^{1}

^{2}

^{3}

^{2}

^{3}

^{2}

^{2}

^{1}

^{4}

^{1}

^{2}

^{3}

^{4}

Using the downhole annular flow measurement system to get real-time information of downhole annular flow is the core and foundation of downhole microflux control drilling technology. The research work of electromagnetic flowmeter in recent years creates a challenge to the design of downhole annular flow measurement. This paper proposes a design and optimization of annular flow electromagnetic measurement system for drilling engineering based on the finite element method. Firstly, the annular flow measuring and optimization principle are described. Secondly, a simulation model of an annular flow electromagnetic measurement system with two pairs of coil is built based on the fundamental equation of electromagnetic flowmeter by COMSOL. Thirdly, simulations of the structure of excitation system of the measurement system are carried out, and simulations of the size of the electrode’s radius are also carried out based on the optimized structure, and then all the simulation results are analyzed to evaluate the optimization effect based on the evaluation indexes. The simulation results show that optimized shapes of the excitation system and electrode size can yield a better performance in the annular flow measurement.

In recent decades, oil and gas exploration is being carried out in some extremely harsh and challenging environmental conditions [

A huge array of flow technology options is on offer which provides options in selecting the correct annular flow measurement for the application of drilling engineering. A broad range of factors regarding the special environment of downhole drilling, such as downhole space, velocity profile, temperature, and fluid properties, should be considered. Electromagnetic measurement has the advantages of simple structure, no moving parts, and no obstruction of fluid flow throttle parts. Also, the flow path does not cause any additional pressure loss, and it does not cause wear or blockage, in particular when measuring slurry with solid particles, sewage and other liquid-solid two-phase bodies, or a variety of viscous slurry, and so on. In addition, because the structure has no moving parts, so any corrosion will be attached to the insulation lining. After selecting corrosion-resistant electrode material with a very good corrosion resistance, it can be used for a variety of corrosive media measurements. In 2017, Liang et al. propose a new method for an annular flow measurement system based on the electromagnetic induction principle [

For the traditional electromagnetic flowmeter used in a round pipe, the signal voltage is dependent on the average flow velocity, the magnetic flux density, and the pipe diameter. The signal voltage is expected to be linearly related to the average flow velocity if the magnetic field is a uniform magnetic field. In this ideal case, the flow rate can be considered to be immune to the velocity profile of the pipe flow, especially when the flow has been fully developed. However, it is difficult for the annular flow electromagnetic measurement system to yield a uniform magnetic field for an annular flow path, and also, the velocity profile of the annular flow also cannot be considered axisymmetrically distributed under this special drilling environment. These effects affect the accuracy of the annular flow electromagnetic measurement system. According to Bevie’s vector weight function theory [

In the past few decades, great efforts about the structure of excitation systems and electrodes have been made to reduce the effects of the velocity profile. Horner B improved the measurement accuracy of flow measurement by increasing the number of electrodes [

Following Faraday’s law, the flow of a conductive liquid through a magnetic field will cause a voltage signal to be sensed by electrodes located on the flow pipe walls. Faraday’s formula can be expressed as

Equation (

The signal voltage is expected to be linearly related to the average flow velocity if the magnetic field is a uniform magnetic field. So (

Here,

However, the annular flow electromagnetic measurement system is difficult to yield a uniform magnetic field for an annular flow path, and the velocity profile of the annular flow also cannot be considered axisymmetrically distributed under this special drilling environment. This affects the accuracy of the annular flow electromagnetic measurement system. So the design and optimization of the downhole annular flow electromagnetic measurement system with two pairs of electrodes cannot be investigated based on the traditional Faraday theory.

According to Bevir’s theory [

Here,

According to Bevie’s vector weight function theory, if the result of the magnetic flux density cross-product density of virtual current is constant, the annular flow electromagnetic measurement system can be considered to be immune to the velocity profile of the annular flow. When the electrode and the structure of the flow path are fixed, the density of virtual current is fixed. Thus, the shape of excitation system can be derived based on this constant condition. Similarly, when the shape of the excitation system and the structure of the flow path are fixed, the density of the virtual current only depends on the electrodes. In this case, the radius of the electrode was selected to be optimized to reduce the distortion caused by the velocity profile.

