the field NRE and τw conditions (NRE > 10,000 and τw < 5 Pa). Referring to Fig. 18, there is one data with 22
̅ < 10% and another data with
̅ between 20 to 30% within that range. Assuming no wax content data available and both data are taken into considerations, the
̅ assumed to be 5%, 15%, and 25% to generate thickness predictions
using Matzain’s model. Fig. 19 shows the thickness prediction results using Matzain’s model without wax content adjustment) and with adjusted wax content. Fig. 19 shows that knowledge on wax content data may improve the thickness prediction significantly. Moreover, note that, at higher wax content, the thickness prediction becomes less sensitive to change in wax content.
Depending on the project
development/ field operating stage, wax content data could be obtained from laboratory flow loop data with same/ similar oil, nearby/ similar field experiences, or samples from previous pigging operation. Based on this study, the experimental data and prediction capability of σ value would be valuable for practical field applications if the scalability of it can be verified. As the σ parameter normalizes the pipe geometry effects, it is expected to have better scalability than separated thickness or wax content data. However, more experimental data are needed to prove this expectation.
Figure 19. MO-02 deposit thickness prediction results with Matzain's model. 7.
Conclusions A total of 70 wax deposition data points from 19 different conditions are compiled in this
manuscript. The normalized wax mass in deposit (σ), deposition flux (J), mass transfer coefficient (
), interfacial concentration gradient (
⁄
|
), and Sherwood number (
) are extracted
from these data points for further analyses. Uncertainty propagation analysis was performed to estimate reasonable ranges of error that may be caused by fluid properties or measurement errors instead of the model formulation itself.
23
Four models are validated with these data points, namely: film mass transfer (FMT), equilibrium model (EM), Matzain, and Venkatesan models. The FMT model regularly over-predicts the deposition rate, which is consistent with previous theoretical and visualization studies. It is observed that EM does not necessarily serve as the lower bound of deposition rate, suggesting that some form of flux reduction effects may exist or the closure relationship for diffusion coefficient may perform poorly.
The
conclusions for EM and FMT performance are also validated by Sherwood number analysis, where the values are compared with the Sherwood numbers estimated by FMT theory and solubility method. A fix set of shear coefficient values in Venketasan’s model is not sufficient to achieve acceptable accuracy in general, indicating that the shear coefficients are likely to be oil and flow conditions dependent. Despite of its inability to predict aging mechanistically, Matzain’s model delivers relatively better performance on deposit wax mass and flux predictions among the four models, although it still unable to predict the deposition trends as a whole. In conclusion, more mechanistically rigorous wax deposition models and flow loop experimental data which decouple NRE, τw, ∆T, and oil composition effects are still needed. Acknowledgements The authors would to acknowledge the financial support from Tulsa University Paraffin Deposition Project (TUPDP) company members for this research effort. Appendix A. Uncertainty Propagation Analysis for Independent Variables. An example of this method is the Reynolds number (NRE) uncertainty calculation, where the three determining variables, density ( ), velocity ( ), and viscosity ( ), are independent to each other. Eqs. A1 and A-2 show the procedure to estimate the uncertainty for NRE knowing the uncertainties of the and
measurements using Taylor’s series. √(
) (
√( ) (
)
)
(
) (
( ) (
)
)
(
(
) (
) (
)
Eq. A-1
)
Eq. A-2 ⁄
Taking MO-01 test as an example:
⁄ , and assuming
, the uncertainty of NRE can be estimated as shown in Eqs. A-3 and A-4. √(
) (
)
(
) (
)
24
(
) (
)
Eq. A-3
Eq. A-4 Appendix B. Uncertainty Propagation Analysis for Dependent Variables. An example of this method is the diffusivity of wax in oil (DWO) uncertainty calculation, where the two of the three determining variables, namely viscosity ( ) and temperature ( ), are inter-related. The correlation used to calculate DWO is the Hayduk-Minhas correlation(Hayduk and Minhas, 1982). Eqs. B-1 to B-4 shows the procedure to estimate the uncertainty for DWO knowing the uncertainties of the , and wax molecular volume (
). In addition to Taylor’s series procedure, the uncertainties of the
correlation itself is considered in this procedure as well.
√
( ⏟
) (
)
) (
(
(
)
(
) (
)
(
) (
)
Eq. B-1
⏟
)
⁄
Eq. B-2 ( [
)
⁄ (
)
⁄
(
)
⁄
(
Eq. B-3
)
]
Eq. B-4 ⁄
Taking MO-01 test as an example: ⁄ , and assuming
, the uncertainty of
estimated to be around 19%. The variable
is
in Eq. B-1 represents the covariance of viscosity on
temperature, which is estimated to be -1.34E-5 for MO-14 fluid. Appendix C. Mathematical Formulation for FMT, EM, Matzain’s and Venkatesan’s Models The governing equation of FMT is essentially the same with Singh et al. (2000) model where heat and mass transfer analogy are used to quantify the wax mass transfer rate from the bulk to interface. A mass balance then applied to distribute the incoming wax mass into aging and growth. Cussler et al. (1988) equation is used to quantify the effective diffusivity into the deposit. This concept is described in Eqs. C-1 to C-3. The calculation of
for turbulent flow is based on an analogy to heat transfer
correlations such as Gnielinski (1976) or Manabe (2001) equations. correlation is commonly used to calculate the
. Currently
Hayduk and Minhas (1982)
is still treated as fitting parameters. The
aging model described in Eqs. C-2 and C-3 is also used in EM and Venkatesan’s model. ⏟
( )
⏟
(
)
|
|
⏟
25
(
)
̅
Eq. C-1
̅
⁄(
Eq. C-2
̅ )
̅
Eq. C-3
EM differs from FMT in the way to quantify the incoming wax mass. As the model assumes that the concentration profile always follows the thermodynamic equilibrium, Eq. C-4 is used to express the incoming wax mass flux. Both EM and FMT apply the same mass balance concept and requires the same fitting parameter. ⏟
(
)
|
|
⏟
(
)
|
|
⏟
(
)
̅
Eq. C-4
Matzain’s model is based on EM with empirical correlation to take account of oil entrapment (
) in deposit (which would enhance the thickness growth rate) and shear stripping (
) effect. Eq. C-
5 to C-9 describe the governing equations of this model for single-phase flow applications. The trapped oil estimation also serves as the wax content prediction in this model, hence Singh et al. (2000) aging model is not used. [ |
| ]
Eq. C-5 Eq. C-6
⁄
Eq. C-7 (
)
Eq. C-8 Eq. C-9
The Venkatesan’s model can be applied to either FMT or EM as a mean to reduce the predicted incoming wax flux based on the in-situ shear stress. In this study, the Venkatesan’s model is applied to FMT as shown in Eq. C-10. The m and n coefficients used in this study is 0.8E-12 and 1.9, respectively. ( ) ̇
Eq. C-10
Appendix D. Sherwood Number Calculation As discussed earlier, for FMT the Sherwood number can be calculated using the analogy of Gnielinski (1976) equation as shown in Eq. D-1. The solubility method is described in Eq. D-2 and D-3, where the heat transfer form of Gnielinski (1976) equation is used to calculate the Nusselt number. (
)
)( (
)
(
Eq. D-1
)
26
| (
Eq. D-2 )
)( (
)
(
Eq. D-3
)
