One-dimensional single-pressure double-fluid model hyperbolic property judgment method based on rooting formula

Abstract

The invention provides a one-dimensional single-pressure double-fluid model hyperbolic property judgment method based on a rooting formula, and belongs to the technical field of multiphase flow, and the method comprises the following steps: establishing a characteristic value equation of a one-dimensional single-pressure double-fluid model, and using the rooting formula for the characteristic value equation to determine the hyperbolic property of the one-dimensional single-pressure double-fluid model; obtaining a solution equation of the gas phase fraction of the one-dimensional single-pressure double-fluid model; basic parameters for hyperbolic judgment of the one-dimensional single-pressure double-fluid model are given, the dimensionless speed is increased from 0, each dimensionless speed and the basic parameters are substituted into a solving equation of the gas phase fraction, and the gas phase fraction meeting the requirement is obtained through solving; and drawing a relational graph of the dimensionless speed and the gas phase fraction, and judging the hyperbolic property of the one-dimensional single-pressure double-fluid model. The one-dimensional single-pressure double-fluid model hyperbolic property judgment method based on the rooting formula has the advantages of being easy to use and visual in result, and can be suitable for any two-phase flow pipeline with known fluid composition and pipeline basic parameters.

Description

Technical Field

[0001] The invention belongs to the field of multiphase flow technology, and in particular is a method for determining the hyperbolicity of a one-dimensional single-pressure two-fluid model based on a root-finding formula. Background Art

[0002] Two-phase flow is widely used in the oil industry. In offshore natural gas pipelines, a two-phase mixture of natural gas and oil is transported from offshore platforms to onshore processing facilities through pipelines. Compared with single-phase flow, two-phase flow has a higher pressure drop and higher transmission energy consumption. In addition, due to the undulating terrain, slug flow may occur in the pipeline, which seriously endangers the safety of pipeline operation. In order to guide the actual production of two-phase flow pipelines and ensure their safe operation, it is necessary to accurately predict parameters such as pressure drop, temperature drop, flow rate, and liquid holdup along the pipeline.

[0003] The most widely used parameter prediction model at present is the one-dimensional single-pressure two-fluid model, which is based on the mass, momentum and energy equations of the oil phase and the gas phase, thus forming a large set of nonlinear partial differential equations. Combined with the numerical solution method, it can accurately predict the flow parameters in the pipeline. However, the model does not have hyperbolicity under specific pressure and volume phase fraction, which means that the eigenvectors corresponding to the model are not linearly independent and do not constitute the basis of the vector space, that is, the numerical solution of the model has no physical meaning and cannot be used to guide field production. If the model has hyperbolicity, it proves that the model has a unique solution that conforms to reality under specific pressure and phase fraction. At this time, the results of the numerical solution are meaningful. Therefore, before numerically solving the one-dimensional single-pressure two-fluid model, it is very important to clarify the hyperbolicity of the model. It not only explains whether the model has hyperbolicity under specific working conditions, but also gives the scope of application of the model. Summary of the invention

[0004] The invention is to solve the hyperbolic problem in numerical solution of a one-dimensional single-pressure two-fluid model and proposes a method for judging the hyperbolicity of a one-dimensional single-pressure two-fluid model based on a root-finding formula.

[0005] The present invention is achieved through the following technical solutions:

[0006] Step 1, establishing an eigenvalue equation of a one-dimensional single-pressure two-fluid model, using a root-finding formula on the eigenvalue equation to obtain a solution equation for the gas phase fraction of the one-dimensional single-pressure two-fluid model;

[0007] Step 2, given the basic parameters for judging the hyperbolicity of the one-dimensional single-pressure two-fluid model, the dimensionless velocity is increased from 0, and each dimensionless velocity and basic parameter is substituted into the solution equation of the gas phase fraction to obtain the gas phase fraction that satisfies the volume fraction constraint condition;

[0008] Step 3, plotting all gas phase fractions that meet the volume fraction constraint condition obtained in step 2 in a graph with the dimensionless velocity as the horizontal coordinate and the gas phase fraction as the vertical coordinate in the order of the dimensionless velocity from small to large, to form a scatter plot of the dimensionless velocity and the gas phase fraction; at this time, the scattered points in the scatter plot will approximately form two curves; according to the relative positions of the two curves in the scatter plot, the curve on the left is called curve I, and the curve on the right is called curve II;

[0009] Step 4, for curve I, take the smallest gas phase fraction in curve I as the starting point, and connect the points on curve I in order from small to large gas phase fraction; for curve II, take the smallest gas phase fraction as the starting point, and connect the points on curve II in order from small to large gas phase fraction; finally, connect curve I and curve II end to end to form a closed area, within which the one-dimensional single-pressure two-fluid model does not have hyperbolicity; outside the closed area, the one-dimensional single-pressure two-fluid model has hyperbolicity.

