Development of a Process Control System for the Production of High-Paraffin Oil

Abstract

This work is aimed at developing methods for increasing the production of heavy crude oil while optimizing energy costs. Various methods have been studied for recovering heavy oil from deep reservoirs. Based on the developed methods, a number of dynamic models have been obtained that describe the behavior of the temperature field in the tubing. Estimations of thermal deformation are carried out. On the basis of dynamic models, fundamentally new devices are obtained and registered in the prescribed manner, providing a subsystem for automated process control systems.

1. Introduction

The huge increase in the production of paraffinic and heavy crude oil due to the constant depletion of conventional oil reserves has attracted considerable attention from researchers. The production of waxy deposits in significant quantities can interfere with the flow ability of these waxy crude oils, which can eventually lead to production shutdowns [1,2]. The deposition and transformation of paraffin molecules into a paraffin gel, which occupies the cross-sectional area of the inner surface of the pipeline, occurs when the bulk oil temperature drops below the paraffin appearance temperature [3,4]. As a result of the accumulation of paraffin on the walls of the pipeline, the area of oil flow decreases. This phenomenon can lead to an emergency situation and a complete cessation of oil transportation through the pipeline [5], which can lead to undesirable consequences of stopping production and large economic losses [6,7,8]. There are such consequences as the accumulation of paraffin deposits in a large amount, leading to the formation of a paraffin gel [9,10], and resulting in aggregation of paraffin in the oil [11,12] with changes in temperature and pressure; gelation of the paraffin layer, which mainly affects the poor flow ability of waxy crude oils [13,14]; layered deposition of wax, which leads to blockage of the flow line [15,16] and which ultimately leads to a complete stop of production processes.

When paraffin is deposited in a pipeline, such mechanisms of molecular diffusion occur [17] as the formation of paraffin crystal nuclei, Brownian motion, and diffusion shift [18]. When using a pour point depressant, such phenomena as eutectic adsorption occurs [19]. For example, in work [20], the authors describe an experiment in which a polarizing microscope was used to observe the morphological lattice of paraffin crystals. In work [21], the authors conducted an experiment in which wax crystals were formed and separated into three stages using a rheometer that was used when changes in the morphological state of oil occur, as a result of which a change in its viscosity occurs and it is necessary to carry out a physicochemical analysis.For this purpose such laboratory instruments as a gas chromatograph and an infrared spectrometer and other technical methods for determining the quality and composition of oil are used, including both physical and chemical methods [22,23,24].

As an example, which is presented in most of the literature, this is a method for removing paraffin from the pipe walls using scrapers [25];this performs the function of the mechanical friction of the scraper against the pipeline wall, as well as heating the pipe cavity [26,27], (Figure 1).

When using pipeline heating, heating detectors and cables are installed on the oil heating pipeline, while the temperature of the heating cable or detector must be higher than the paraffin precipitation temperature [28,29]. Although it is widely used in oil fields [30,31,32], the heating furnace has a large heat loss, which is not beneficial from either an economic [33] or environmental point of view [34]. Physicochemical methods [35] use the addition of some inhibitors to oil to prevent the formation of paraffin deposits [36,37]; these inhibitors and depressants improve the rheological properties of crude oil during pipeline transport [38,39,40].

Physico-chemical methods [35] use the addition of certain inhibitors to oil to prevent the formation of paraffin deposits [36,37]; these inhibitors and pour point depressants [38,39,40] such as comb polymers, polyvinyl acetate, amorphous polymers, surfactants and nanohybrids, improve the rheological properties of crude oil in pipeline transport [41,42,43]. These paraffins are a mixture of alkanes [44,45,46], which consist of straight and branched chains [47,48], solid or liquid state [49,50]. These include microcrystalline paraffin deposits [51] with a carbon chain from C₁₆ to C₄₀ (alkenyl radicals), as well as isoparaffin hydrocarbons, the content of which is from C₃₀ to C₆₀ [52,53] and which crystallizes into the amorphous structure of the naphthenic ring (Figure 2).

The main tasks aimed at improving the economic efficiency of oil production are the study of linear automatic oil production systems, the construction of their optimal structures, the search for methods and control algorithms [54], and the patterns of optimal functioning of the technological process.

The purpose of this work is to develop a control system for the process of pulsed heating of a high-paraffin oil flow in the tubing of low-rate oil wells, aimed at reducing the cost of oil production by preventing the formation of asphalt, tar, and paraffin deposits.