References Agarwal, J., 2016. Experimental Study of Wax Deposition in Turbulent Flow Conditions by Using Model Oil (MO-14), MS Thesis, The University of Tulsa, Tulsa, OK, 77 pp. Chi, Y., 2015. Investigation of Wax Inhibitors on Wax Deposition Based on Flow Loop Testing, MS Thesis, The University of Tulsa, Tulsa, OK, 75 pp. Cussler, E.L., Hughes, S.E., Ward III, W.J. and Rutherford, A., 1988. Barrier Membranes. Journal of Membrane Sciences, 388: 161-174. Dieck, R.H., 2007. Measurement Uncertainty: Methods and Applications. ISA, North Carolina (USA). Dwivedi, P., Sarica, C. and Shang, W., 2013. Experimental Study on Wax Deposition Characteristics of a Waxy Crude Under Single-Phase Turbulent Conditions. Oil Gas Facil., 2(04): 61-73. Gnielinski, V., 1976. New equations for heat and mass transfer in turbulent pipe and channel flow. Int. Chem. Eng., 16(2): 359-368. Hayduk, W. and Minhas, B.S., 1982. Correlations of Predictions of Molecular Diffusivities in Liquids. The Canadian Journal of Chemical Engineering, 60: 295-299. Hernandez, O. et al., 2003. Improvements in Single-Phase Paraffin Deposition Modeling. In: SPE (Editor), SPE Annual Technical Conference and Exhibition. SPE, Denver, CO. Huang, Z., Lee, H.S., Senra, M. and Fogler, H.S., 2011. A Fundamental Model of Wax Deposition in Subsea Oil Pipelines. AIChE J., 57(11): 2955-2964. Huang, Z., Zheng, S. and Fogler, H.S., 2015. Wax Deposition: Experimental Characterizations, Theoretical Modeling, and Field Practices. CRC Press, Boca Raton, FL. Lee, H.S., 2008. Computational and Rheological Study of Wax Deposition and Gelation in Subsea Pipelines, Ph.D. Dissertation, The University of Michigan, Ann Arbor, 113 pp. Lu, Y., Huang, Z., Hoffman, R., Amundsen, L. and Fogler, H.S., 2012. Counterintuitive Effects of Oil Flowrate in Wax Deposition. Energy Fuels, 26: 4091-4097. Lubansky, A.S., Yeow, Y.L., Leong, Y.-K., Wickramasinghe, S.R. and Han, B., 2006. A General Method in Computing the Derivative of Experimental Data. AIChE J., 52(1): 323-332. Manabe, R., 2001. A Comprehensive Mechanistic Heat Transfer Model for Two-Phase Flow with High Pressure Flow Pattern Validation, Ph.D Dissertation, The University of Tulsa, Tulsa, OK, 160 pp. Matzain, A., 1999. Multiphase Flow Paraffin Deposition Modeling, Ph.D. Dissertation, The University of Tulsa, Tulsa, OK, 235 pp. Niesen, V.G., 2002. The real cost of subsea pigging, E&P Mag, pp. 97. Panacharoensawad, E., 2012. Wax Deposition Under Two-Phase Oil-Water Flowing Conditions, Ph.D. Dissertation, The University of Tulsa, Tulsa, OK, 422 pp. Panacharoensawad, E. and Sarica, C., 2013. Experimental Study of Single-Phase and Two-Phase Waterin-Crude-Oil Dispersed Flow Wax Deposition in a Mini Pilot-Scale Flow Loop. Energy & Fuels, 27(9): 5036–5053. Rittirong, A., 2014. Paraffin Deposition under Two-Phase Gas Oil Slug Flow in Horizontal Pipes, Ph.D. Dissertation, The University of Tulsa, Tulsa, OK, 344 pp.
27
Rittirong, A., Panacharoensawad, E. and Sarica, C., 2014. An Experimental Study of Paraffin Deposition under Two-Phase Gas-Oil Slug Flow in Horizontal Pipes, Offshore Technology Conference. SPE, Houston. Sarica, C. and Panacharoensawad, E., 2012. Review of Paraffin Deposition Research under Multiphase Flow Conditions. Energy Fuels, 26: 3968-3978. Singh, A., 2013. Experimental and Field Verification Study of Wax Deposition in Turbulent Flow Condition, MS Thesis, The University of Tulsa, Tulsa, OK, 151 pp. Singh, A., Lee, H.S., Singh, P. and Sarica, C., 2011. Validation of Wax Deposition Models Using Field Data from Subsea Pipeline, Proc. - Annu. Offshore Technol. Conf. SPE, Houston, TX. Singh, P., Venkatesan, R., Fogler, H.S. and Nagarajan, N., 2000. Formation and Aging of Incipient Thin Film Wax-Oil Gels. AIChE J., 46(5): 1059-1074. Singh, P., Venkatesan, R., Fogler, H.S. and Nagarajan, N., 2001. Morphological Evolution of Thick Wax Deposits during Aging. AIChE J., 47(1): 6-18. Soedarmo, A.A., Daraboina, N., Lee, H.S. and Sarica, C., 2016a. Microscopic Study of Wax Precipitation: Static Conditions. Energy & Fuels, 30: 954-961. Soedarmo, A.A., Daraboina, N. and Sarica, C., 2016b. Microscopic Study of Wax Deposition: Mass Transfer Boundary Layer and Deposit Morphology. Energy & Fuels, 30(4): 2674-2686. Venkatesan, R., 2004. The Deposition and Rheology of Organic Gels, Ph.D. Dissertation, University of Michigan, Ann Arbor, MI, 226 pp. Venkatesan, R. and Creek, J.L., 2007. Wax Deposition During Production Operations: SOTA, Proc. - Annu. Offshore Technol. Conf. SPE, Houston, TX. Venkatesan, R. and Fogler, H.S., 2004. Comments on Analogies for Correlated Heat and Mass Transfer in Turbulent Flow. AIChE J., 50(7): 1623-1626.
Highlights
Film mass transfer model (FMT) significantly over-predicts all turbulent wax deposition data. Equilibrium model (EM) does not necessarily serve as lower wax deposition limit. Matzain’s model delivers superior performance, but need to be validated with longer experiments. Experimental data to decouple NRE, τw, ∆T, and oil composition effects are still insufficient. More mechanistically rigorous wax deposition models are still needed.
28
www.elsevier.com/locate/petrol
PII: DOI: Reference:
S0920-4105(16)30613-1 http://dx.doi.org/10.1016/j.petrol.2016.10.017 PETROL3677
To appear in: Journal of Petroleum Science and Engineering Received date: 12 May 2016 Revised date: 3 October 2016 Accepted date: 6 October 2016 Cite this article as: Auzan A. Soedarmo, Nagu Daraboina and Cem Sarica, Validation of Wax Deposition Models with Recent Laboratory Scale Flow Loop Experimental Data, Journal of Petroleum Science and Engineering, http://dx.doi.org/10.1016/j.petrol.2016.10.017 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Validation of Wax Deposition Models with Recent Laboratory Scale Flow Loop Experimental Data Auzan A. Soedarmo, Nagu Daraboina, Cem Sarica McDougall School of Petroleum Engineering, The University of Tulsa Abstract Experimental data at various conditions are important for improving wax deposition prediction and upscaling efforts. This study consolidates several single-phase wax deposition data acquired in recent years and validates available wax deposition models for a comprehensive range of experimental conditions. The film mass transfer (FMT), equilibrium (EM), Matzain (1999), and Venkatesan (2004) models were tested against 70 experimental data points (19 sets of initial operating conditions), obtained with four different oils and flow loop facilities. Uncertainty propagation analysis was performed to estimate reasonable ranges of error in the prediction that may be caused by fluid properties or standard correlations instead of the model formulation itself. The FMT model regularly over-predicts the deposition rate, which is consistent with previous theoretical, experimental, and visualization studies. It is observed that EM does not necessarily serves as the lower bound of deposition rate, suggesting that some form of flux reduction effects may exist. Sherwood number analysis was performed to validate conclusions from FMT and EM model assessment. A fix set of shear coefficient values in Venketasan’s model is not sufficient to achieve acceptable accuracy in general, indicating that the shear coefficients are likely to be oil and flow conditions dependent. Matzain’s model delivers relatively superior performance on deposit wax mass and flux predictions among the four models, although it is still unable to predict aging mechanistically and the deposition trends as a whole. In conclusion, more mechanistically rigorous wax deposition models and experimental data are still needed. Keywords: wax deposition models, flow loop experiments, model verification 1.