[0010] In step 1, further technical solutions include the following:

[0011] S11, the one-dimensional single-pressure two-fluid model, takes into account the compressibility of the oil phase and the gas phase;

[0012] S12, the characteristic value equation of the one-dimensional single pressure two-fluid model is formula (1):

[0013]

[0014] (1) In the formula, a = -sχ-1, and are dimensionless pressure, dimensionless velocity and dimensionless eigenvalue respectively; χ and s are coefficients; u g is the gas phase flow rate; u o is the oil phase flow rate; c g is the gas phase sound velocity; c o is the oil phase sound velocity; ρ g is the gas phase density; ρ o is the density of the oil phase; α g is the gas phase fraction; α L is the oil phase fraction; P cj is the pressure correction term of the oil-gas interface, expressed as formula (2):

[0015]

[0016] In the formula (2), h o A is the oil phase liquid level height; ois the cross-sectional area of ​​the pipeline occupied by the oil phase, θ is the pipeline inclination, g is the gravity acceleration, and x is the pipeline length;

[0017] S13, the root-finding formula is (3),

[0018] B 2 -4AC=0 (3)

[0019] In the formula (3), A=D 2 -3F; B = DF-9E 2 ; C=F 2 -3DE 2 ; D = 3b 2 -8ac;

[0020] E=-b 3 +4abc-8a 2 d; F = 3b 4 +16a 2 c 2 -16ab 2 c+16a 2 bd-64a 3 e; the rest of the symbols, definitions and calculation methods are consistent with those in (1) and (2).

[0021] In step 2, further technical solutions include the following:

[0022] S21, the basic parameters include: pipeline pressure, gas phase temperature, oil phase temperature, gas phase density, oil phase density, gas phase sound velocity, oil phase sound velocity;

[0023] S22, the steps for solving the gas phase fraction of the volume fraction constraint condition are as follows: input the basic parameters and the dimensionless velocity into formula (3) to solve the gas phase fraction; in formula (3), there are multiple solutions for the gas phase fraction, only the solutions with the gas phase fraction in the interval of 0 to 1 are retained, that is, the phase fraction that satisfies the volume fraction constraint condition, and the solutions exceeding the interval of 0 to 1 are discarded; keep the basic parameters unchanged, increase the dimensionless velocity, and continue to substitute into formula (3) to solve the gas phase fraction; when the dimensionless velocity is increased to the point that all the solutions for the gas phase fraction in formula (3) exceed the interval of 0 to 1, stop the calculation.

[0024] The present invention adopts the above technical solution to achieve the following beneficial effects:

[0025] (1) The present invention proposes a method for determining the hyperbolicity of a one-dimensional single-pressure two-fluid model based on a root-finding formula, which determines the hyperbolicity of the one-dimensional single-pressure two-fluid model through a root-finding formula and clarifies the method for giving the model boundary conditions.

[0026] (2) The method for determining the hyperbolicity of a one-dimensional single-pressure two-fluid model based on a root-finding formula proposed in the present invention has the advantages of being simple to use and having intuitive results, and can be applied to any two-phase flow pipeline with known fluid composition and basic pipeline parameters. BRIEF DESCRIPTION OF THE DRAWINGS

[0027] Figure 1 It is a step diagram of a method for determining the hyperbolicity of a one-dimensional single-pressure two-fluid model based on a root-finding formula according to the present invention.

[0028] Figure 2 It is a scatter plot of dimensionless velocity and gas phase fraction in the two-phase flow pipeline under working condition 1 in an embodiment of the present invention.