2. Materials and Methods

2.1. Existing Solution Method

Analyzing the oil field as an object of management, it is possible to identify a five-level structure of information interaction in the oil field, which has three administrative levels and two operational levels [55]. The cluster measuring and computing center, which is at the head of the operational level, supplies the central engineering service in real time with operational information about the technical condition and modes of operation of the exploited field, Figure 3. In the process of applying actions to remove Asphalt-resinous and paraffin deposits in this information scheme, nothing changes, since the technological process for removing Asphalt-resinous and paraffin deposits is not part of the general technological process, but is a third-party, one-time procedure. We will carry out a deep modernization of the technological process by integrating means preventing the formation of Asphalt-resinous and paraffin deposits into the technological process.

Thus, the fifth level of the conceptual model will take the form shown in Figure 4.

The solution to the key element of this upgrade is the introduction of a pulsed heating element into the system. With this element, the classical mathematical model of the tubing has the form:

$$\begin{array}{l}\frac{\partial T}{\partial t}=a^2\left(\frac{{\partial }^{2}T}{\partial x^2}+\frac{{\partial }^{2}T}{\partial y^2}+\frac{{\partial }^{2}T}{\partial z^2}\right);\\ 0<x<l_x; 0<y<l_y; 0<z<l_z.\end{array}$$

(1)

$$\begin{array}{l}T(x,y,L_z,\tau )=U(x,y,\tau ); \frac{\partial T(x,y,0,\tau )}{\partial z}=0;\\ T(x,0,z,\tau )=T(x,l_y,z,\tau )=T(0,y,z,\tau )=T(l_x,y,z,\tau )=0;\\ T(x,y,z,0)=0.\end{array}$$

(2)

The equation can be represented as a Green’s function.

2.2. Mathematical Model Development

We will obtain dynamic mathematical models to describe the temperature field of oil flow in tubing.

A one-dimensional equation is presented:

$$\begin{array}{l}T(x_j,t)={\displaystyle \sum _{i=1}^d}{\displaystyle \sum _{n=1}^k\frac{2}{l}\exp \left[-{\left(\frac{\pi na}{l}\right)}^{2}t\right]}\sin \frac{\pi n}{l}{x}_j\sin \frac{\pi n}{l}{\xi }_i\\ +{\displaystyle \sum _p}{\displaystyle \sum _{n=1}^k\frac{2}{l}}\exp \left[-{\left(\frac{\pi na}{l}\right)}^2\left(t-{\tau }_p\right)\right]\sin \frac{\pi n}{l}{x}_j\sin \frac{\pi n}{l}{\xi }_{z\left(p\right)}.\end{array}$$

(3)

Two-dimensional equation:

$$\begin{array}{l}T(x_j,y_j,t)={\displaystyle \sum _{i=1}^d}{\displaystyle \sum _{k,m=1}^{\infty }\frac{4}{l_1\cdot l_2}\exp \left[-a^2{\pi }^2\cdot t\cdot \left(\frac{k^2}{l_1^2}+\frac{m^2}{l_2^2}\right)\right]\cdot \sin \left(\frac{k\cdot \pi \cdot x_j}{l_1}\right)}\cdot \sin \left(\frac{k\cdot \pi \cdot {\mathsf{\rho }}_i}{l_1}\right)\\ \times \sin \left(\frac{m\cdot \pi \cdot y_j}{l_2}\right)\cdot \sin \left(\frac{m\cdot \pi \cdot {\mathsf{\nu }}_i}{l_2}\right)+{\displaystyle \sum _p{\displaystyle \sum _{k,m=1}^{\infty }\frac{4}{l_1\cdot l_2}\cdot }}\exp \left[-a^2{\pi }^2\cdot (t-{\tau }_p)\cdot \left(\frac{k^2}{l_1^2}+\frac{m^2}{l_2^2}\right)\right]\\ \times \sin \left(\frac{m\cdot \pi \cdot y_j}{l_2}\right)\cdot \sin \left(\frac{k\cdot \pi \cdot x_j}{l_1}\right)\cdot \sin \left(\frac{k\cdot \pi \cdot {\mathsf{\rho }}_{z(p)}}{l_1}\right)\cdot \sin \left(\frac{m\cdot \pi \cdot {\mathsf{\nu }}_{z(p)}}{l_2}\right).\end{array}$$

(4)