Figure

The schematic diagram of the annular four-electrode flow electromagnetic measurement system and the flow path.

To simplify the boundary conditions, we suppose that the electronic conductivity of the outer surface of the annular flow path is much smaller than the flow. The partial differential equation and the boundary conditions of the outer surface of the annular flow path and electrodes can be written as follows:

By using the segregation variable method, (

When

Equation (

If A1 and B1 are the positive electrodes and A2 and B2 are the negative electrodes, the boundary conditions of the inner surface of the annular flow path and electrodes can be written as follows:

Based on boundary condition (

Thus,

Here, we suppose that

The density of the virtual current can be derived using the gradient method based on (

To avoid winding the complicated coil, the iron core was introduced to the excitation part of the annular flow electromagnetic measurement system. Considering the special downhole environment, the schematic diagram of the excitation part of annular flow electromagnetic measurement system is shown in Figure

The schematic diagram of the excitation part of the annular flow electromagnetic measurement system.

The 3D model of the annular four-electrode flow electromagnetic measurement system and the flow path.

In this paper, the structure of the iron core and the size of the coil among the annular domain between the inner and outer surfaces of the system were investigated, and an optimized excitation structure was designed. Therefore, the width of the core and the height of core protrusion could be used as variables. In the simulations, the round electrodes were chosen with a radius of 0.7 cm, and the length of the iron core was 10 cm. The excitation coils were modified with the width of the core changing from 2 cm to 5 cm and the height of core protrusion from 5.5 cm to 8.5 cm. The width of the core was incremented every 1 cm, while the height of core protrusion every 1 cm.

To validate the effectiveness of the optimization, some performances of the selected optimum annular flow electromagnetic measurement system were compared. Compared simulation results of the magnetic flux vector and weight function vector are shown from Figures

The simulation diagram of the magnetic flux vector and weight function vector with core width equal to 2 cm.

When the height of core protrusion is 5.5 cm

When the height of core protrusion is 7.5 cm

The simulation diagram of the magnetic flux vector and weight function vector with core width equal to 3 cm.

When the height of core protrusion is 5.5 cm

When the height of core protrusion is 7.5 cm

The simulation diagram of the magnetic flux vector and weight function vector with core width equal to 4 cm.

When the height of core protrusion is 5.5 cm

When the height of core protrusion is 7.5 cm

The simulation diagram of the magnetic flux vector and weight function vector with core width equal to 5 cm.

When the height of core protrusion is 5.5 cm

When the height of core protrusion is 7.5 cm

Figures

After the structure of the excitation system of the measurement system was optimized and in order to improve the accuracy of the system, some simulations about the radius of the electrodes were used to improve the system performance. The electrodes were modified, with the radius changing from 1 cm to 5 cm, and the radius of the electrodes was incremented every 0.5 cm. Compared simulation results of the virtual current vector and weight function vector are shown in Figures

The simulation diagram of virtual current density with electrode radius equal to 1 cm, 3 cm, and 5 cm separately.

Radius equal to 1 cm in the

Radius equal to 1 cm in the

Radius equal to 3 cm in the

Radius equal to 3 cm in the

Radius equal to 5 cm in the

Radius equal to 5 cm in the

The simulation diagram of weight function vector with electrode radius equal to 1 cm, 3 cm, and 5 cm separately.