[0029] Figure 3 1 is a diagram showing the relationship between the dimensionless velocity and the gas phase fraction closed area of ​​the two-phase flow pipeline under working condition 1 in an embodiment of the present invention. DETAILED DESCRIPTION

[0030] 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.

[0031] like Figure 1 As shown, a method for determining the hyperbolicity of a one-dimensional single-pressure two-fluid model based on a root-finding formula comprises the following steps:

[0032] Step 1, establishing an eigenvalue equation of a one-dimensional single-pressure two-fluid model, using a root-finding formula on the eigenvalue equation to obtain a solution equation for the gas phase fraction of the one-dimensional single-pressure two-fluid model;

[0033] Step 2, given the basic parameters for judging the hyperbolicity of the one-dimensional single-pressure two-fluid model, the dimensionless velocity is increased from 0, and each dimensionless velocity and basic parameter is substituted into the solution equation of the gas phase fraction to obtain the gas phase fraction that satisfies the volume fraction constraint condition;

[0034] Step 3, plotting all gas phase fractions that meet the volume fraction constraint condition obtained in step 2 in a graph with the dimensionless velocity as the horizontal coordinate and the gas phase fraction as the vertical coordinate in the order of the dimensionless velocity from small to large, to form a scatter plot of the dimensionless velocity and the gas phase fraction; at this time, the scattered points in the scatter plot will approximately form two curves; according to the relative positions of the two curves in the scatter plot, the curve on the left is called curve I, and the curve on the right is called curve II;

[0035] Step 4, for curve I, take the smallest gas phase fraction in curve I as the starting point, and connect the points on curve I in order from small to large gas phase fraction; for curve II, take the smallest gas phase fraction as the starting point, and connect the points on curve II in order from small to large gas phase fraction; finally, connect curve I and curve II end to end to form a closed area, within which the one-dimensional single-pressure two-fluid model does not have hyperbolicity; outside the closed area, the one-dimensional single-pressure two-fluid model has hyperbolicity. Specific embodiments

[0037] There is a two-phase flow pipeline. Now, according to the method of the present invention, the hyperbolicity of the one-dimensional single-pressure two-fluid model is judged. The implementation steps are as follows:

[0038] Step 1, establishing a solution equation for the gas phase fraction of a one-dimensional single-pressure two-fluid model;

[0039] Step 2,

[0040] S21, given the basic parameters for judging the hyperbolicity of the one-dimensional single-pressure two-fluid model, as shown in Table 1:

[0041] Table 1 Basic parameters of a two-phase flow pipeline

[0042]

[0043] S22, increase the dimensionless velocity from 0, substitute each dimensionless velocity and basic parameter into the solution equation of the gas phase fraction, and solve the gas phase fraction that meets the volume fraction constraint condition. The solution results are shown in Table 2:

[0044] Table 2 Gas phase fraction solution results

[0045]

[0046]

[0047] Step 3, according to the results in Table 2, a scatter plot of dimensionless velocity and gas phase fraction is drawn with dimensionless velocity as the horizontal coordinate and gas phase fraction as the vertical coordinate. The drawing result is as follows: Figure 2 As shown in the figure, two curves composed of scattered points can be clearly observed. According to the relative positions of the two curves, the curve on the left is named curve I, and the curve on the right is named curve II.

[0048] Step 4, for curve I, starting from the smallest gas phase fraction in curve I, connect the points on curve I in order of gas phase fraction from small to large; for curve II, starting from the smallest gas phase fraction, connect the points on curve II in order of gas phase fraction from small to large; finally, connect curve I and curve II end to end to form a closed area, and draw the result as shown in Figure 3 As shown; within the closed area, the one-dimensional single-pressure two-fluid model does not have hyperbolicity, and outside the closed area, the one-dimensional single-pressure two-fluid model has hyperbolicity.