Equation of a Line in Three Dimensions

$$G(x,y,z,\mathsf{\rho },\mathsf{\nu },\mathsf{\vartheta },t)=\frac{8}{l_1\cdot l_2\cdot l_3}\cdot {\displaystyle \sum _{k,m,n=1}^{\infty }{B}_{k,m,n}}(\cdot )\cdot \exp \left[-a^2{\pi }^2\cdot t\cdot \left(\frac{k^2}{l_1^2}+\frac{m^2}{l_2^2}+\frac{n^2}{l_3^2}\right)\right]$$

(5)

$$B_{k,m,n}(\cdot )=\sin \left(\frac{k\cdot \pi \cdot x}{l_1}\right)\cdot \sin \left(\frac{m\cdot \pi \cdot y}{l_2}\right)\cdot \sin \left(\frac{m\cdot \pi \cdot z}{l_3}\right)\cdot \sin \left(\frac{k\cdot \pi \cdot \mathsf{\rho }}{l_1}\right)\cdot \sin \left(\frac{k\cdot \pi \cdot v}{l_2}\right)\cdot \sin \left(\frac{n\cdot \pi \cdot \mathsf{\vartheta }}{l_3}\right).$$

(6)

The dependences obtained allow us to carry out numerical simulation of the behavior of the temperature field in time [37].

2.3. Simulation Method

In this work, programming in the DELPHI environment was used for a two-dimensional object for controlling the temperature field of oil flow between paraffin molecules in tubing. The graph of the function is shown in Figure 5a–c.

Figure 5 shows that the generated temperature field has a cyclic nature, and the obtained control algorithms do not use all the heating elements located on the control object [38,39]. This Figure was made according to the developed mathematical model based on the Green’s function which allows you to create a point temperature effect of unit power. The existing Green’s function is one-dimensional. The author obtained a two-dimensional model of the optimal oil production regime.

Thus, it is possible to determine the optimal (smallest) number of heating elements required to maintain a given temperature regime [40].

3. Results

3.1. Numerical Example

For this, a mathematical model was obtained to determine the places and moments of switching on the heating elements based on the formulated optimality criterion for the two-dimensional case.

$$x=\arcsin \frac{\frac{4}{l_1{l}_2}{\displaystyle \sum _{k,m=1}^{\infty }\sin \left(\frac{m\pi y}{l_2}\right)\sin \left(\frac{k\pi \mathsf{\rho }}{l_1}\right)\sin \left(\frac{m\pi \mathsf{\nu }}{l_2}\right)\exp \left[-a^2{\pi }^{2}t\left(\frac{k^2}{l_1^2}+\frac{m^2}{l_2^2}\right)\right]}}{G\left(x,y,\mathsf{\rho },\mathsf{\nu },t\right)}\left(\frac{l_1}{k\pi }\right);$$

(7)

$$y=\arcsin \frac{\frac{4}{l_1{l}_2}{\displaystyle \sum _{k,m=1}^{\infty }\sin \left(\frac{m\pi x}{l_1}\right)\sin \left(\frac{k\pi \mathsf{\rho }}{l_1}\right)\sin \left(\frac{m\pi \mathsf{\nu }}{l_2}\right)\exp \left[-a^2{\pi }^{2}t\left(\frac{k^2}{l_1^2}+\frac{m^2}{l_2^2}\right)\right]}}{G\left(x,y,\mathsf{\rho },\mathsf{\nu },t\right)}\left(\frac{l_2}{m\pi }\right).$$

(8)

$$t=\frac{\ln \left(\frac{\frac{4}{l_1{l}_2}{\displaystyle \sum _{k,m=1}^{\infty }\sin \left(\frac{k\pi x}{l_1}\right)\sin \left(\frac{m\pi y}{l_2}\right)\sin \left(\frac{k\pi \mathsf{\rho }}{l_1}\right)\sin \left(\frac{m\pi \mathsf{\nu }}{l_2}\right)\exp \left[-a^2{\pi }^{2}t\left(\frac{k^2}{l_1^2}+\frac{m^2}{l_2^2}\right)\right]}}{G\left(x,y,\mathsf{\rho },\mathsf{\nu },t\right)}\right)\left(\frac{l_1^2}{k^2}+\frac{l_2^2}{m^2}\right)}{a^2{\pi }^2}.$$

(9)

