Characteristic value polynomial-based three-fluid model suitability determination method

Abstract

The invention provides a three-fluid model suitability determination method based on a characteristic value polynomial, and belongs to the technical field of multiphase flow, and the method comprises the following steps: giving starting point and terminal point parameters of a three-phase flow pipeline, and basic parameters of the pipeline; inputting given parameters into a characteristic value polynomial of the three-fluid model, and calculating characteristic values of a pipeline starting point and a pipeline ending point; according to the calculation result of the characteristic value, the suitability of the three-fluid model at the pipeline starting point and the pipeline end point is judged, and starting point and end point boundary conditions for numerical solution of the three-fluid model are specified. The three-fluid model suitability determination method based on the characteristic value polynomial has the advantages of being easy to use and visual in result, and can be suitable for any three-phase flow pipeline with known fluid composition and pipeline basic parameters. The method is of great significance in determining the applicability of the model, specifying the boundary condition of numerical solution, accurately predicting the parameters of the three-phase flow pipeline and guaranteeing the safe operation of the pipeline.

Description

Technical Field

[0001] The invention belongs to the field of multiphase flow technology, and in particular is a method for determining the suitability of a three-fluid model based on an eigenvalue polynomial. Background Art

[0002] Oil is one of the most important energy sources in the world. In the process of oil extraction and pipeline transportation, it is inevitable to encounter three-phase flow composed of oil, gas and water. Compared with the two-phase flow of oil and gas, the flow pattern of three-phase flow is more diverse, and it has a larger pressure drop and higher liquid holdup, which makes the pipeline face more complex flow assurance problems. In order to perform flow assurance analysis on the three-phase flow pipeline, flow parameters such as pipeline pressure, temperature, flow rate and liquid holdup are required, which can be calculated and predicted through the three-fluid model. Compared with using metering instruments and data acquisition systems to obtain flow parameters, this parameter acquisition method based on flow mechanism is more efficient and can save a lot of investment.

[0003] However, the three-fluid model consists of a large number of partial differential equations, and its analytical solution is basically impossible to obtain, so it needs to be solved by numerical methods. When solving numerically, the model must be well-posed. A well-posed model means that it has a unique solution that conforms to reality under specific flow conditions. An ill-posed model may not accurately capture the physical properties of the fluid, and the solution is prone to oscillation or non-physical solutions during the solution process, which ultimately leads to solution failure. However, the three-fluid model is ill-posed under most flow conditions. Therefore, before numerically solving the three-fluid model, the well-posedness of the model must be judged. In addition, the judgment result of well-posedness can also provide guidance for the selection of boundary conditions of the model during numerical solution.

[0004] According to literature research, among the existing patents, the multiphase flow transient calculation method proposed in patent CN109635405A "A method and system for transient calculation of multiphase flow based on space conservation" can predict the flow parameters of three-phase flow, but does not judge the suitability of the model; patent CN111400950A "Hydrate slurry multiphase pipeline transient flow simulation method and device" achieves accurate prediction of pipeline pressure drop, temperature drop and other flow parameters by establishing a transient model of hydrate slurry multiphase flow, and also ignores the judgment of the suitability of the model. At present, there is no method for judging the suitability of the three-fluid model. The judgment method proposed in this patent, as the basis for solving the three-fluid model, is of great significance for clarifying the applicability of the model, specifying the boundary conditions of the numerical solution, accurately predicting the parameters of the three-phase flow pipeline, and ensuring the safe operation of the pipeline. Summary of the invention

[0005] The present invention is to solve the ill-posed problem in numerical solution of a three-fluid model and proposes a method for determining the well-posedness of a three-fluid model based on eigenvalue polynomials.

[0006] The present invention is achieved through the following technical solutions:

[0007] The above-mentioned method for determining the suitability of a three-fluid model based on an eigenvalue polynomial comprises the following steps:

[0008] Step 1, giving the starting point and end point parameters of the three-phase flow pipeline, as well as the basic parameters of the pipeline; the starting point parameters include: pressure, gas phase temperature, oil phase temperature, water phase temperature, gas phase velocity, oil phase velocity, water phase velocity, gas phase sound velocity, oil phase sound velocity, water phase sound velocity, gas phase fraction, oil phase fraction and water phase fraction; the end point parameters include: pressure, gas phase temperature, oil phase temperature, water phase temperature, gas phase velocity, oil phase velocity, water phase velocity, gas phase sound velocity, oil phase sound velocity, water phase sound velocity, gas phase fraction, oil phase fraction and water phase fraction; the basic parameters of the pipeline include: pipeline diameter and pipeline length;