[0049] The present invention provides a method for judging the hyperbolicity of a one-dimensional single-pressure two-fluid model based on a root-finding formula. The method judges the hyperbolicity of a one-dimensional single-pressure two-fluid model through a root-finding formula and a dimensionless velocity and gas phase fraction diagram. The method is simple to use, the results are intuitive, the scope of application of the model is clarified, and a solid foundation is laid for the numerical solution of the model. The above 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 hyperbolicity of a one-dimensional single-pressure two-fluid model based on a root-finding formula, characterized in that: The following steps are involved: Step 1, establishing an eigenvalue equation of a one-dimensional single-pressure two-fluid model, using a root-finding formula on the eigenvalue equation to obtain a solution equation for the gas phase fraction of the one-dimensional single-pressure two-fluid model; Step 2, given the basic parameters for judging the hyperbolicity of the one-dimensional single-pressure two-fluid model, the dimensionless velocity is increased from 0, and each dimensionless velocity and basic parameter is substituted into the solution equation of the gas phase fraction to obtain the gas phase fraction that satisfies the volume fraction constraint condition; Step 3, plotting all gas phase fractions that meet the volume fraction constraint condition obtained in step 2 in a graph with the dimensionless velocity as the horizontal coordinate and the gas phase fraction as the vertical coordinate in the order of the dimensionless velocity from small to large, to form a scatter plot of the dimensionless velocity and the gas phase fraction; at this time, the scattered points in the scatter plot will approximately form two curves; according to the relative positions of the two curves in the scatter plot, the curve on the left is called curve I, and the curve on the right is called curve II; Step 4, for curve I, starting from the smallest gas phase fraction in curve I, connect the points on curve I in order from the smallest gas phase fraction to the largest; For curve II, take the smallest gas phase fraction as the starting point, and connect the points on curve II in order from small to large gas phase fraction; finally, connect curve I and curve II end to end to form a closed area. Within this closed area, the one-dimensional single-pressure two-fluid model does not have hyperbolicity; outside this closed area, the one-dimensional single-pressure two-fluid model has hyperbolicity.

2. A method for determining the hyperbolicity of a one-dimensional single-pressure two-fluid model based on a root-finding formula as claimed in claim 1, characterized in that: The step 1 includes the following contents: S11, the one-dimensional single-pressure two-fluid model, takes into account the compressibility of the oil phase and the gas phase; S12, the characteristic value equation of the one-dimensional single pressure two-fluid model is formula (1): (1) In the formula, a = -sχ-1, and are dimensionless pressure, dimensionless velocity and dimensionless eigenvalue respectively; χ and s are coefficients; u g is the gas phase flow rate; u o is the oil phase flow rate; c g is the gas phase sound velocity; c o is the oil phase sound velocity; ρ g is the gas phase density; ρ o is the density of the oil phase; α g is the gas phase fraction; α L is the oil phase fraction; P cj is the pressure correction term of the oil-gas interface, expressed as formula (2): In the formula (2), h o A is the oil phase liquid level height; o is the cross-sectional area of ​​the pipeline occupied by the oil phase, θ is the pipeline inclination, g is the gravity acceleration, and x is the pipeline length; S13, the root-finding formula is (3), B 2 -4AC=0 (3) In the formula (3), A=D 2 -3F; B = DF-9E 2 ; C=F 2 -3DE 2 ; D = 3b 2 -8ac; E=-b 3 +4abc-8a 2 d; F = 3b 4 +16a 2 c 2 -16ab 2 c+16a 2 bd-64a 3 e; the rest of the symbols, definitions and calculation methods are consistent with those in (1) and (2).

3. The method for determining the hyperbolicity of a one-dimensional single-pressure two-fluid model based on a root-finding formula as claimed in claim 1, characterized in that: The step 2 includes the following contents: S21, the basic parameters include: pipeline pressure, gas phase temperature, oil phase temperature, gas phase density, oil phase density, gas phase sound velocity, oil phase sound velocity; S22, the steps for solving the gas phase fraction that satisfies the volume fraction constraint condition are as follows: input the basic parameters and the dimensionless velocity into formula (3) to solve the gas phase fraction; in formula (3), there are multiple solutions for the gas phase fraction, and only the solutions with the gas phase fraction in the interval of 0 to 1 are retained, that is, the phase fraction that satisfies the volume fraction constraint condition, and the solutions exceeding the interval of 0 to 1 are discarded; keep the basic parameters unchanged, increase the dimensionless velocity, and continue to substitute into formula (3) to solve the gas phase fraction; when the dimensionless velocity is increased to the point that all the solutions for the gas phase fraction in formula (3) exceed the interval of 0 to 1, stop the calculation.

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