Development of a New Model for the Formation of Wax Deposits through the Passage of Crude Oil within the Well

Abstract

Wax deposits related to flow assurance are a costly problem in oil production in many fields around the world. Modeling of this process is the main tool for creating and optimizing methods to deal with this problem. This paper considers a new empirical model for the formation of these deposits, based on the results of an array of laboratory studies, theoretical data and technological calculations are presented. The created technique takes into account the conditions of oil flow, data from laboratory studies, and the water cut of the product. The experience of the industrial operation of the technique showed a high convergence of the calculated and actual deposit profiles. Based on a comparison of the calculated and actual deposit profiles, it was concluded that the standard deviation of maximum wax thickness is 6.0%, and the depth with the greatest wax thickness is 3.5%, which is a fairly high result. The use of this technique makes it possible to optimize the depth of mechanical cleaning of the well, the installation of heating cables, as well as the parameters of hot flushing, which increases their efficiency and reduces the cost of combating the formation of wax deposition.

1. Introduction

The formation of wax deposits is one of the most common complications in oil production in the territory of many oil fields around the world [1].

At present, this problem is becoming more and more urgent in the context of a decline in hydrocarbon reserves and an increase in the distance of fields that are moving to the later stages of development [2,3]. As a result of these processes, there is a change in the technological parameters of oil production, namely: an increase in the water cut of the product, a decrease in bottomhole and reservoir pressure, and a change in the component composition of the produced product [4,5,6]. Such dynamics of indicators contribute to the earlier formation of organic deposits [7] and often lead to a shift in the place where paraffin formation begins from the lift pipes to the pore space of productive formations.

1.1. Preventing and Removing of Wax Deposits

The fight against the formation of deposits is carried out in two ways: the removal or prevention of the formation of these deposits. Methods for removing deposits include mechanical cleaning, flushing with hot agents (oil, water, and steam) or hydrocarbon solvents [8,9]. Methods for preventing the formation of deposits include the use of heating cable lines, track heaters, dosing of chemical reagents, the use of smooth coatings, physical fields (magnetic, acoustic) [10,11,12]. Deposit prevention methods are preferable because they allow valuable high molecular weight components to be retained in the oil composition. At the same time, scientific works [13,14] have proven that for the effective application of various methods to combat the formation of wax deposition, a correct assessment of the depth and profile of deposits is required. Next, the main aspects of wax deposit education are considered. Thus, for the use of well flushing with hot agents, it is necessary to assess the profile of wax deposits in an oil-producing well, since this type of treatment is effective only at shallow depths [15]. A similar disadvantage is the use of a heating cable, since in the absence of information about the interval of the formation of wax deposition, it is not possible to determine the installation interval of the heating cable.

1.2. Digitalization of the Oil Industry

A lot of research in this area is aimed at developing a methodology for assessing the spatiotemporal distribution of these deposits [16,17]. However, at the moment there is no universal technique [18]. At the same time, in the context of intensive digitalization of the oil field, work is underway to create digital twins of oil fields [19,20]. To accomplish this task, it is necessary to be able to simulate all technological processes in an oil field, including the formation of wax deposits, since their accumulation leads to a change in pressure and temperature in the oil collection and transportation system [21,22].

1.3. Mechanisms for the Formation of Wax Deposition

The scientific literature considers several main approaches to modeling the formation of deposits: mass transfer and phase transition [23,24]. At the same time, most scientific works describe the first approach. Mass transfer models are based on the concentration difference between the bulk and the interface, usually assuming Fick’s first law [25,26]. Phase transition models, on the other hand, assume cooling and transition from a liquid to a solid state, by analogy with the formation of ice [27]. The main mechanisms used in mass transfer models are molecular diffusion and shear dispersion [28]. A detailed description of these mechanisms is provided in [29] based on which it can be concluded that these mechanisms make little contribution to modeling the formation of wax deposition. The phase transition model is based on the assumption of a constant interfacial temperature, so the process of deposit formation is related to the phase transition problem. At the same time, in the published work [30], it is indicated that a similar system of equations can be applied to simulate the formation of wax deposition as for modeling the formation of ice. A detailed review of phase transition models is provided in [31]. In favor of the validity of the application of the phase transition mechanism, the results of studies pointing to a constant temperature of the phase interface [32,33] are in favor. Moreover, this mechanism makes it possible to explain the phenomenon of a decrease in the rate of formation of wax deposition with an increase in the Reynolds number [34].