[0009] Step 2, inputting the pipeline start point, pipeline end point and basic parameters of the pipeline into the eigenvalue polynomial of the three-fluid model to calculate the eigenvalues ​​of the pipeline start point and pipeline end point;

[0010] Step 3: According to the calculation results of the eigenvalues, determine the well-posedness of the three-fluid model at the starting point and the end point of the pipeline, and specify the starting point and end point boundary conditions for the numerical solution of the three-fluid model.

[0011] In step 2, the pipeline start point, pipeline end point and basic parameters of the pipeline are input into the eigenvalue polynomial of the three-fluid model, and the eigenvalues ​​of the pipeline start point and pipeline end point are calculated as follows:

[0012] S21, the characteristic polynomial of the three-fluid model is shown in formula (1):

[0013]

[0014] In the formula (1), λ is the characteristic value; u g ,u o ,u w Represent the flow rates of gas phase, oil phase and water phase respectively; ρ g , ρ o , ρ w Represent the density of gas phase, oil phase and water phase respectively; α g , α o , α w Represent the phase fractions of gas phase, oil phase and water phase respectively; c g 、c o 、c w represent the sound speed of gas phase, oil phase and water phase respectively; P ci , P cjRepresents the pressure correction term of the oil-water interface and the oil-gas interface. Its calculation formula is shown in formula (2) and formula (3):

[0015]

[0016] In the formula (2), h w is the height of the liquid level of the water phase; D is the pipe diameter; θ is the pipe inclination; g is the gravitational acceleration; x is the pipe length; (3) where h o is the liquid level of the oil phase; D is the pipe diameter; θ is the pipe inclination; g is the gravitational acceleration; and x is the pipe length.

[0017] S22, substituting the pipeline start point, pipeline end point and basic parameters of the pipeline into the formula (1) to solve the eigenvalues ​​of the three-fluid model at the pipeline start point and pipeline end point; in the formula (1), there are 9 solution results of the eigenvalue λ, which are λ1, λ2, λ3, λ4, λ5, λ6, λ7, λ8 and λ9;

[0018] In step 3, based on the calculation results of the eigenvalues, the well-posedness of the three-fluid model at the start and end of the pipeline is determined, and the start and end boundary conditions for the numerical solution of the three-fluid model are specified as follows:

[0019] S31, at the starting point of the pipeline, if the eigenvalues ​​obtained by solving equation (1) are 9 real eigenvalues, the three-fluid model is well-posed at the starting point of the pipeline; if the number of real eigenvalues ​​obtained by solving equation (1) is less than 9, or there are imaginary numbers among the 9 eigenvalues ​​obtained by solving equation (1), the three-fluid model is ill-posed at the starting point of the pipeline;

[0020] S32, at the end of the pipeline, if the eigenvalues ​​obtained by solving equation (1) are 9 real eigenvalues, the three-fluid model is well-posed at the end of the pipeline; if the number of real eigenvalues ​​obtained by solving equation (1) is less than 9, or there are imaginary numbers among the 9 eigenvalues ​​obtained by solving equation (1), the three-fluid model is ill-posed at the end of the pipeline;

[0021] S33, at the starting point of the pipeline, if the eigenvalues ​​obtained by solving equation (1) are 9 real eigenvalues, and are 8 negative eigenvalues ​​and 1 positive eigenvalue, then 8 boundary conditions need to be given at the starting point of the pipeline during numerical solution, including: gas phase flow rate, oil phase flow rate, water phase flow rate, gas phase temperature, oil phase temperature, water phase temperature, gas phase fraction, and water phase fraction; at the starting point of the pipeline, if the eigenvalues ​​obtained by solving equation (1) are 9 real eigenvalues, and are 7 negative eigenvalues ​​and 2 positive eigenvalues, then 8 boundary conditions need to be given at the starting point of the pipeline during numerical solution, including: gas phase flow rate, oil phase flow rate, water phase flow rate, gas phase temperature, oil phase temperature, water phase temperature, gas phase fraction, and water phase fraction. Eigenvalues, then 7 boundary conditions need to be given at the starting point of the pipeline during numerical solution, including: gas phase flow rate, oil phase flow rate, water phase flow rate, gas phase temperature, oil phase temperature, water phase temperature, and gas phase fraction; at the starting point of the pipeline, if the eigenvalues ​​obtained by solving equation (1) are 9 real eigenvalues, and there are 6 negative eigenvalues ​​and 3 positive eigenvalues, then 6 boundary conditions need to be given at the starting point of the pipeline during numerical solution, including: gas phase flow rate, oil phase flow rate, water phase flow rate, gas phase temperature, and oil phase temperature;