Introduction Lack of mechanistic understanding has been highlighted as one of the primary shortcomings in
available wax deposition models (Dwivedi et al., 2013; Sarica and Panacharoensawad, 2012; Venkatesan and Creek, 2007). Flow loop testing serves as the primary mean for model benchmarking prior to field application (Huang et al., 2015) and, to some extent, explain the deposition physics. It is important to organize available data and validate existing models with more comprehensive range of experimental conditions. Accurate prediction of single-phase wax deposition is the key enabler to develop predictive tools for more complex multiphase cases. Due to complex nature of wax deposition experiments, models are usually validated against limited data set. Recently, wax deposition flow loop data have been generated extensively, which involve different testing fluids, expanded experimental conditions range, and improved measurement techniques in general. Therefore, it is necessary to continue the models validation efforts with these newer sets of data. Moreover, with time constraints in collecting statistically meaningful data set, an effort is needed to consolidate available experimental data and enhance their
1
accessibility to the best extent possible. A consolidated data also enables more comprehensive data analyses and identification of current knowledge gaps to direct future studies. The wax deposition models need to perform two main tasks in series: quantification of the total wax mass flux coming onto the cold interface, and distribution of this quantity into deposit aging and growth. Error from the first task will conceivably be propagate to the second one. Therefore, it is imperative to prioritize the wax mass flux prediction accuracy as the first step of model improvement/ validation efforts. The second task itself is dependent on the crystal aspect ratio (Hernandez et al., 2003; Sarica and Panacharoensawad, 2012; Soedarmo et al., 2016a), for which a proper closure relationship is still not available. To systematically assess the models, this study focuses on model-to-experiment comparison of deposit wax mass per deposition area (σ), which is more directly related to the performance of models for flux prediction compared to separated thickness or wax content data. The flux quantification depends on the physical assumptions inherent in the models regarding solid-liquid equilibrium state and flow effects to the diffusion process. This manuscript covered the film mass transfer (FMT), equilibrium (EM), Matzain (1999), and Venkatesan (2004) models. By evaluating the capability of these for predicting σ, one could assess the applicability of the physical assumptions made during their formulation. The FMT model assumes independent heat and mass transfer processes which lead to maximum super-saturation and wax concentration gradient near the interface. On the other hand, EM assumes thermodynamic equilibrium condition, which leads to much smaller concentration gradient than FMT. Matzain’s model modifies the EM diffusion equation with empirical correlation to take account of shear stripping and trapped oil in the deposit. The Venkatesan’s model uses an empirical relationship with two coefficients to quantify the shear effect in wax mass flux reduction. With the exception of Matzain’s model, other models use Singh et al. (2000) model to mechanistically predict the increase of wax content in deposit over time (aging). Matzain’s model is the only self-sufficient model among the four and does not need any arbitrary fitting coefficients. However, it is important to note that the empirical constants inherent in this model are determined solely based on South Pelto crude experimental data, hence the model’s performance for other oils at different operating conditions remains to be seen. For convenience, the mathematical formulations of these models are described briefly in the appendix. 2.
Experimental Data Collection and Interpretation To conduct analyses based on σ, experiments producing both raw deposit mass (including the
trapped oil) and wax content data are more reliable. These experiments obtain deposit mass data by pigging the whole pipe or a removable spool piece section upon completion of each test. Mass of the collected deposit is then converted to deposit thickness, knowing the deposit density and test section 2
dimension. Panacharoensawad and Sarica (2013), Singh (2013), Rittirong et al. (2014), Chi (2015), and Agarwal (2016) have performed a total of 19 single-phase experiments (70 data points) with this method using four oils and four different experimental flow loop facilities. In these studies, wax content data from high temperature gas chromatography (HTGC) are available. The operating range for these experiments and the specifications of the four facilities are described in Table 1. The C17+ carbon number distributions and fluid properties of the oils used in these experiments are shown in Figs. 1 and 2, respectively. These oils are the model oil (MO-14), Garden Banks condensate (GB), South Pelto oil (SP), and Caspian Sea condensate (CS). Table 1. Operating conditions of the experiments covered in this study, * denote the length of the removable spool piece only instead of the whole test section.
Panacharoensawad (2012)
Singh (2013)
Rittirong et al. (2014) Chi (2015)
Agarwal (2016)
Fluid Test #ID SP-01 SP-02 SP SP-02R SP-03 GB-01 GB-02 GB GB-03 GB-04 GB-05 GB GB-06 CS-01 CS CS-02 MO-01 MO-02 MO-03 MO-04 MO-05 MO-06 MO-07 MO-08 MO-14
Study
ID [in] 0.651 0.651 0.651 0.651 0.651 0.651 0.651 0.651 2.067 2.067 1.05 1.05 0.622 1.61 0.622 1.049 1.61 0.622 1.049 1.61
L [m] 0.84 0.84 0.84 0.84 0.84 0.84 0.84 0.84 1.52* 1.52* 1,00 1.00 2.44 2.44 2.44 2.44 2.44 2.44 2.44 2.44
Tb [°F] 105 105.4 105.4 105.4 77.6 78.5 78.3 79.2 76.6 77.2 95 95 95 95 95 95 95 95 95 95
3
Ti,ini[°F] 85 85 85 85 61 61 60 61 62 63 77 78 91 91 91 91 91 91 91 91
NRE [-] 2732 4232 4232 5210 7423 11282 15000 18989 3788 11448 1064 5471 12907 12643 5031 5059 4986 8532 8431 8451
τw [Pa] 7.2 15.1 15.1 21.6 13.5 28.2 47.8 72.2 0.4 2.7 0.4 6.9 25.5 3.4 4.7 1.6 0.7 12.0 3.9 1.6
Time points [h] 4, 16, 32, 48 4, 16, 32, 48 4, 16, 32, 48 4, 16, 32, 48 2.67, 8, 16, 24 2.67, 8, 16, 24 2.67, 8, 16, 24 2.67, 8, 16, 24 4, 12, 24 4, 12, 24 2, 8, 20, 24, 48, 72 2, 8, 20, 24, 72 2, 4, 12, 24 2, 4, 12, 18, 24 2, 4, 12, 24 2, 4, 12, 24 2, 4, 12, 24 2, 4, 12, 24 2, 4, 12, 24 2, 4, 12, 24
Figure 1. C17+ carbon number distribution of GB (Rittirong, 2014; Singh, 2013), SP (Panacharoensawad, 2012), CS (Chi, 2015), and MO-14 (Agarwal, 2016) oils.
Figure 2. Viscosity (a) and solubility (b) curves of MO-14 (Agarwal, 2016), CS (Chi, 2015), GB (Rittirong et al., 2014; Singh, 2013), and SP (Panacharoensawad and Sarica, 2013) oils. It can be seen from Table 1 and Fig. 2 that all of the tests were performed under sub-cooled conditions, hence the thermal restriction effect can be neglected (Venkatesan, 2004). The uncertainty of wax content measurement using HTGC is reported to be 1.5%-wt. (Panacharoensawad and Sarica, 2013), while the relative uncertainty of direct mass measurement is expected to be less than 1% (Panacharoensawad, 2012). Note that since the wax content uncertainty is given in absolute value, the uncertainty ranges for data with lower wax content would be greater. The uncertainties caused by procedural error is assumed minimal owing to good experimental practice. The uncertainties caused by 4
errors due to operating conditions settings (flow rates and temperatures) will be covered in the uncertainty propagation analysis for the model predictions instead of the experimental data. The experimental data used for this analysis is summarized in Table 2. Errors in deposition prediction accumulate over time. Given this consideration, a model analysis solely based on σ parameter might become vague at later time although it will still serve as a valuable performance indicator in a hindsight. Therefore, the experimental wax mass flux ( ̇
) values were
extracted from the deposit mass data to exclude the time effects. To further normalize the driving force discrepancies, the actual mass transfer coefficient ( ⁄
(
|
) and interfacial concentration gradient
) are back-calculated from experimental data. These values indirectly represent the actual
wax concentration gradient at the interface for FMT and EM models, respectively.