1.4. Mathematical Models of the Formation of Wax Deposition

To date, three models of wax deposition formation are widely used in commercial multiphase flow simulators: Rygg, Rydahl, and Rønningsen (RRR), and Matzain and Thermal Analogy (Heat Analogy). A detailed description of these models is provided in [35]. In [36], a review of the existing models for the formation of wax deposition was carried out. As a result, it was found that models based on correlations and physical laws have insufficient accuracy and cannot be used in various conditions. An example of an empirical model of the formation of wax deposition is the work [37]; however, the applied model did not take into account a number of important factors, as a result of which its industrial application was difficult.

1.5. Oil Flow Modes

When oil flows, two modes of oil flow are observed: “hot” and “cold” (Figure 1). The “hot” flow regime implies the oil flow at a temperature exceeding the wax appearance temperature (WAT), and in the “cold” flow regime, the oil temperature is below WAT [38,39]. In the “hot” flow regime, wax deposits are formed from the near-wall oil layer, which cools below WAT [40,41]. In the “cold flow” mode, the formation of deposits occurs in the volume of oil, creating a suspension of “oil-wax deposits” [42,43].

Practical confirmation of the correctness of this model of sediment distribution is presented in several scientific works, for example, in [44,45]. As part of this work, the authors combined the results of laboratory studies and the experience of past studies of authors from various countries to create a comprehensive empirical system for predicting the distribution of wax deposits in linear pipelines and lift wells.

2. Materials and Methods

2.1. “Wax Flow Loop” Installation

Laboratory studies in the framework of this work were carried out on the laboratory stand “Wax Flow Loop” (DIUS-LAB LLC). Research on this stand consists of the circulation of the studied oil at a given temperature along a closed circuit. Oil is heated by a circulation thermostat. To start the test, the temperature of the test section is reduced to create a temperature gradient and start the formation of deposits. The pressure in the system is controlled by supplying gaseous nitrogen to the raw material tank. The inner diameter of the test section is 5.5 mm, the length is 1.3 m, and the wall thickness is 0.89 mm. The thickness of the deposits is estimated based on the dynamics of the increase in the pressure drop between the inlet and outlet of the test section according to the Poiseuille equation (Equation (1)).

$$d_{in}={\left(\frac{Q·128·\mu ·l}{\mathsf{\pi }·\Delta P}\right)}^{1/4}$$

(1)

where $\Delta P$ is the pressure drop in the test section, MPa; $Q$ is the volume flow rate of oil, m³/s; $\mu$ is the oil viscosity, mPa·s; $l$ is the length of the test section, m; and $d_{in}$ is the inner diameter of the test section, m.

Determination of the dynamic viscosity of oil takes place before the study when the oil is circulating in the absence of a temperature gradient. However, ref. [46] describes a method for determining the dynamics of changes in the rheological properties of oil during the formation of paraffin deposits in installations of the “Wax Flow Loop” type, used in this work.

As a result of the execution and processing of the study, a wax deposition formation curve is obtained, a graph of the dependence of the thickness of deposits on time.

2.2. Definition of WAT

For a detailed analysis of studies and further development of a methodology for predicting the formation of deposits, it is necessary to evaluate one of the most important values, wax appearance temperature (WAT). This parameter is also evaluated on the Wax Flow Loop. To do this, the oil and the test section are heated to a temperature above WAT. Then, the wall temperature is gradually reduced and then held for 2 h. If, after this time, no increase in the temperature gradient is registered, then the temperature of the oil is again reduced until a layer of deposits appears. According to the manufacturer’s data, the error of the differential pressure gauge is ±0.075 kPa, which corresponds to the error in the determined thickness of the wax deposition ± 0.006 mm.