[0022] S34, at the end point of the pipeline, if the eigenvalue obtained by solving formula (1) is 9 real eigenvalues, and is 8 negative eigenvalues ​​and 1 positive eigenvalue, then one boundary condition needs to be given at the end point of the pipeline during the numerical solution, including: pressure; at the end point of the pipeline, if the eigenvalue obtained by solving formula (1) is 9 real eigenvalues, and is 7 negative eigenvalues ​​and 2 positive eigenvalues, then two boundary conditions need to be given at the end point of the pipeline during the numerical solution, including: pressure and water phase fraction; at the end point of the pipeline, if the eigenvalue obtained by solving formula (1) is 9 real eigenvalues, and is 6 negative eigenvalues ​​and 3 positive eigenvalues, then three boundary conditions need to be given at the end point of the pipeline during the numerical solution, including: pressure, gas phase fraction, and water phase fraction.

[0023] The present invention adopts the above technical solution to achieve the following beneficial effects:

[0024] (1) The present invention proposes a method for determining the well-posedness of a three-fluid model based on an eigenvalue polynomial, which determines the well-posedness of the three-fluid model through the characteristic polynomial of the three-fluid model and clarifies the method for giving the model boundary conditions.

[0025] (2) The method for determining the suitability of a three-fluid model based on an eigenvalue polynomial proposed in the present invention has the advantages of being simple to use and having intuitive results, and can be applied to any three-phase flow pipeline with known fluid composition and basic pipeline parameters. BRIEF DESCRIPTION OF THE DRAWINGS

[0026] Figure 1 It is a step diagram of a method for determining the suitability of a three-fluid model based on an eigenvalue polynomial according to the present invention. DETAILED DESCRIPTION

[0027] In order to make the purpose, technical solution and advantages of the present invention more clearly understood, the present invention is further described below in conjunction with the accompanying drawings of the embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention.

[0028] like Figure 1 As shown, a method for determining the suitability of a three-fluid model based on an eigenvalue polynomial comprises the following steps: Step 1, giving the starting point and end point parameters of a three-phase flow pipeline, and basic parameters of the pipeline; the starting point parameters include: pressure, gas phase temperature, oil phase temperature, water phase temperature, gas phase velocity, oil phase velocity, water phase velocity, gas phase sound velocity, oil phase sound velocity, water phase sound velocity, gas phase fraction, oil phase fraction and water phase fraction; the end point parameters include: pressure, gas phase temperature, oil phase temperature, water phase temperature, gas phase velocity, oil phase velocity, water phase velocity, gas phase sound velocity, oil phase sound velocity, water phase sound velocity, gas phase fraction, oil phase fraction and water phase fraction; the basic pipeline parameters include: pipeline diameter and pipeline length;

[0029] Step 2, by inputting the given parameters into the eigenvalue polynomial of the three-fluid model, the eigenvalues ​​of the pipeline start point and the pipeline end point are calculated;

[0030] Step 3: According to the calculation results of the eigenvalues, determine the well-posedness of the three-fluid model at the starting point and the end point of the pipeline, and specify the starting point and end point boundary conditions for the numerical solution of the three-fluid model. Specific embodiments

[0032] There is a three-phase flow pipeline with a pipe diameter of 450 mm and a length of 50 km. Now, the well-posedness of the three-fluid model is judged according to the method of the present invention. The implementation steps are as follows:

[0033] Step 1, assume the boundary parameters of the pipeline inlet and outlet, the results are shown in Table 1:

[0034] Table 1 Boundary parameter values ​​of a pipeline inlet and outlet

[0035]

[0036]