With these
procedures, one could assess each model performance in predicting mass transfer rate more thoroughly. The procedures to extract the ̇
,
, and
⁄
|
values from experimental data are shown in
Eqs. 1 and 2. One should be aware that these values may have greater uncertainties than σ due to the inherent uncertainties in derivative operation of experimental data and wax precipitation curves. (
(
̅ )
̅ )
( ) (
)
(
)
( )
̇
( ̇
|
( ))
Eq. 1
|
Eq. 2
PVT simulations was used to generate the wax precipitation curves (as shown in Fig. 2b). To illustrate the magnitude of uncertainties in the wax precipitation curves, Fig. 3 provides the MO-14 precipitation curves generated with various methods (Agarwal, 2016). At the experimental conditions (Tb = 95° F and Ti, ≈ 91 to 94.8° F), the uncertainties of
and
⁄
| values are estimated to be 29.3
– 33.1% and 33 – 39%, respectively (assuming no error in temperature measurements/ calculations). Due to unavailability of data, it is not possible to conduct similar analysis for other oils. Therefore, the uncertainty values for MO-14 are assumed applicable for other oils discussed in this study.
Table 2. Summary of the experimental data (Agarwal, 2016; Chi, 2015; Panacharoensawad and Sarica, 2013; Rittirong et al., 2014; Singh, 2013). ̅
σ [g/m2]
m [g]
δ [mm]
25.6
0.37
11.0
23.3
29.3
0.43
12.8
31.0
12
57.1
0.87
14.6
68.9
24
61.2
0.93
20.4
103.0
Test t [h] 2 4 12 18 24 MO-02
MO-01
Test t [h] 2 4
5
m [g] δ [mm]
̅
σ [g/m2]
116.0
0.67
5.9
21.7
154.9
0.91 1.37
11.9 11.5
58.8 84.6
1.58
12.4
107.5
1.64
12.3
111.4
230.9
271.0 282.9
79.3
107.2
1.73
10.8
95.8
314.3
1.90
4.6
46.3
356.8
2.16
5.4
60.9
511.3
3.17
5.1
83.4
579.0
3.63
6.2
113.9
110.4
0.98
5.1
27.4
124.1
1.11
8.9
54.3
183.1
1.68
10.7
95.7
186.1
1.71
13.0
118.6
9.7
0.22
6.1
13.6
23.8
0.61
9.1
49.7
36.1
0.96
12.1
100.2
44.0
1.19
12
121.1
11.7
0.28
8.1
21.6
34.7
0.92
12.6
100.1
42.8
1.14
18.3
179.4
38.4
1.03
23
202.2
7.1
0.19
8.0
13.1
11.7
0.32
11.1
29.7
15.4
0.42
11.8
41.6
16.6
0.46
16.3
62.2
7.6
0.21
10.0
17.4
10.6
0.29
16.9
41.0
11.4
0.31
24.6
64.3
12.8
0.35
35.5
104.1
29.1
0.17
6.8
7.8
78.9
0.46
7.1
20.6
138.4
0.81
7.2
36.4
68.0
0.86
9.6
78.2
175.4
2.35
9.7
203.4
250.1
3.53
10.8
321.3
283.5
4.10
10.9
370.2
389.1
6.20
11.5
532.7
433.3
7.15
11.5
595.6
MO-04
56.9
9.0
MO-06
8.0
1.71
MO-08
1.37
SP-02
85.8 106.2
SP-03
39.0
GB-02
7.1
GB-04
1.06
GB-06
66.5
CS-02
MO-03 MO-05 MO-07 SP-01 SP-02R GB-01 GB-03 GB-05 CS-01
2 4 12 24 2 4 12 24 2 4 12 24 4 16 32 48 4 16 32 48 2.67 8 16 24 2.67 8 16 24 4 12 24 2 8 20 24 48 72
6
2 4 12 24 2 4 12 24 2 4 12 24 4 16 32 48 4 16 32 48 2.67 8 16 24 2.67 8 16 24 4 12 24 2 8 20
147.9
1.37
5.4
39.4
171.1
1.59
5.7
47.4
255.6
2.44
7.4
92.3
282.2
2.71
8.9
122.9
32.7
0.49
9.0
24.4
51.6
0.79
9.9
42.3
76.2
1.19
13.8
87.0
85.2
1.34
15.7
110.2
214.5
1.24
5.9
40.1
231.2
1.34
11.9
87.8
338.0
1.99
11.5
123.9
396.0
2.36
12.4
157.0
11.7
0.28
8.1
21.6
37.5
1.00
14.1
121.0
42.3
1.16
18.1
175.5
42.6
1.15
22
214.9
15.9
0.39
10.8
39.4
25.8
0.66
19.8
117.1
28.5
0.74
23.2
151.6
30.6
0.80
28.6
200.6
7.2
0.20
9.2
15.3
10.8
0.29
15.7
38.8
11.9
0.33
21.1
57.5
14.0
0.38
25.2
80.8
7.3
0.20
11.5
19.2
8.8
0.24
23.2
46.8
10.8
0.29
30.0
74.3
11.1
0.30
35.7
90.7
29.5
0.17
16.0
18.4
48.1
0.29
21.9
41.9
67.6
0.39
24.7
66.4
45.0
0.55
12.66
68.0
66.5
0.82
13.92
110.6
94.4
1.18
14.86
167.3
24
97.4
1.22
16.22
188.5
72
186.0
2.48
18.53
411.4
Figure 3. MO-14 wax precipitation curves based on PVT simulations and DSC measurements (Agarwal, 2016). It is generally accepted that direct numerical differentiation amplifies the noises in experimental data and may lead to unreliable results (Lubansky et al., 2006). Singh (2013) performed the flux analysis with first central difference method suppressing the noise, however several computations still yielded peculiar trends, which show increasing flux with increasing time. These trends are not realistic since the deposition rate is expected to decrease over time due to self-insulation effects or diminishing driving force (Huang et al., 2015; Lu et al., 2012; Venkatesan, 2004). The power law curve fitting to correlate the ̅ data with respect to time is used in this study to suppress the data noise. The wax mass flux for a
given test condition is then calculated based on the derivative of the power law equation. However, due to the plateauing nature of wax deposition process (the power law coefficient is less than unity), the analytical derivative is not doable at t = 0. Therefore, the initial fluxes are excluded from the analyses performed in this manuscript. That being said, the application of power law curve fitting equation have improved the continuity of wax deposition rate trends compared to central time differentiation by suppressing data noises as illustrated in Fig. 4.
7
Figure 4. Examples of wax deposition rate quantification using Central Time Difference and Power Law Curve-Fit methods: (a) CS-01 test wax mass as a function of time, (b) CS-01 test wax deposition rate as a function of time; (c) GB-02 test wax mass as a function of time; (d) GB-02 test wax deposition rate as a function of time. 3.
Uncertainty Propagation and Model Sensitivity Analyses To objectively assess the individual model performance, it is necessary to separate the
uncertainties originated from the model inputs (fluid properties, measurements) and from the model formulation. Uncertainty propagation analyses (Dieck, 2007), provided in the appendix, are performed to serve this purpose. The assumptions of variable uncertainties used in this study are shown in Table 3, while the results are shown in Fig. 5. Based on these results, the tolerance ranges for FMT and EM are set at 60%. From Fig. 5, one could see that the sensitivity of EM in general is slightly higher than FMT. Sensitivity analyses were performed to identify the most influential input parameters in wax deposition model. Fig. 6 shows the results of sensitivity analyses, which conclude that prediction results (particularly EM model) are affected the most by the accuracy of temperature measurement.
