Gas state equation construction method suitable for deep coal seam nanoscale pores

Abstract

The invention provides a gas state equation construction method suitable for deep coal seam nanoscale pores, and belongs to the technical field of natural gas low-carbon mining. The theoretical problems of unconventional physical property characteristics and phase behavior laws of gas in nano-scale pores of the deep coal seam are solved; comprising the following steps: determining a hypothesis condition; the method comprises the following steps: establishing mutual potential energy between gas / solid molecules, obtaining a total potential energy function of gas molecules in nanoscale pores, deducing a gas density non-uniform distribution function of the nanoscale pores, and establishing a gas state equation suitable for the nanoscale pores of the deep coal bed gas. According to the gas state equation, factors such as nanoscale space, solid-phase molecule size, gas molecule size and gas-solid interaction strength are considered, the method is suitable for calculation of gas physical properties and phase state characteristics in different coal seams and different-scale pores, and a more accurate gas state equation is provided for evaluation of the occurrence state and gas content of deep coal bed gas. The method is applied to deep coal bed gas resource exploitation.

Description

Technical Field

[0001] The invention provides a method for constructing a gas state equation applicable to nanometer-scale pores in deep coal seams, belonging to the technical field of low-carbon natural gas mining. Background Art

[0002] my country is rich in coalbed methane resources, and coalbed methane has become an important successor to energy reserves and production. Coalbed methane buried more than 1,500 meters deep is usually called deep coalbed methane.

[0003] Nanoscale pores are well developed in deep coal seams, and about 90% of natural gas is stored in nanoscale pores. Deep coalbed methane has the characteristics of "high gas content, high saturation, and free gas". Due to the characteristics of ultra-low porosity and ultra-low permeability of deep coal seams, it is difficult to obtain high yields by conventional drainage and pressure reduction mining methods. It is necessary to undergo large-scale extreme fracturing to construct a high-permeability diversion channel of matrix-fracture-wellbore to form industrial production capacity. However, the existing experimental results show that after fracturing of deep coal seams, the production characteristics of single wells are high in the initial stage and rapidly declining. This is because the amount of methane desorption gas is insufficient to make up for the rapid production of free gas. Therefore, natural gas is super-stable in nanoscale pores. Clarifying the physical properties and phase characteristics of natural gas in nanoscale pores is of great significance to promoting the desorption of deep coalbed methane and improving coalbed methane production capacity.

[0004] Gas in nanoscale pores is affected by the nano-space confinement effect, and its critical temperature value deviates significantly from the theoretical calculated value, showing unconventional thermodynamic properties. Therefore, the traditional thermodynamic state equations that describe the physical properties and phase characteristics of gases, such as the vdW state equation, the PR state equation, and the SRK state equation, are difficult to apply to the exploration and development process of deep coalbed methane resources. It is urgent to construct a gas state equation applicable to nanoscale pores in deep coal seams. Summary of the invention

[0005] In response to the theoretical problems existing in the existing prediction of the occurrence state of deep coalbed methane, the present invention provides a method for constructing a gas state equation suitable for nanoscale pores in deep coal seams. Based on the theory of intermolecular interactions, the potential energy function of gas molecules in nanoscale pores under the influence of solid walls is obtained, and the non-uniform distribution function of gas density on the nanopore cross section is derived. Combined with the theory of thermodynamics, the gas state equation suitable for nanoscale pores in deep coal seams is constructed.

[0006] The technical solution adopted by the present invention is: a method for constructing a gas state equation suitable for nanoscale pores in deep coal seams, comprising the following steps:

[0007] S1: Determine the assumptions;

[0008] S2: Establish the total potential energy function of gas molecules in nanopores: Use the van der Waals intermolecular interaction theory to establish the interaction potential energy between gas molecules and the interaction potential energy between gas and solid molecules, and superimpose them to obtain the total potential energy function of gas molecules in nanopores;

[0009] S3: Establish the non-uniform distribution function of gas density in nanopores: Combined with the Boltzmann distribution theory, the non-uniform distribution function of gas density in nanopores is derived based on function replacement and simplification;

[0010] S4: Construct the gas state equation for nanopores: Based on the thermodynamic relationship between pressure and gas density, derive the gas state equation applicable to nanoscale pores in deep coal seams.