2.3. Fluid Properties and Operational Parameters of Oil Wells

To develop a model for the formation of paraffin deposits, studies performed on oil samples “A” and “B” were used. The viscosity of oil under surface conditions was determined using a Rheotest RN 4.1 rotary viscometer, and the density was determined using an Anton Paar laboratory density meter.

Their physical and chemical properties are presented in

Table 1. Physicochemical properties of the studied fluids.

Parameter		Dimension	Value
			«A»	«B»
Oil density	in reservoir *	kg/m3	819	860
	degassed **		859	815
Oil dynamic viscosity	in reservoir *	mPa·s	7.0	0.9
	degassed **		45.6	2.8
Content in oil	Asphaltenes	%	1.7	0.5
	Resins		34.9	5.2
	Paraffins		7.2	6.3
Wax appearance temperature		°C	26	21

* For the oil sample, the reservoir temperature is 27.0 °C; for the “B” sample, 24.1 °C; ** determined under normal conditions (20 °C, 1 atm).

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Within the framework of this work, the results of field studies for 10 oil-producing wells are also presented. The parameters of the operation of these wells and the properties of the extracted fluids are presented in

Table 2. Operating parameters of oil wells and fluid properties.

Parameter		Dimension	Value
			No. 1	No. 2	No. 3	No. 4	No. 5	No. 6	No. 7	No. 8	No. 9	No. 10
Well operation parameters
Well flow rate	for oil	t/day	5.1	8.1	4.1	20.5	14.9	6.8	4.1	7.2	6.8	14.9
	for liquid	m3/day	14.3	23.4	6.3	56	34	9.8	7.1	11.9	10.3	19.5
Product water cut		%	53.6	62.5	21.9	56	50	16	34.2	60	23	12
Pressure	Reservoir	MPa	10.3	13.1	8.7	11.7	15.6	12.3	10.2	10.3	15.8	8.9
	Bottomhole		8.2	5.6	6.4	9.2	9.54	1.39	9.06	8.2	8.4	4.43
	Buffer		1.2	0.7	0.5	1.8	0.7	0.7	1.9	1.2	0.8	2.1
	Annulus		1.2	0.7	0.72	1.8	0.7	0.7	1.9	1.20	0.6	2.1
Gas factor		m3/t	9	9	37.2	37.2	40	40	66.5	9.0	131.2	66.5
Oil saturation pressure with associated petroleum gas		MPa	12.1	12.1	10.1	10.1	9.2	9.2	10.7	12.1	15.8	10.7
Reservoir temperature		$°\mathrm{C}$	31	31	24	24	25	25	26	29.5	30	27
Type of downhole pumping equipment		-	Rod pump	ESP	Rod pump	ESP	ESP	Rod pump	Rod pump	Rod pum	ESP	ESP
Pump descent depth		m	937	1438	1397	1357	1500	1350	1198	1480	1904	1500
Reservoir depth			1710	1580	1436	1405	1538	1386	1567	1530	2547	1644
Dynamic level			850	905	731	663	632	1317	841	980	1648	1357
Fluid properties
Oil density	in reservoir *	kg/m3	914	914	834	834	870	870	813	841	745	814
	degassed **		919	920	884	900	917	899	860	873	821	857
Oil dynamic viscosity	in reservoir *	mPa·s	87.1	87.1	6.3	6.3	12.3	12.3	3.73	7.0	1.25	4.3
	degassed **		104.5	101.49	30.07	20.47	105.3	80.65	30.3	18.5	4.04	10.66
Content in oil	Asphaltenes	%	3.88	4.93	2.64	2.75	7.36	4.98	2.80	1.6	1.35	2.2
	Resins		31.77	30.6	24.62	30.73	28.29	26.46	13.52	17.6	10.88	21.64
	Paraffins		2.31	3.17	2.4	2.9	2	1.66	6.84	2.1	3.23	3.31
Wax appearance temperature		$°C$	20	20	35	35	34	34	20	16	25	20

* Reservoir conditions are described in part «Well operation parameters»; ** determined under normal conditions (20 °C, 1 atm).