EM
prediction results may be heavily affected by temperature measurement errors especially if the oil has a steep precipitation curve slope at the experimental conditions. Temperature measurement uncertainty of ± 2°F, which is approximately ± 2-4% of the typical temperature range in the experiments, may results in wax prediction error up to 50%. The effect of precipitation curve uncertainty is uniform for all models and oils assuming that no significant uncertainty arises from the derivative operation of the precipitation 8
curve, which in reality might be too optimistic. This assumption however, was reasonable for MO-14 experiments as described earlier. Note that while this analysis may not provide an exact uncertainty value, it still provide a useful insight on the effect of input values uncertainties to the wax deposition prediction results. Referring to Niesen (2002), it can be inferred that 100% conservatism in wax deposition prediction may impose up to $13M yearly revenue deferral due to over-pigging (in 2002 fiscal year condition), depending on the wax deposition case severity. Unfortunately, for wax deposition case, 100% uncertainty of wax deposition prediction is still somewhat common. The uncertainty propagation analyses show that the expected uncertainty due to fluid properties and measurement may reach 60%, even in a relatively controlled laboratory environment. This uncertainty range may expand for field applications, where subsea temperature and heat transfer coefficient are usually based on rule of thumb at best. Seasonal temperature changes may complicate the deposition prediction further for long period cases. The diffusivity of wax in oil to date is still estimated based on correlation developed for binary mixture. However, efforts to develop necessary correlations for crude oil may not be easily justified due requirement of complex and extensive testing. The best chance to reduce the range of model uncertainty induced by fluid properties inputs might be through better measurement of wax solubility/ precipitation curve and viscosity. Table 3. Assumptions on model input uncertainties. Fluid Properties ε-ρ ± 5% ε-µ ± 20% ε-CP ± 5% ε-k ± 5% ε-Vw ± 5% ε-Cws(T)a ± 33% ε-hcoolant ± 10%
Measurement ε-velocity ± 2% U-Tb ± 0.5°F ± 1.5%-wt. U- ̅ ε-m ± 1% c ε-δwax ± 10%
Correlation ε-Hayduk-Minhasd ε-Maxwell ε-H/M transport
20% 10% 5%
a) Based on MO-14 precipitation curves information b) Additional tolerance is added on top of reported 0.2° F value (Panacharoensawad, 2012) c) Panacharoensawad (2012) reported 5-6% uncertainty for deposit mass less than 50g d) Additional tolerance is added to the reported 3.4% value (Hayduk and Minhas, 1982) to take account of non-binary system effects Note: unless noted, the values in this table are simply arbitrary estimation by the authors and should not be taken as facts, better values from experiments might be available in the future and should be used in-lieu of these values.
9
Figure 5. Estimated FMT and EM models uncertainties caused by fluid properties or measurement errors.
Figure 6. FMT and EM model sensitivity to uncertainties in temperature (a), viscosity (b), wax precipitation curve (c), and Hayduk-Minhas correlation (d). 10
4.
Model Performance Assessment The assessment was performed using TUWAX software developed by Tulsa University Paraffin
Deposition Projects (TUPDP). This software is equipped with four models used in this study. Following assumptions were made for this study.
The “end-effects” are negligible (the flow is assumed to be fully developed thermally and hydrodynamically throughout the test section).
Manabe (2001) model is used for heat transfer equation
For EM, FMT, and Venkatesan’s models, the value of Kα (aspect ratio proportionality constant) is assumed to be 5. It is assumed that the change in aspect ratio has negligible impact to the total incoming flux quantity.
For Venkatesan’s model, the values for shear stripping coefficients m and n are set at 0.8E-12 and 1.9, respectively. These are the default TUWAX values suggested by Singh et al. (2011). The mass transfer coefficient is calculated based on FMT. The FMT model performance for σ and ̇ predictions are summarized in Fig. 7. This figure
shows that FMT consistently over-predicts the wax deposit mass and deposition rate. This observation is consistent with flow loop experiments (Venkatesan, 2004), heat/ mass transfer theory (Lee, 2008; Venkatesan and Fogler, 2004) and microscopic observation in the boundary layer (Soedarmo et al., 2016b). This suggests that the assumption of complete super-saturation (independent heat-mass transfer) inherent in FMT model is not realistic most of the times. The CS data sets show a distinctive difference between laminar and turbulent flow. The FMT prediction for CS-01 (laminar, denoted with letter “L”) is observed to be closer to experimental values compared to CS-02 (turbulent), which is consistent with literature (Singh et al., 2000; Singh et al., 2001). Fig. 8 shows the comparison between the mass transfer coefficient back-calculated from experimental data (
) and the ones determined based on FMT
theory. Both Gnielinski (1976) equation (
) and back-calculation from TUWAX FMT
simulation (
) results are used to determine the FMT mass transfer coefficients. These two
values would differ greatly at later time due to accumulation of errors over time on TUWAX. The mass transfer coefficient is proportional to the concentration gradient at the interface (inversely proportional to the boundary layer thickness). Consistent with the conclusion from Fig. 7, Fig. 8 shows that FMT (both calculation methods) over-predict the mass transfer coefficient.
11
Figure 7. FMT model performance in predicting: (a) the normalized wax mass in deposit (σ) and (b) wax mass flux ( )̇ .
Figure 8. FMT model(s) performance in predicting the convective mass transfer coefficient value; (a) forward calculation using Gnielinski equation and (b) back-calculation from TUWAX simulations. The EM model performance for σ and ̇ predictions are summarized in Fig. 9. Assuming the thermodynamic equilibrium (instantaneous precipitation) in the boundary layer is expected to result in lower limit of deposition rate (Huang et al., 2011; Lee, 2008). However, Fig.9 shows EM may over12
predict the deposition rate as well. It is important to note that EM over-predicts MO-14 and CS data only at relatively high shear stress conditions (τw,ini > 25 Pa for MO-14 and τw,ini > 7 Pa for CS), while it overpredicts all of GB data. Hence, it can be inferred that the onset of EM over-prediction appears to be oil dependent. Note that EM is able to predict all SP data accurately. By comparing Fig. 9 and Fig.7, one could infer that although EM produces more scattered behavior, it is considerably more accurate than FMT.
Figure 9. EM model performance in predicting: (a) the normalized wax mass in deposit (σ) and (b) wax mass flux ( )̇ . The interfacial wax precipitation curve derivatives ( ⁄
experimental data. Fig. 10 shows the values of ⁄
calculated from TUWAX EM simulation ( information (
⁄
|
). The
⁄
|
|
|
⁄
|
) are back-calculated from the
, compared to the same parameter back) and calculated from the fluid properties
values serve as direct indication of wax super-saturation
near the interface (in the boundary layer). If the
⁄
|
is greater than
⁄
|
, it indicates that
super-saturation exists near the interface, leading to greater concentration gradient than thermodynamic equilibrium prediction. Theoretically, the opposite behavior of the abovementioned statement indicates that some form of flux reduction mechanisms such as: shear reduction (Matzain, 1999; Venkatesan, 2004) or shifts in solid-liquid equilibrium condition at the deposit interface may exist. Figs. 9 and 10 show that although EM under-predict the interfacial concentration gradient and σ for majority of the data, it still over-predicts all GB data and some of high shear stresses CS and MO-14 data. Other possibilities for the
13
cause of EM over-prediction are excessively poor performance of the diffusivity correlation (Sarica and Panacharoensawad, 2012) and/ or inaccurate precipitation curves beyond the expected uncertainty range.
Figure 10. EM model(s) performance in predicting the interfacial wax precipitation curve gradient value; (a) back-calculation from TUWAX simulation and (b) based on fluid properties data. Matzain’s model uses EM principle thermodynamically, with empirical coefficients to take account of shear stripping and thickness enhancement due to trapped oil in deposit. Fig. 11 shows the performance of Matzain’s model in predicting the normalized wax mass in deposit (σ) and wax mass flux ( ̇). Fig. 11 shows that, despite being empirically developed for South Pelto crude, Matzain’s model performs satisfactory for MO-14. Matzain’s model prediction for GB data is somewhat similar to EM with more scattered behavior. On the other hand, the prediction for CS data appears to be less scattered. Fig. 11(b) implies that the Matzain model tends to over-predict the early time fluxes (the larger values in each data series) and under-predict the late time ones (the smaller values in each data series), this eventually evens out resulting in generally reasonable σ prediction. This phenomena will be discussed later when all models are compared to each other.
14
Figure 11. Performance of Matzain’s model in predicting: (a) the normalized wax mass in deposit (σ) and (b) wax mass flux ( )̇ .