[0011] Furthermore, the assumptions include:

[0012] 1) Ignore the polarization phenomenon of gas molecules caused by gas-solid interaction;

[0013] 2) The translational and rotational kinetic energy of gas molecules themselves is only affected by temperature;

[0014] 3) Ignore the influence of short-range forces between molecules;

[0015] 4) The solid surface is an ideal surface, that is, the surface roughness is ignored.

[0016] Furthermore, the interaction potential energy μ g-g The expression is as follows:

[0017]

[0018] In the formula: C represents the intermolecular van der Waals interaction constant, σ represents the gas molecule size, ρ represents the gas molecule number density, and A represents the intermolecular interaction parameter;

[0019] A gas molecule at a distance Z from the left wall of the pore is subject to the additional potential energy μ generated by the interaction with the pore wall. g-s (Z) is expressed as:

[0020] Where: C' represents the van der Waals interaction constant between gas molecules and solid molecules, D represents the characteristic size of nanoscale pores, ρ s It represents the number density of solid molecules per unit volume;

[0021] The total potential energy μ(Z) of gas molecules in the nanopore at a distance Z from the left wall of the pore is:

[0022]

[0023] Furthermore, the expression of the non-uniform distribution function of gas density in nanopores is:

[0024]

[0025] Where: Z represents the number density of gas molecules at position Z from the left wall of the pore, ρ0 represents the number density of bulk gas molecules not affected by gas-solid interaction, μ0 represents the potential energy of bulk gas molecules not affected by gas-solid interaction, k represents the Boltzmann constant, and T represents the temperature.

[0026] Furthermore, the expression of the gas state equation applicable to nanoscale pores in deep coal seams is:

[0027]

[0028] Where: P represents pressure, v represents gas molar volume, A represents intermolecular interaction parameter, B represents parameter in the derivation process, a represents internal pressure correction due to intermolecular interaction of gas, b represents gas molecular volume correction, c represents internal pressure correction caused by the action of solid wall on gas molecules, d represents external pressure correction caused by nanospace of nanoscale pores.

[0029] The beneficial effects of the present invention compared to the prior art are as follows: the present invention obtains the potential energy of gas molecules under the action of solid walls in nanoscale pores through intermolecular interaction potential energy analysis. Further combined with the Boltzmann distribution function, the non-uniform distribution function of gas density in nanopores is derived on the basis of function replacement and model simplification. On the basis of the thermodynamic relationship between pressure and gas density, a gas state equation suitable for deep nanoscale pores is derived. The gas state equation takes into account factors such as nanoscale space, solid phase molecular size, gas molecular size, and gas-solid interaction strength. It is suitable for the calculation of gas properties and phase characteristics in different coal seams and pores of different scales, and provides a more accurate gas state equation for the evaluation of deep coalbed methane occurrence state and gas content, and facilitates the theoretical development and technical optimization of efficient development of unconventional natural gas. BRIEF DESCRIPTION OF THE DRAWINGS

[0030] The present invention will be further described below in conjunction with the accompanying drawings:

[0031] Figure 1 Schematic diagram of gas-solid molecular interaction in nanoscale pores of deep coal seams of the present invention;

[0032] Figure 2 This is a gas density distribution diagram in nanometer-scale pores of deep coal seams of the present invention;

[0033] Figure 3 This is a graph showing the variation trend of the critical temperature offset factor of gas in nanopores of different sizes under different gas-solid interaction strength conditions of the present invention. DETAILED DESCRIPTION

[0034] like Figures 1 to 3 As shown, the present invention provides a method for constructing a gas state equation applicable to nanoscale pores in deep coal seams, comprising the following steps:

[0035] S1: Determine the assumptions;

[0036] In order to establish the gas state equation of nanoscale pores and apply it to the prediction of deep coalbed methane content and occurrence state, the following assumptions are made:

[0037] 1) Ignore the polarization of gas molecules caused by gas-solid interaction;

[0038] 2) The kinetic energy of gas molecules such as translation and rotation is only affected by temperature;

[0039] 3) Ignore the effect of short-range forces (i.e. repulsion) between molecular interactions;

[0040] 4) The solid surface is an ideal surface, that is, the surface roughness is ignored.