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3. Intermediate Results of Creating a Wax Deposition Formation Model

Due to the considerable duration and complexity of creating a wax deposition formation model, this process is presented in stages. For ease of perception of the graphs, the research parameters are encrypted in the signature according to the following template “XX/YY–ZZ”: XX: oil temperature during the study, °C; YY: wall temperature during the study, °C; ZZ: mass flow rate of oil during the study, kg/h. As part of the development of the methodology, more than 100 studies were conducted on the Wax Flow Loop installation. The most representative variants are presented in the text of the work.

Before presenting analysis of research results, an important remark should be made. The oil sample entering the test section is intensively cooled. Moreover, according to studies [47], cooling in the initial part of the test section is very intense. As a result, it must be considered that the actual oil temperature in the test section is lower than at the inlet. During the analysis of the performed studies, it was determined that in the absence of sediment breakdown, the effect of velocity is limited by changes in the parameters of heat and mass transfer.

3.1. A series of Laboratory Studies on Fluid Sample “A”

Figure 2 shows a series of studies performed on one oil sample.

Analyzing these graphs, a number of important dependencies can be noted.

• Analysis of Figure 2a indicates a decrease in the rate of wax formation with an increase in the oil flow rate. At high oil temperature and movement speed, the liquid does not have time to cool down to achieve WAT and the highest intensity of wax deposition formation. As a result, a decrease in the flow rate leads to its cooling and the flow reaches WAT with an increase in the intensity of wax formation.
• At the same time, the opposite dependence is observed in Figure 2b, an increase in the flow rate leads to an increase in the intensity of wax deposition formation. This is due to the fact that the oil temperature during the study is equal to WAT. This means that with an increase in the speed of oil movement, the oil temperature in the test section remains higher, respectively, the intensity of the sediment formation process increases.
• When considering Figure 2c, it can be noted that at an oil velocity of 4 kg/h, the greatest intensity of wax deposition formation, is observed at a temperature corresponding to WAT. At the same time, Figure 2d shows studies at a lower speed (1 kg/h), at which the greatest intensity of wax deposition formation is achieved at a temperature exceeding WAT (30 °C). This dependence can also be explained by oil cooling. In the study of oil with a temperature above WAT (30 °C) at high speed, this temperature is maintained, as a result of which the formation of deposits is not so intense. At a low temperature of the oil flow, it cools, reaching the highest intensity at a temperature of 25 °C. At the same time, studies at 25 °C and a flow rate of 1 kg/h lead to rapid cooling of oil and a sharp decrease in the intensity of the sediment formation process.

The results obtained allow us to assert that when creating a model, it is extremely important to take into account the oil flow regimes.

3.2. Application of the Theory of “Hot” and “Cold” Flow

The change in the rate of deposit formation in the performed studies is similar to the theoretical curve presented in Figure 1. To compare the theoretical and practical curve for oil sample “A”, several additional studies were carried out at a constant wall temperature and a flow rate of 4 and 5 kg/h. These velocities were chosen because they correlate with the speed of oil movement along the 62 mm tubing at a flow rate of 13.6–16.9 m³/day, which is the average well flow rate in the target fields. To construct Figure 3 values, the values of the thickness of the deposits were obtained at one time during the study of one oil sample under different temperature conditions. The obtained thicknesses are recorded as relative values with respect to the thickness of deposits at an oil temperature equal to WAT. For ease of perception, the graphs are shown in reverse order of temperature, similar to Figure 1.

3.3. Studies with a Change in Wall Temperature

The next series of studies is provided at a constant oil temperature and flow rate, but with a change in wall temperature. The studies were carried out on sample “B”, the properties of which are also presented in Table 1. The results of the studies are shown in Figure 4.