15
The performance of Venkatesan’s model in predicting σ and wax mass flux is shown in Fig. 12. This figure shows the model’s performance with constant shear coefficients is rather unpredictable although most of the σ wax mass flux data are still over-predicted. This is conceivable as the model only takes account of shear stress in determining the flux reduction; hence, it is less influential in low shear stress data. Comparing Fig. 12 with Fig. 7, for example, the correction effects on GB-01 to GB-04 data (indicated by yellow diamonds) which were generated under high shear stresses (τw > 13 Pa) are more pronounced than GB-05 and GB-06 (indicated by red diamonds), which were generated under less than 2.7 Pa shear stresses. These results implies that shear stress may not be only determining factor in quantifying the flux reduction from FMT theory, and/ or that shear reduction coefficients are indeed not constant (changing with different oils and flow conditions).
Figure 12. Venkatesan’s model performance in predicting: (a) the normalized wax mass in deposit (σ) and (b) wax mass flux ( )̇ . The summary of models performance in predicting the wax mass flux is shown in Table 4, where the highlighted cells indicate the error values within expected uncertainty range caused by fluid properties and measurement errors. Table 4 coarsely shows that Matzain’s model delivers relatively better performance compared to the other models. All models over-predict GB data in considerably different magnitudes compared to their prediction for other oils. In addition to model formulation deficiencies, this behavior may be influenced by inaccurate fluid properties data or mass transport correlations as well. With the exception of GB, performances of EM and Matzain’s models are less sensitive to the type of oil compared to those of FMT and Venkatesan’s models.
16
Table 4 also shows that the performance of Matzain’s model is relatively insensitive to changes in initial NRE and τw while FMT and EM uncertainties are expanding with increase in both parameters. The behavior of FMT model is consistent with available literature (Singh et al., 2000; Soedarmo et al., 2016b; Venkatesan, 2004), but further study is needed to explain the behavior exhibited by EM. This table also confirms that Matzain’s model tends to over-predict early time fluxes and under-predict the late time fluxes. The time effects on EM and FMT models behavior appears to be minimal based on Table 4. The behavior of Venkatesan’s model with changes in initial NRE and τw does not exhibit any noticeable pattern. This model exhibits slight reduction in the flux over-prediction at later time, which might be caused by increasing magnitude of shear reduction term over time as the effective pipe diameter is reduced by the deposition. Table 4. Overall model performance summary. Wax Mass Flux Prediction Errors
Time
Initial NRE
Initial τw
Oil
FMT MO-14 SP GB CS < 5 Pa 5-10 Pa 10-20 Pa 20-30 Pa > 30 Pa < 5000 5000 – 10000 10000-15000 > 15000 0-8 h 8-20 h > 20 h
Avg. |ε| 295% 673% 2453% 384% 371% 807% 1019% 1372% 2634% 387% 839% 1246% 2634% 925% 992% 978%
EM Avg. ε 295% 673% 2453% 384% 371% 807% 1019% 1372% 2634% 387% 839% 1246% 2634% 925% 992% 978%
Avg. |ε| 62% 18% 339% 59% 62% 75% 128% 206% 368% 36% 119% 190% 394% 133% 126% 120%
Matzain Avg. ε 34% 17% 339% 56% 36% 75% 127% 206% 368% 18% 105% 187% 394% 107% 123% 120%
Avg. |ε| 20% 40% 107% 37% 61% 38% 51% 34% 31% 64% 39% 47% 33% 55% 42% 17%
Avg. ε -9% -25% 96% -19% 34% 5% -3% -5% 2% 25% -1% 31% 33% 44% 9% -18%
Venkatesan Avg. |ε| 105% 65% 337% 86% 227% 117% 82% 112% 95% 164% 95% 286% 160% 184% 165% 86%
Avg. ε 104% 29% 337% 86% 226% 117% 58% 95% 95% 151% 87% 286% 160% 178% 156% 108%
With the data sets presented in this manuscript, decoupling of time and ΔT effects from NRE and τw is not doable yet. However, oil specific analyses are still feasible to some extent by exploring the fact that temperature difference between bulk oil and cold interface (∆T) diminishes as the time progresses. This phenomena theoretically leads to reduction of equivalent cooling rate and super-saturation in the boundary layer over time. Consequently, it is conceivable that FMT over-prediction expands and EM under-prediction decreases over time. This theory has been supported by visual observation on MO-14 oil (Soedarmo et al., 2016b). Fig.13 shows the trends of
(mass transfer coefficient ratio) over the
deposition time. Fig.13 (a) supports the referred visual observation as it consistently shows that the mass transfer coefficient ratio values for MO-14 decrease with time, indicating an increase in FMT over17
prediction magnitude. Fig.13 (b) shows that only GB’s and several SP data exhibit the similar tendency as MO-14 (with less magnitude), while CS data shows relatively constant mass transfer coefficient ratio over time. Conversely, Fig. 14 shows the change in (
⁄
|
) (
⁄
|
) values (precipitation
curve derivative ratio) over time. Fig 14 (a) shows that for MO-14, most of precipitation curve derivative ratio values are greater than one (except the high shear stress case), indicating that the EM under-predicts the deposition rate. The precipitation curve derivative ratio values tend to be closer to one as time progresses, indicating agreement with visual observation. However, Fig. 14(b) shows that other oils do not necessarily follow the same pattern. These analyses indicate that crude oils may demonstrate different behavior/ sensitivity with respect to change in ∆T compared to model oils.
Figure 13. Mass transfer coefficient ratio value with respect to deposition time.
Figure 14. Solubility curve derivative ratio value with respect to deposition time. 18
The early time over-prediction of Matzain’s model is also demonstrated in examples in Fig. 15. It is likely due to the fact that Matzain’s model does not model aging mechanistically. Instead, it empirically assigns a value for trapped oil fraction in deposit for a given NRE condition. Consequently, the wax fraction predictions increase sharply at early time and tend to be relatively flat afterward, as the NRE changes throughout a single experiment are relatively modest. EM model on other hand, typically predicts gradual increase over time such that it predicts the early time data better compared to Matzain. However, Matzain’s model actually predicts the end of test data slightly better than EM in general (except for several GB cases).
Figure 15. Examples of simulation results for CS (a), SP (b), GB (c), and MO-14 (d) data. 5.
Sherwood Number Analysis This analysis offer a way to validate the TUWAX simulation results for FMT and EM by with
two available Sherwood number calculation methods, which directly related to the prediction of concentration profile near the interface. Huang et al. (2011) presented a model with precipitation kinetics 19
term. In this model, the deposition rate prediction is bounded by FMT (referred as independent heat/ mass transfer or IMHT in the original publication) as upper limit and instantaneous precipitation theory, also referred as solubility method (Venkatesan and Fogler, 2004), as lower limit. The solubility method is essentially similar with EM. These two methods differ in the Sherwood number calculations. For convenience, mathematical formulations of these two methods are provided in the appendix. experimental Sherwood number is back-calculated from
The
obtained from Eq. 1 and compared to the
Sherwood number predicted by these two methods. Gnielinski (1976) equation is used in the Sherwood number calculation for FMT and Nusselt number calculation for solubility method. This comparison is shown in Fig. 16. Fig. 16 shows that the FMT over-predicts the Sherwood number, which is consistent with FMT model performance assessment described earlier. On the other hand, although majority of the experimental Sherwood are under-predicted by solubility method, some of the data are still overpredicted.
This observation agrees with EM model performance assessment described earlier, and
indicates that the thermodynamic equilibrium assumption may not necessarily serves as lower bound of deposition rate prediction.
Figure 16. Comparison between experimental and predicted Sherwood number values. Although neither methods predict the experimental Sherwood number accurately, the average absolute error of solubility method (102%) is considerably lower than FMT (960%). The superior overall performance of solubility method is illustrated further with some examples in Fig. 17. Fig.17 (a) shows two experimental data (MO-01 and MO-02) which were generated under similar NRE (~13,000) but different initial wall shear stresses (25 Pa and 3.4 Pa, respectively). This figure shows that at low shear stress condition, the solubility method under-predicts the experimental Sherwood number. However, at higher shear stress, the experimental Sherwood number is much closer to that of the solubility method 20
prediction, indicating the effect of shear stress in wax thermodynamic equilibrium state near the interface. Fig. 17 (b) shows that solubility method performs well for SP oil, supporting the information provided in Table 4, which states that the EM works best for SP oil data. Fig. 17 (c) supports the EM performance analysis earlier by showing that even the solubility method still over-predicts the Sherwood number value for GB data. Given the consistency of the data sets from two different investigators, it is fair to assume that the data quality is acceptable. Fig. 17 (d) shows the analysis for CS-01 (the only laminar data set) and demonstrates that FMT model is satisfactory for laminar condition. This observation is in agreement with well-established conclusions from previous studies (Singh et al., 2000; Singh et al., 2001). However, for turbulent data (CS-02), the solubility method prediction is considerably more accurate than FMT.