[0037] Step 2, by inputting the given starting point and end point parameters into the eigenvalue polynomial of the three-fluid model, the eigenvalues ​​of the pipeline starting point and the pipeline end point are calculated; Table 3 is the calculation result of the eigenvalue at the pipeline starting point, and Table 4 is the calculation result of the eigenvalue at the pipeline end point;

[0038] Table 3 Calculation results of characteristic values ​​at the starting point of the pipeline

[0039]

[0040] Table 4 Calculation results of characteristic values ​​at the end point of the pipeline

[0041]

[0042]

[0043] Step 3: According to the calculation results of the eigenvalues, determine the well-posedness of the three-fluid model at the start and end of the pipeline, and specify the start and end boundary conditions of the numerical solution of the three-fluid model. The steps for determining the well-posedness and specifying the boundary conditions are as follows:

[0044] S31, if the calculation results of the characteristic values ​​at the starting point of the pipeline are as shown in Table 3, then the three-fluid model is well-posed at the starting point of the pipeline in working conditions 1, working conditions 2, working conditions 3 and working conditions 4;

[0045] S32, if the calculation results of the characteristic value at the end point of the pipeline are as shown in Table 4, then the three-fluid model is well-posed at the end point of the pipeline in working conditions 1, working conditions 2, working conditions 3 and working conditions 4;

[0046] S33, if the calculation results of the characteristic values ​​at the starting point of the pipeline are as shown in Table 3, in working conditions 1, 2, and 3, 6 boundary conditions need to be specified at the starting point of the pipeline for numerical solution, including: gas phase flow rate, oil phase flow rate, water phase flow rate, gas phase temperature, and oil phase temperature; in working condition 4, 8 boundary conditions need to be specified at the starting point of the pipeline for numerical solution, including: gas phase flow rate, oil phase flow rate, water phase flow rate, gas phase temperature, oil phase temperature, gas phase fraction, and water phase fraction;

[0047] S34, if the calculation results of the characteristic values ​​at the end of the pipeline are as shown in Table 4, in working conditions 1, 2, and 3, three boundary conditions need to be specified at the end of the pipeline for numerical solution, including: gas phase fraction, water phase fraction, and pressure; in working condition 4, one boundary condition needs to be specified at the end of the pipeline for numerical solution, including: pressure.

[0048] The present invention provides a method for determining the well-posedness of a three-fluid model based on an eigenvalue polynomial. The method determines the well-posedness of the three-fluid model by using the characteristic polynomial of the three-fluid model. The method is simple to use, the results are intuitive, and the method for determining the boundary conditions of the model is clarified, laying a solid foundation for the numerical solution of the model.

[0049] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included in the protection scope of the present invention.

Claims

1. A method for determining the suitability of a three-fluid model based on an eigenvalue polynomial, characterized in that: The following steps are involved: Step 1, giving the starting point and end point parameters of the three-phase flow pipeline, as well as the basic parameters of the pipeline; the starting point parameters include: pressure, gas phase temperature, oil phase temperature, water phase temperature, gas phase velocity, oil phase velocity, water phase velocity, gas phase sound velocity, oil phase sound velocity, water phase sound velocity, gas phase fraction, oil phase fraction and water phase fraction; the end point parameters include: pressure, gas phase temperature, oil phase temperature, water phase temperature, gas phase velocity, oil phase velocity, water phase velocity, gas phase sound velocity, oil phase sound velocity, water phase sound velocity, gas phase fraction, oil phase fraction and water phase fraction; the basic pipeline parameters include: pipeline diameter and pipeline length; Step 2, inputting the pipeline start point, pipeline end point and basic parameters of the pipeline into the eigenvalue polynomial of the three-fluid model to calculate the eigenvalues ​​of the pipeline start point and pipeline end point; Step 3: According to the calculation results of the eigenvalues, determine the well-posedness of the three-fluid model at the starting point and the end point of the pipeline, and specify the starting point and end point boundary conditions for the numerical solution of the three-fluid model.