Considering the obtained results, it can be seen that the rate of formation of deposits decreases sharply when the wall is heated above 10 °C. At a temperature of 20 °C, the formation of deposits practically stops (at a wall temperature of 25 °C no change in pressure was recorded). By processing the obtained curves, by dividing the value of the deposit thickness for all curves by the value for the curve 35/5–4, it is possible to obtain curves of the relative intensity of deposit formation. Moreover, the site is accepted for consideration after 20 h. Based on the results of this assessment, it is possible to determine the coefficients for recalculating the wax deposition formation curves. For wall temperature 5 °C: 1; for 10 °C: 0.63; for 15 °C: 0.15; and for 20 °C: 0.04.

3.4. Transfer of Laboratory Research to Field Conditions

To perform modeling of the formation of wax deposits on real oil field facilities, it is necessary to conduct studies on a sample of the target fluid. Samples are examined on the “Wax Flow Loop” installation, and the wax deposition formation curve is approximated by one or more laws (Figure 5).

4. Wax Deposit Formation Prediction Model

On the basis of the studies performed, the authors formed a model for the formation of organic deposits. Its step-by-step application is described in this chapter. As an example, we simulate the formation of deposits in well No. 8.

• Performing a “basic” study on the “Wax Flow Loop” installation, it is necessary to determine the shape of the curve for the formation of organic deposits. After the study, it must be approximated for use as a mathematical function. Let us provide, as an example, the result of a laboratory study of an oil sample taken from well No. 8. The properties of the oil sample from the target oil well and parameters of this well are shown in Table 2. Figure 5 shows the result of the study of this sample on the “Wax Flow Loop” installation.
• In addition, a study to estimate the WAT of this oil is necessary. For this sample this value is 16 °C.
• The study performed on the Wax Flow Loop installation is valid only for the conditions in the test section. However, during the movement of oil through the lift column or a linear oil pipeline, it cools and the conditions for wax deposition formation change significantly. It is important to consider that the constancy of the temperature gradient does not mean the constancy of the intensity of wax deposition formation. As described earlier, oil cooling can lead to wax crystallization inside the stream, as a result of which the intensity of wax deposition formation is significantly reduced. To take into account the conditions of oil movement in the developed model, two coefficients are used.
• A coefficient is needed that considers the change in oil temperature ($K_{oil}$). To determine the value of this coefficient at each moment of time, it is necessary to construct a curve (basic curve), the shape of which was obtained in Section 3.2. This curve is plotted against the WAT value of each oil. At the same time, it is necessary to consider the conditions for performing the study. So, in this example, the study on the Wax Flow Loop installation was performed at an oil temperature of 25 °C. In this regard, the curve must be modified by dividing all values by the $K_{oil}$ value of the base curve at the temperature of the study. This process is shown in Figure 6.
• The coefficient for the wall temperature ($K_{surf})$ is built based on the coefficients presented after the analysis of Figure 4. It is obvious that above WAT, the formation of deposits does not occur, and as the temperature decreases, the dynamics of the increase in intensity is also assumed to be identical for all the considered oils. An example of the curve is shown in Figure 7.
• The simulation time is set, the temperature profile along the oil wellbore is calculated (Figure 8a). Based on the simulation time, the approximation law is determined and, by supplying the simulation time to it, the base thickness of the deposits is determined. For example, when performing simulation for a period of 936 h using the logarithmic according to the law described in Figure 5, the base deposit thickness is 2.93 mm.
• Based on the calculated wall temperature and the oil flow temperature, the coefficients $K_{oil}$ and $K_{surf}$ are determined. An example is shown in Figure 8b.
• The influence of water cut is assumed to be linear, since only the initial and final points of the influence of water cut are reliably known, and it is obvious that with its increase, the intensity of wax deposition formation decreases.

As a result, the equation for determining the thickness of deposits in a particular section of the test section contains several multipliers and is presented in Equation (2).

$$\delta =K_{oil}·K_{surf}·\left(C_1·t+C_2\right)·\left(1-w_c\right)$$

(2)

where $K_{oil}$ is the coefficient considering changes in oil temperature; $K_{surf}$ is the coefficient considering the change in wall temperature; $C_1$ is the approximation law constant, mm/h; $C_2$ is the approximation law constant, mm; $t$ is the simulation time, c; and $w_c$ is the water cut, unit fraction.