And three-dimensional case

$$x=\frac{l_1}{\pi }\arcsin \frac{\frac{8}{l_1{l}_2{l}_3}\exp \left[-a^2{\pi }^{2}t\left(\frac{1}{l_1^2}+\frac{1}{l_2^2}+\frac{1}{l_3^2}\right)\right]\cdot \sin \frac{\pi }{l_2}y\cdot \sin \frac{\pi }{l_3}z\cdot {\displaystyle \sum _{i=1}^d\sin \frac{\pi }{l_1}}{\mathsf{\rho }}_i\cdot \sin \frac{\pi }{l_2}{\mathsf{\nu }}_i\cdot \sin \frac{\pi }{l_3}{\mathsf{\vartheta }}_i}{T(x,y,z,t)}\cdot$$

(10)

$$\begin{array}{l}y=\frac{l_2}{\pi }\cdot \arcsin \frac{\frac{8}{l_1{l}_2{l}_3}\exp \left[-a^2{\pi }^{2}t\left(\frac{1}{l_1^2}+\frac{1}{l_2^2}+\frac{1}{l_3^2}\right)\right]\cdot \sin \frac{\pi }{l_1}x\cdot \sin \frac{\pi }{l_3}z\cdot {\displaystyle \sum _{i=1}^d\sin \frac{\pi }{l_1}}{\mathsf{\rho }}_i\cdot \sin \frac{\pi }{l_2}{\mathsf{\nu }}_i\cdot \sin \frac{\pi }{l_3}{\mathsf{\vartheta }}_i}{T(x,y,z,t)}.\\ z=\frac{l_3}{\pi }.\arcsin \frac{\frac{8}{l_1{l}_2{l}_3}\exp \left[-a^2{\pi }^{2}t\left(\frac{1}{l_1^2}+\frac{1}{l_2^2}+\frac{1}{l_3^2}\right)\right]\cdot \sin \frac{\pi }{l_1}x\cdot \sin \frac{\pi }{l_2}y\cdot {\displaystyle \sum _{i=1}^d\sin \frac{\pi }{l_1}}{\mathsf{\rho }}_i\cdot \sin \frac{\pi }{l_2}{\mathsf{\nu }}_i\cdot \sin \frac{\pi }{l_3}{\mathsf{\vartheta }}_i}{T(x,y,z,t)}\cdot \end{array}$$

(11)

On the basis of these equations, a large number of computer and natural experiments were carried out, which showed the occurrence of thermal deformation of the tubing. To analyze thermal deformation, we will analyze the mathematical model.

$$\frac{d{T}_1(x,r,\mathsf{\Theta },\tau )}{d\tau }=a_1\cdot (\frac{d^2{T}_1(x,r,\mathsf{\Theta },\tau )}{d{r}^2}+\frac{1}{r}\cdot \frac{d{T}_1(x,r,\mathsf{\Theta },\tau )}{dr}+\frac{d^2{T}_1(x,r,\mathsf{\Theta },\tau )}{d{x}^2}+\frac{1}{r}\cdot \frac{d^2{T}_1(x,r,\mathsf{\Theta },\tau )}{d{\mathsf{\Theta }}^2}),$$

(12)

$$0<x<L, R_2<r<R_1, 0<\Theta <{360}^{°}.$$

(13)

Temperature field of oil flow

$$\frac{d{T}_2(x,r,\mathsf{\Theta },\tau )}{d\tau }=a_2\cdot (\frac{d^2{T}_2(x,r,\mathsf{\Theta },\tau )}{d{r}^2}+\frac{1}{r}\cdot \frac{d{T}_2(x,r,\mathsf{\Theta },\tau )}{dr}+\frac{d^2{T}_2(x,r,\mathsf{\Theta },\tau )}{d{x}^2}+\frac{1}{r}\cdot \frac{d^2{T}_2(x,r,\mathsf{\Theta },\tau )}{d{\mathsf{\Theta }}^2}),$$

(14)

$$0<x<L, 0<r<R_2, 0<\Theta <{360}^{°}.$$

(15)

Boundary conditions for the phase variable T₁

$${\lambda }_1\frac{d{T}_1(x,R_1,\mathsf{\Theta },\tau )}{dr}={\lambda }_B\frac{d{T}_B(x,R_1,\mathsf{\Theta },\tau )}{dr},$$

(16)

$$T_1(x,R_1,\mathsf{\Theta },\tau )=T_B(x,R_1,\mathsf{\Theta },\tau );0<x<L, 0<\mathsf{\Theta }<{360}^{\mathrm{o}}.$$

(17)

Boundary conditions for the phase variable T₂

$${\lambda }_1\frac{d{T}_1(x,R_2,\mathsf{\Theta },\tau )}{dr}={\lambda }_2\frac{d{T}_2(x,R_2,\mathsf{\Theta },\tau )}{dr};$$