Radius equal to 1 cm in the

Radius equal to 1 cm in the

Radius equal to 3 cm in the

Radius equal to 3 cm in the

Radius equal to 5 cm in the

Radius equal to 5 cm in the

Figures

The ideal excitation system design is to make the weight function vector to be a constant, and this is helpful to improve the measurement accuracy of the system. However, the weight function vector is not easily designed to be a constant. In order to design a magnetic field that makes the weight function vector as constant as possible, it is necessary to define a design quantity that measures the degree of nonuniformity of the weight function vector over the cross-sectional area of the annular domain. The definition of the design quantity was discussed by Dennis and Wyatt in 1972 [

To evaluate the homogeneity range in the annular area, the ratio of the homogeneity range is used as an evaluation criterion [

If a finite element meets (_{1} and the number of homogeneous finite elements is _{2}, the homogeneity range ratio can be defined as follows:

The coefficient of variation is a useful statistic for comparing the degree of variation from one data series to another, even if the means are drastically different from one another. A coefficient of variation of the weight function vector is a statistical measure of the dispersion of the weight function vector data points in a data series around the mean. It is calculated as follows:

The fourth performance criterion is the sensitivity of the output voltage which gives the strength of the system output. In this research, we assure that the turns and current of the coil are constant, and we get the sensitivity of the output voltage while the average flow velocity is 1 meter per second.

During the optimal design and quantitative evaluation process, smaller standard deviation of the weight function vector is better, as is a bigger homogeneity range ratio of the weight function vector is better, smaller coefficient of variation of the weight function vector is better, and bigger sensitivity of the output voltage is better. In these four evaluation indexes, the standard deviation of the vector weight function is the most important evaluation index, and the other 3 indexes are auxiliary reference indexes.

In order to intuitively analyze and compare the effect of excitation structure optimization, the value of the weight function vector standard deviation is analyzed by the definition of the design quantity. To validate the effectiveness of the optimization, the results of the weight function vector standard deviation are compared in Figure

The curve of weight function vector standard deviation and structure of excitation system.

In Figure

Using Figure

The curve of the homogeneity range ratio and structure of the excitation system.

The curve of the coefficient of variation of weight function and structure of the excitation system.

The curve of the sensitivity of the output voltage and the structure of excitation system.

It can be seen from Figures

Considering that the standard deviation of the vector weight function is the most important evaluation index, optimum results of annular flow electromagnetic measurement system under the double pairs of coil excitation structure and current finite structure space can be obtained, with core_X = 5.5 cm and Core_w = 2 cm being the best coil excitation structures. However, from the range of numerical magnitude of evaluation indexes, we can find that changing the width of the core and height of core protrusion has a limited effect on improving the system measurement performance.

In order to intuitively analyze and compare the effect of excitation system optimization, the value of the weight function vector standard deviation can be analyzed using the definition of the design quantity. To validate the effectiveness of the optimization, the results of the weight function vector standard deviation are compared in Figure

The curve of the weight function vector and electrode radius.

Similarly, based on simulation data and mathematical operations, the variation curve of the homogeneity range ratio, the coefficient of variation of the weight function vector, and the sensitivity of output voltage are shown from Figures

The curve of the homogeneity range ratio and electrode radius.

The curve of the coefficient of variation of weight function and electrode radius.

The curve of the sensitivity of the output voltage and electrode radius.

This paper described the design and optimization of annular flow electromagnetic measurement system for drilling engineering. The following conclusions can be drawn according to the above-mentioned analysis:

The theory of annular flow electromagnetic measuring and optimization principles were described, and an annular flow electromagnetic measurement system simulation model was built based on this theory by COMSOL.

Simulations on the structure of excitation system of measurement system were carried out, as well as some simulations on the size of the electrode’s radius were also carried based on the optimized structure, and then all the simulation results were analyzed to evaluate the optimization effects based on the evaluation indexes. The simulation results showed that the optimized structure of excitation system and electrode radius size can yield a better performance during the annular flow measurement process.

The authors declare that there is no conflict of interest regarding the publication of this article.

This work is supported by Open Fund (OGE201702-19) of Key Laboratory of Oil & Gas Equipment of Ministry of Education (Southwest Petroleum University), the National Natural Science Foundation (no. 51504211), and the State Administration of National Security (no. sichuan-0011-2016AQ).