Figure 17. Sherwood number analysis examples for MO-14 (a), SP (b), GB (c), and CS (d) data. 6.
Field Implications
21
This study provides a review of several simple wax deposition models performance to predict recent laboratory/ bench-scale flow loop data. It is observed that the EM and Matzain Model are able to predict majority of the normalized wax mass in deposit (σ) data within the expected uncertainty range, with the exception of GB data. However, using the EM as the base case for flow assurance strategy development increases the risk of being overly optimistic. It is also observed that Matzain’s model predicts the end-of-test (later time) data better than EM, which is conceivably more relevant to field applications. That being said, as Matzain’s model is still unable predict the deposition trends as a whole; it is uncertain whether the model is applicable for longer test durations. Accordingly, it implies that more experimental/ field data and more mechanistically rigorous models are still needed. The σ prediction has not covered the whole wax deposition modeling objectives, as it is still needed to be translated into deposit thickness (δ) and wax content ( ̅ ). Nevertheless, the benefit of reasonably accurate σ predictions could not be undermined. From modeling perspective, reliable aging/ growth prediction is not plausible if the total mass prediction is inaccurate. For practical purposes, if a reasonable range of wax content could be estimated then thickness prediction can be extracted from σ. Fig. 18 shows the range distribution of
̅ data with respect to initial NRE and τw, where
̅ is defined
as the total wax content increase from initial condition (wax content of oil) to the end of test condition. Fig. 18 may not necessarily represent the field cases wax content, as there are other parameters aside from NRE and τw, which differentiate the typical flow loop and field conditions (e.g.: ∆T, duration). However, it illustrates that if more experimental data are available, a plausible range of wax content might be reasonably estimated with greater confidence. Availability of more wax content data is also essential for aging modeling as the next step of comprehensive wax deposition models improvement.
Figure 18. Wax content distribution with respect to initial NRE and τw. MO-02 is one of the data sets which exhibit a resemblance to the field NRE and τw conditions (NRE > 10,000 and τw < 5 Pa). Referring to Fig. 18, there is one data with 22
̅ < 10% and another data with
̅ between 20 to 30% within that range. Assuming no wax content data available and both data are taken into considerations, the
̅ assumed to be 5%, 15%, and 25% to generate thickness predictions
using Matzain’s model. Fig. 19 shows the thickness prediction results using Matzain’s model without wax content adjustment) and with adjusted wax content. Fig. 19 shows that knowledge on wax content data may improve the thickness prediction significantly. Moreover, note that, at higher wax content, the thickness prediction becomes less sensitive to change in wax content.
Depending on the project
development/ field operating stage, wax content data could be obtained from laboratory flow loop data with same/ similar oil, nearby/ similar field experiences, or samples from previous pigging operation. Based on this study, the experimental data and prediction capability of σ value would be valuable for practical field applications if the scalability of it can be verified. As the σ parameter normalizes the pipe geometry effects, it is expected to have better scalability than separated thickness or wax content data. However, more experimental data are needed to prove this expectation.
Figure 19. MO-02 deposit thickness prediction results with Matzain's model. 7.
Conclusions A total of 70 wax deposition data points from 19 different conditions are compiled in this
manuscript. The normalized wax mass in deposit (σ), deposition flux (J), mass transfer coefficient (
), interfacial concentration gradient (
⁄
|
), and Sherwood number (
) are extracted
from these data points for further analyses. Uncertainty propagation analysis was performed to estimate reasonable ranges of error that may be caused by fluid properties or measurement errors instead of the model formulation itself.
23
Four models are validated with these data points, namely: film mass transfer (FMT), equilibrium model (EM), Matzain, and Venkatesan models. The FMT model regularly over-predicts the deposition rate, which is consistent with previous theoretical and visualization studies. It is observed that EM does not necessarily serve as the lower bound of deposition rate, suggesting that some form of flux reduction effects may exist or the closure relationship for diffusion coefficient may perform poorly.
The
conclusions for EM and FMT performance are also validated by Sherwood number analysis, where the values are compared with the Sherwood numbers estimated by FMT theory and solubility method. A fix set of shear coefficient values in Venketasan’s model is not sufficient to achieve acceptable accuracy in general, indicating that the shear coefficients are likely to be oil and flow conditions dependent. Despite of its inability to predict aging mechanistically, Matzain’s model delivers relatively better performance on deposit wax mass and flux predictions among the four models, although it still unable to predict the deposition trends as a whole. In conclusion, more mechanistically rigorous wax deposition models and flow loop experimental data which decouple NRE, τw, ∆T, and oil composition effects are still needed. Acknowledgements The authors would to acknowledge the financial support from Tulsa University Paraffin Deposition Project (TUPDP) company members for this research effort. Appendix A. Uncertainty Propagation Analysis for Independent Variables. An example of this method is the Reynolds number (NRE) uncertainty calculation, where the three determining variables, density ( ), velocity ( ), and viscosity ( ), are independent to each other. Eqs. A1 and A-2 show the procedure to estimate the uncertainty for NRE knowing the uncertainties of the and
measurements using Taylor’s series. √(
) (
√( ) (
)
)
(
) (
( ) (
)
)
(
(
) (
) (
)
Eq. A-1
)
Eq. A-2 ⁄
Taking MO-01 test as an example:
⁄ , and assuming
, the uncertainty of NRE can be estimated as shown in Eqs. A-3 and A-4. √(
) (
)
(
) (
)
24
(
) (
)
Eq. A-3
Eq. A-4 Appendix B. Uncertainty Propagation Analysis for Dependent Variables. An example of this method is the diffusivity of wax in oil (DWO) uncertainty calculation, where the two of the three determining variables, namely viscosity ( ) and temperature ( ), are inter-related. The correlation used to calculate DWO is the Hayduk-Minhas correlation(Hayduk and Minhas, 1982). Eqs. B-1 to B-4 shows the procedure to estimate the uncertainty for DWO knowing the uncertainties of the , and wax molecular volume (
). In addition to Taylor’s series procedure, the uncertainties of the
correlation itself is considered in this procedure as well.