[0041] S2: Using the van der Waals intermolecular interaction theory, the additional potential energy function of the pore wall on the gas molecules is established, and the total potential energy function of the gas molecules in the nanopores is obtained by superposition. Specifically, the total potential energy function of the gas molecules in the nanopores is obtained by establishing the interaction potential energy between gas molecules and the interaction potential energy between gas and solid molecules;

[0042] The specific implementation steps are as follows:

[0043] The intermolecular interaction potential energy function ω(r) in the van der Waals theory system is expressed as:

[0044] ω(r)=-C / r n (1);

[0045] In formula (1), C represents the intermolecular van der Waals interaction constant, with the unit of J·m 6 ; r represents the distance between the centers of interacting atoms or molecules, with the unit of m; n represents the number of bond energies between interacting atoms or molecules.

[0046] The interaction potential μ between two adjacent gas molecules g-g for:

[0047]

[0048] In formula (2), ρ represents the number density of gas molecules, with the unit of 1 / m 3 ; σ represents the size of gas molecules, in m. For the intermolecular van der Waals interaction force, the bond energy number between the interacting atoms or molecules is 6. Therefore, equation (2) becomes:

[0049]

[0050] In formula (3), A represents the intermolecular interaction parameter.

[0051] In nanoscale pores, the potential energy of gas molecules consists of two parts: the potential energy of gas-gas interaction and the potential energy of gas-solid interaction. The potential energy of gas-gas interaction can be represented by formula (3). The potential energy of gas-solid interaction can be represented by formula (1), but due to the dense number of solid molecules on the pore wall, the potential energy composition needs to be analyzed specifically. The additional potential energy generated by the gas molecules in nanoscale pores due to the action of the solid wall is as follows: Figure 1 shown.

[0052] The potential energy of a gas molecule at a distance Z from the left wall of the pore consists of two parts: the potential energy μ generated by the interaction with the surrounding gas molecules g-g , the additional potential energy μ generated by the interaction with the two pore walls g-s The potential energy μ generated by the interaction between gas molecules g-g It can be calculated by formula (3). It should be noted that due to the interaction between gas and solid molecules, the number density of gas molecules is unevenly distributed near the solid surface, that is, the number density ρ Z is a function of the position Z.

[0053] The potential energy of the interaction between gas molecules and pore walls is analyzed as follows: For a single wall, a ring-shaped unit cell with a cross-sectional area of ​​dxdz and a radius of x is taken in the solid, such as Figure 1 As shown. The volume of this unit cell is 2πxdxdz, and the number of solid molecules it contains is 2πρ s xdxdz, where ρ s is the number density of solid molecules per unit volume. Therefore, the additional potential energy generated by a gas molecule at a distance Z from the left wall of the pore due to the action of the left wall of the pore is μ g-s1 :

[0054]

[0055] In formula (4), C' represents the van der Waals interaction constant between gas molecules and solid molecules, with the unit of J·m 6 Here, the number density of solid molecules per unit volume is s The size of the solid molecule d w The relationship between s =6 / π / d w 3 .

[0056] Similarly, the additional potential energy generated by a gas molecule at a distance Z from the left wall of the pore due to the action of the right wall of the pore is μg-s2 :

[0057]

[0058] In formula (5), D represents the characteristic size of nanoscale pores, and the unit is m.