(18)

$$T_1(x,R_2,\mathsf{\Theta },\tau )=T_2(x,R_2,\mathsf{\Theta },\tau );$$

(19)

$$0<x<L, 0<\Theta <360°$$

(20)

End faces of the object

$$\begin{array}{l}{T}_1(0,r,\mathsf{\Theta },\tau )=T_1(L,r,\mathsf{\Theta },\tau )=0; R_2<r<R_1;\\ T_2(0,r,\mathsf{\Theta },\tau )=T_2(L,r,\mathsf{\Theta },\tau )=0; 0<r<R_2;\end{array}$$

(21)

Then, the final equation for estimating the thermal deformation of the tubing wellbore has the form

$$f={\displaystyle \sum _{j=1}^N\mathrm{arctg}\left(\right(\Delta x_j-\Delta {x)/(2R}_3\left)\right)}.$$

(22)

3.2. Thermodynamic Properties

The final step in the assessment of the regulatory system is the determination of sustainability [41]. It should be noted that the control system is non-linear, and therefore there are no methods for assessing stability [42,43]. We adapt the Nyquist stability criterion for an impulsive distributed system [44,45]. To do this, we obtain the transfer function of an open-loop control system, which is written as follows.

$$\begin{array}{c}{W}^{*}{}_P\left(G_{\eta ,\gamma },s\right)=(1-\exp (-\mathrm{st}) \cdot \frac{1}{\mathrm{s}}\cdot \frac{1}{t}\cdot {\displaystyle \sum _{r=-\infty }^{r=\infty }(}(E_1\cdot \left[\frac{n_1-1}{n_1}+\frac{1}{n_1}\cdot G_{\eta ,\gamma }\right]+E_4\cdot \left[\frac{n_4-1}{n_4}+\frac{1}{n_4}\cdot G_{\eta ,\gamma }\right]\cdot \frac{1}{s}\\ +E_2\cdot \left[\frac{n_2-1}{n_2}+\frac{1}{n_2}\cdot G_{\eta ,\gamma }\right]\cdot s)\cdot \frac{\exp \left(\beta (G_{\eta ,\gamma }\cdot z^{*}\right)+\exp \left(-\beta (G_{\eta ,\gamma }\cdot z^{*}\right)}{\lambda \cdot \beta (G_{\eta ,\gamma })\cdot (\exp \left(\beta (G_{\eta ,\gamma })\cdot z_L\right)-\exp \left(-\beta (G_{\eta ,\gamma })\cdot z_L\right))}),\\ \beta ={\left(\frac{s+jr{\omega }_u}{a}+G_{\eta ,\gamma }\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.},\left(\eta ,\gamma =\overline{1,\infty }\right)\end{array}$$

(23)

By passing from an infinite number of circles of unit radius, we obtain the hodograph of a spatially distributed impulse control system, Figure 6.

Methods for the analysis and synthesis of a closed system for controlling the temperature field of tubing have been obtained. Based on the methods obtained, a number of thermal drills were designed and tested on an industrial scale, performing various specific tasks of geological exploration and thermal drilling in the conditions of the Arctic zone. Experimental studies were carried out on an electron microscope in various modes, demonstrating uniform heating of the metal. The results of the experiment confirm the absence of destruction of the metal structure caused by pulsed heating.

4. Conclusions

This article discusses the technological process of cleaning downhole equipment from asphalt and paraffin deposits. The cause-and-effect relationships of the formation of deposits of various components of paraffin and depressant molecules during the phase transition of paraffinic oil are considered. Based on the work done, conclusions were drawn about the effectiveness of the application of thermal control methods. For these purposes, a mathematical model of temperature fields and a synthesis of pulsed control of the temperature field using the Green’s function have been developed to automate the technological process of oil production from fields; also, a method for determining absolute stability based on the Nyquist criterion for impulsive distributed systems in the form of a Green’s function has been developed. The presented mathematical model takes into account the spatial coordinates of the location and the spatial orientation of the object of study.

Future developments in the field of heavy oil production should be based on the results of pilot and pilot plants in order to offer companies a reliable and adapted technology. In the near future, for the rational use of technological advances, the expected need for the extraction of heavy and extra-heavy oil will be brought into line. This may require the convergence of several technologies to meet the requirements of the oil industry. Fundamental research in the oil industry plays a key role in this regard.