As a result of the above studies, a model was created that makes it possible to predict the intensity of wax deposition formation in the production string of an oil well (Figure 8c).

5. Results of Application of the Developed Model

As can be seen, the result of applying the developed technique made it possible to reproduce with high accuracy the results of direct measurements of the thickness of wax deposits. To confirm the work of the above method, it is necessary to carry out several such comparisons. Using the above algorithm, a comparison was made between the calculated and actual profiles for 10 oil wells. To assess the correctness of the model, two parameters were used: the deviation of the maximum thickness and the deviation of the depth at which the largest layer of sediments is formed. The evaluation results are shown in

Table 3. The results of modeling the formation of wax deposition.

No.	Maximum Wax Thickness, mm			Depth with Maximum Wax Thickness, m
	Actual	Calculated	Deviation, %	Actual	Calculated	Deviation, %
1	3.5	4.5	28.6 *	200	20	−90.0 *
2	7.5	8.27	10.3	170	20	−88.2 *
3	9	10.16	12.9	260	265	1.9
4	30	24.7	−17.7	260	265	1.9
5	8.6	10.7	24.4 *	109	0	- *
6	7	9.5	35.7 *	90	100	11.1
7	12.6	13.26	5.2	160	0	- *
8	4	3.98	−0.5	260	282	8.5
9	1.68	1.96	16.7	770	792	2.9
10	6.8	7.1	4.4	150	160	6.6
Standard Deviation	-	-	6.03	-	-	3.5

* These values are not accepted for calculating the average deviation and are considered separately.

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The results of applying this technique on 10 oil wells allow us to speak about the high accuracy of this technique. However, the authors consider it important to discuss emerging deviations (wells 1 and 5), as well as technological limitations of the applied technique.

6. Discussion

So, despite the positive result of applying the developed methodology, it is important to discuss the limitations of this methodology. Next, we describe the main limitations of the technique point by point.

• At present, the technique has not been explored in batch wells. The reason for this is the peculiarities of the operating mode of these wells;
• The technique considers the effect of the water cut as a linear one since it is not possible to correctly assess the effect of the water phase on the Wax Flow Loop installation;
• The methodology does not consider the change in oil properties when associated petroleum gas is dissolved in it, which is a technological limitation of the installation;
• It is also important to consider in detail the results with the largest deviation. As an example, in Figure 9, we present the results of comparing the wax profiles for wells No. 1 and 5.

Based on the presented results, the main reason for the deviation of the values of the maximum thickness and depth of the formation of the wax deposition is a deviation in the wellhead part. Similar patterns occur in several other simulated wells operated with sucker rod pumps (No. 2, 7), but in these wells they are the most obvious. There can be several reasons for this phenomenon, but the most likely cause is crushing and removal of deposits when lifting the tubing rods. In wells operated with electrical submersible pumps, there is no such specific feature.

7. Conclusions

• Based on laboratory studies, the authors developed a new empirical model of the formation of paraffin deposits. The developed model is based on the results of laboratory studies, water cut in the product, temperature of the oil flow and the internal surface of the tubing. The model considers the phenomena of hot and cold flow, as well as oil WAT.
• As a result of the application of the developed methodology on several oil wells, it was obtained that the standard deviation of maximum wax thickness is 6.0%, and the depth with the greatest wax thickness is 3.5%, which is a fairly high result. The paper also provides examples of simulation results, where there is a significant deviation of the calculated thickness from the actual one. After analyzing these examples, the possible reasons for the deviation are described, which are the technological features of the well repair.
• The developed model makes it possible to simulate the spatial and temporal distribution of organ deposits along the length of the elevator column to assess dangerous areas and periods of well cleaning from the formed deposits.
• The use of this technique for predicting the formation of deposits in an oil well in industrial conditions has shown that it allows: optimizing existing methods of sediment control, choosing optimal methods, and preventing emergencies. These results are obtained due to the possibility of evaluating the optimal intervals for installing heating cables, mechanical cleaning of the well, and conditions for hot flushing.