2. The method for determining the suitability of a three-fluid model based on an eigenvalue polynomial as claimed in claim 1, characterized in that: The step 2 includes the following contents: S21, the characteristic polynomial of the three-fluid model is shown in formula (1): In the formula (1), λ is the eigenvalue; u g ,u o ,u w Represent the flow rates of gas phase, oil phase and water phase respectively; ρ g , ρ o , ρ w Represent the density of gas phase, oil phase and water phase respectively; α g , α o , α w Represent the phase fractions of gas phase, oil phase and water phase respectively; c g 、c o 、c w represent the sound speed of gas phase, oil phase and water phase respectively; P ci , P cj represents the pressure correction term of the oil-water interface and the oil-gas interface. Its calculation formula is shown in formula (2) and formula (3): In the formula (2), h w is the height of the liquid level of the water phase; D is the pipe diameter; θ is the pipe inclination; g is the gravitational acceleration; x is the pipe length; (3) where h o is the liquid level of the oil phase; D is the pipe diameter; θ is the pipe inclination; g is the gravitational acceleration; x is the pipe length; S22, substituting the pipeline starting point, pipeline end point and basic parameters of the pipeline in step 1 into the formula (1) to solve the eigenvalues ​​of the three-fluid model at the pipeline starting point and pipeline end point; in the formula (1), there are 9 solution results of the eigenvalue λ, which are λ1, λ2, λ3, λ4, λ5, λ6, λ7, λ8 and λ9.

3. The method for determining the suitability of a three-fluid model based on an eigenvalue polynomial as claimed in claim 1, characterized in that: The step 3 includes the following contents: S31, at the starting point of the pipeline, if the eigenvalues ​​obtained by solving equation (1) are 9 real eigenvalues, then the three-fluid model is well-posed at the starting point of the pipeline; If the number of real eigenvalues ​​obtained by solving equation (1) is less than 9, or there are imaginary numbers among the 9 eigenvalues ​​obtained by solving equation (1), the three-fluid model is ill-posed at the starting point of the pipeline; S32, at the end of the pipeline, if the eigenvalues ​​obtained by solving equation (1) are 9 real eigenvalues, the three-fluid model is well-posed at the end of the pipeline; if the number of real eigenvalues ​​obtained by solving equation (1) is less than 9, or there are imaginary numbers among the 9 eigenvalues ​​obtained by solving equation (1), the three-fluid model is ill-posed at the end of the pipeline; S33, at the starting point of the pipeline, if the eigenvalues ​​obtained by solving equation (1) are 9 real eigenvalues, and are 8 negative eigenvalues ​​and 1 positive eigenvalue, then 8 boundary conditions need to be given at the starting point of the pipeline during numerical solution, including: gas phase flow rate, oil phase flow rate, water phase flow rate, gas phase temperature, oil phase temperature, water phase temperature, gas phase fraction, and water phase fraction; at the starting point of the pipeline, if the eigenvalues ​​obtained by solving equation (1) are 9 real eigenvalues, and are 7 negative eigenvalues ​​and 2 positive eigenvalues, then 8 boundary conditions need to be given at the starting point of the pipeline during numerical solution, including: gas phase flow rate, oil phase flow rate, water phase flow rate, gas phase temperature, oil phase temperature, water phase temperature, gas phase fraction, and water phase fraction. Eigenvalues, then 7 boundary conditions need to be given at the starting point of the pipeline during numerical solution, including: gas phase flow rate, oil phase flow rate, water phase flow rate, gas phase temperature, oil phase temperature, water phase temperature, and gas phase fraction; at the starting point of the pipeline, if the eigenvalues ​​obtained by solving equation (1) are 9 real eigenvalues, and there are 6 negative eigenvalues ​​and 3 positive eigenvalues, then 6 boundary conditions need to be given at the starting point of the pipeline during numerical solution, including: gas phase flow rate, oil phase flow rate, water phase flow rate, gas phase temperature, and oil phase temperature; S34, at the end point of the pipeline, if the eigenvalue obtained by solving formula (1) is 9 real eigenvalues, and is 8 negative eigenvalues ​​and 1 positive eigenvalue, then one boundary condition needs to be given at the end point of the pipeline during the numerical solution, including: pressure; at the end point of the pipeline, if the eigenvalue obtained by solving formula (1) is 9 real eigenvalues, and is 7 negative eigenvalues ​​and 2 positive eigenvalues, then two boundary conditions need to be given at the end point of the pipeline during the numerical solution, including: pressure and water phase fraction; at the end point of the pipeline, if the eigenvalue obtained by solving formula (1) is 9 real eigenvalues, and is 6 negative eigenvalues ​​and 3 positive eigenvalues, then three boundary conditions need to be given at the end point of the pipeline during the numerical solution, including: pressure, gas phase fraction, and water phase fraction.

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