√
( ⏟
) (
)
) (
(
(
)
(
) (
)
(
) (
)
Eq. B-1
⏟
)
⁄
Eq. B-2 ( [
)
⁄ (
)
⁄
(
)
⁄
(
Eq. B-3
)
]
Eq. B-4 ⁄
Taking MO-01 test as an example: ⁄ , and assuming
, the uncertainty of
estimated to be around 19%. The variable
is
in Eq. B-1 represents the covariance of viscosity on
temperature, which is estimated to be -1.34E-5 for MO-14 fluid. Appendix C. Mathematical Formulation for FMT, EM, Matzain’s and Venkatesan’s Models The governing equation of FMT is essentially the same with Singh et al. (2000) model where heat and mass transfer analogy are used to quantify the wax mass transfer rate from the bulk to interface. A mass balance then applied to distribute the incoming wax mass into aging and growth. Cussler et al. (1988) equation is used to quantify the effective diffusivity into the deposit. This concept is described in Eqs. C-1 to C-3. The calculation of
for turbulent flow is based on an analogy to heat transfer
correlations such as Gnielinski (1976) or Manabe (2001) equations. correlation is commonly used to calculate the
. Currently
Hayduk and Minhas (1982)
is still treated as fitting parameters. The
aging model described in Eqs. C-2 and C-3 is also used in EM and Venkatesan’s model. ⏟
( )
⏟
(
)
|
|
⏟
25
(
)
̅
Eq. C-1
̅
⁄(
Eq. C-2
̅ )
̅
Eq. C-3
EM differs from FMT in the way to quantify the incoming wax mass. As the model assumes that the concentration profile always follows the thermodynamic equilibrium, Eq. C-4 is used to express the incoming wax mass flux. Both EM and FMT apply the same mass balance concept and requires the same fitting parameter. ⏟
(
)
|
|
⏟
(
)
|
|
⏟
(
)
̅
Eq. C-4
Matzain’s model is based on EM with empirical correlation to take account of oil entrapment (
) in deposit (which would enhance the thickness growth rate) and shear stripping (
) effect. Eq. C-
5 to C-9 describe the governing equations of this model for single-phase flow applications. The trapped oil estimation also serves as the wax content prediction in this model, hence Singh et al. (2000) aging model is not used. [ |
| ]
Eq. C-5 Eq. C-6
⁄
Eq. C-7 (
)
Eq. C-8 Eq. C-9
The Venkatesan’s model can be applied to either FMT or EM as a mean to reduce the predicted incoming wax flux based on the in-situ shear stress. In this study, the Venkatesan’s model is applied to FMT as shown in Eq. C-10. The m and n coefficients used in this study is 0.8E-12 and 1.9, respectively. ( ) ̇
Eq. C-10
Appendix D. Sherwood Number Calculation As discussed earlier, for FMT the Sherwood number can be calculated using the analogy of Gnielinski (1976) equation as shown in Eq. D-1. The solubility method is described in Eq. D-2 and D-3, where the heat transfer form of Gnielinski (1976) equation is used to calculate the Nusselt number. (
)
)( (
)
(
Eq. D-1
)
26
| (
Eq. D-2 )
)( (
)
(
Eq. D-3
)
References Agarwal, J., 2016. Experimental Study of Wax Deposition in Turbulent Flow Conditions by Using Model Oil (MO-14), MS Thesis, The University of Tulsa, Tulsa, OK, 77 pp. Chi, Y., 2015. Investigation of Wax Inhibitors on Wax Deposition Based on Flow Loop Testing, MS Thesis, The University of Tulsa, Tulsa, OK, 75 pp. Cussler, E.L., Hughes, S.E., Ward III, W.J. and Rutherford, A., 1988. Barrier Membranes. Journal of Membrane Sciences, 388: 161-174. Dieck, R.H., 2007. Measurement Uncertainty: Methods and Applications. ISA, North Carolina (USA). Dwivedi, P., Sarica, C. and Shang, W., 2013. Experimental Study on Wax Deposition Characteristics of a Waxy Crude Under Single-Phase Turbulent Conditions. Oil Gas Facil., 2(04): 61-73. Gnielinski, V., 1976. New equations for heat and mass transfer in turbulent pipe and channel flow. Int. Chem. Eng., 16(2): 359-368. Hayduk, W. and Minhas, B.S., 1982. Correlations of Predictions of Molecular Diffusivities in Liquids. The Canadian Journal of Chemical Engineering, 60: 295-299. Hernandez, O. et al., 2003. Improvements in Single-Phase Paraffin Deposition Modeling. In: SPE (Editor), SPE Annual Technical Conference and Exhibition. SPE, Denver, CO. Huang, Z., Lee, H.S., Senra, M. and Fogler, H.S., 2011. A Fundamental Model of Wax Deposition in Subsea Oil Pipelines. AIChE J., 57(11): 2955-2964. Huang, Z., Zheng, S. and Fogler, H.S., 2015. Wax Deposition: Experimental Characterizations, Theoretical Modeling, and Field Practices. CRC Press, Boca Raton, FL. Lee, H.S., 2008. Computational and Rheological Study of Wax Deposition and Gelation in Subsea Pipelines, Ph.D. Dissertation, The University of Michigan, Ann Arbor, 113 pp. Lu, Y., Huang, Z., Hoffman, R., Amundsen, L. and Fogler, H.S., 2012. Counterintuitive Effects of Oil Flowrate in Wax Deposition. Energy Fuels, 26: 4091-4097. Lubansky, A.S., Yeow, Y.L., Leong, Y.-K., Wickramasinghe, S.R. and Han, B., 2006. A General Method in Computing the Derivative of Experimental Data. AIChE J., 52(1): 323-332. Manabe, R., 2001. A Comprehensive Mechanistic Heat Transfer Model for Two-Phase Flow with High Pressure Flow Pattern Validation, Ph.D Dissertation, The University of Tulsa, Tulsa, OK, 160 pp. Matzain, A., 1999. Multiphase Flow Paraffin Deposition Modeling, Ph.D. Dissertation, The University of Tulsa, Tulsa, OK, 235 pp. Niesen, V.G., 2002. The real cost of subsea pigging, E&P Mag, pp. 97. Panacharoensawad, E., 2012. Wax Deposition Under Two-Phase Oil-Water Flowing Conditions, Ph.D. Dissertation, The University of Tulsa, Tulsa, OK, 422 pp. Panacharoensawad, E. and Sarica, C., 2013. Experimental Study of Single-Phase and Two-Phase Waterin-Crude-Oil Dispersed Flow Wax Deposition in a Mini Pilot-Scale Flow Loop. Energy & Fuels, 27(9): 5036–5053. Rittirong, A., 2014. Paraffin Deposition under Two-Phase Gas Oil Slug Flow in Horizontal Pipes, Ph.D. Dissertation, The University of Tulsa, Tulsa, OK, 344 pp.
27
Rittirong, A., Panacharoensawad, E. and Sarica, C., 2014. An Experimental Study of Paraffin Deposition under Two-Phase Gas-Oil Slug Flow in Horizontal Pipes, Offshore Technology Conference. SPE, Houston. Sarica, C. and Panacharoensawad, E., 2012. Review of Paraffin Deposition Research under Multiphase Flow Conditions. Energy Fuels, 26: 3968-3978. Singh, A., 2013. Experimental and Field Verification Study of Wax Deposition in Turbulent Flow Condition, MS Thesis, The University of Tulsa, Tulsa, OK, 151 pp. Singh, A., Lee, H.S., Singh, P. and Sarica, C., 2011. Validation of Wax Deposition Models Using Field Data from Subsea Pipeline, Proc. - Annu. Offshore Technol. Conf. SPE, Houston, TX. Singh, P., Venkatesan, R., Fogler, H.S. and Nagarajan, N., 2000. Formation and Aging of Incipient Thin Film Wax-Oil Gels. AIChE J., 46(5): 1059-1074. Singh, P., Venkatesan, R., Fogler, H.S. and Nagarajan, N., 2001. Morphological Evolution of Thick Wax Deposits during Aging. AIChE J., 47(1): 6-18. Soedarmo, A.A., Daraboina, N., Lee, H.S. and Sarica, C., 2016a. Microscopic Study of Wax Precipitation: Static Conditions. Energy & Fuels, 30: 954-961. Soedarmo, A.A., Daraboina, N. and Sarica, C., 2016b. Microscopic Study of Wax Deposition: Mass Transfer Boundary Layer and Deposit Morphology. Energy & Fuels, 30(4): 2674-2686. Venkatesan, R., 2004. The Deposition and Rheology of Organic Gels, Ph.D. Dissertation, University of Michigan, Ann Arbor, MI, 226 pp. Venkatesan, R. and Creek, J.L., 2007. Wax Deposition During Production Operations: SOTA, Proc. - Annu. Offshore Technol. Conf. SPE, Houston, TX. Venkatesan, R. and Fogler, H.S., 2004. Comments on Analogies for Correlated Heat and Mass Transfer in Turbulent Flow. AIChE J., 50(7): 1623-1626.
Highlights
Film mass transfer model (FMT) significantly over-predicts all turbulent wax deposition data. Equilibrium model (EM) does not necessarily serve as lower wax deposition limit. Matzain’s model delivers superior performance, but need to be validated with longer experiments. Experimental data to decouple NRE, τw, ∆T, and oil composition effects are still insufficient. More mechanistically rigorous wax deposition models are still needed.
28