[0059] Therefore, a gas molecule at a distance Z from the left wall of the pore is subject to the additional potential energy μ generated by the interaction with the pore wall. g-s It can be expressed as:

[0060]

[0061] Therefore, the total potential energy of the gas molecules in the nanopore at a distance Z from the left wall of the pore is μ(Z):

[0062]

[0063] S3: Derive the non-uniform distribution function of gas density in nanopores;

[0064] The specific process is as follows:

[0065] According to the Boltzmann distribution theory, the number density of gas molecules at a distance Z from the wall is ρ Z It can be calculated by the following formula:

[0066] ρ Z =ρ0exp[-(μ(Z)-μ0) / kT] (8);

[0067] In formula (8), k represents the Boltzmann constant, k = 1.380649 × 10 -23 J / K, T represents temperature, ρ0 represents the number density of bulk gas molecules not affected by gas-solid interaction; μ0 represents the potential energy of bulk gas molecules not affected by gas-solid interaction, which can be obtained by the following formula:

[0068]

[0069] Combining equations (7) to (9), the number density of gas molecules at position Z away from the left wall of the pore is:

[0070] ρ Z =ρ0exp[-(μ g-g (Z)+μ g-s (Z)+μ g-s (DZ)-μ0) / kT] (10);

[0071] Combining equations (3) to (5), equation (10) can be transformed into:

[0072]

[0073] Formula (11) is a transcendental function. To facilitate the solution, the following substitutions and simplifications are made:

[0074]

[0075] In formula (12), the value range of x is [0,1), and equivalent function replacement is performed within this range:

[0076]

[0077] Through error analysis, it can be seen that the correlation between the replacement function and the objective function is 0.905, which means that equivalent replacement can be performed in the range of [0,1). Therefore, formula (11) can be rewritten as:

[0078] (2.5μ0+kT)ln(1-x)-μ g-s =0 (14);

[0079] Solving equation (14) yields:

[0080]

[0081] Therefore, the number density of gas molecules at position Z from the left wall of the pore is:

[0082]

[0083] According to formula (16), the gas density distribution diagram in the nanopores can be drawn, as follows: Figure 2 As shown in the figure, the ordinate represents ρ / ρ0.

[0084] Integrating equation (16) yields the nanopore gas density ρ: D for:

[0085]

[0086] In formula (17), ξ represents the calculation coefficient of nanopore gas density.

[0087] S4: Establish a gas state equation applicable to nanoscale pores in deep coal seams. The specific process is as follows:

[0088] According to thermodynamics, under certain temperature T conditions, the relationship between system pressure P and gas molecule number density ρ can be expressed as:

[0089]

[0090] Therefore, pressure can be expressed as:

[0091]

[0092] Combining equations (7) and (16), the partial differential of the gas molecule potential energy μ with respect to the molecular number density ρ can be expressed as:

[0093]

[0094] Where B is a parameter in the derivation process and has no practical meaning;

[0095] therefore,

[0096]

[0097] Since Bρ<1, the logarithmic part of equation (21) can be simplified as:

[0098]

[0099] also,

[0100]

[0101] Therefore, the gas state equation applicable to nanoscale pores in deep coal seams is:

[0102]

[0103] In the state equation (24), v represents the molar volume of the gas, which is used to correct the parameters a, b, c, and d. Parameter a represents the internal pressure correction caused by the interaction between gas molecules, parameter b represents the volume correction of gas molecules, parameter c represents the internal pressure correction caused by the action of the solid wall on the gas molecules, and parameter d represents the external pressure correction caused by the nanospace of the nanoscale pores.

[0104] Completing steps S1 to S4 can accurately give the gas density distribution curve in the nanoscale pores of the deep coal seam (such as Figure 2 as shown) and the critical temperature of gas in nanoscale pores in deep coal seams.

[0105] Taking a coalbed methane field as an example, the coal seam depth is 1300 meters, the pressure is 13.5MPa, and the temperature is 65.5℃. Through testing, it is known that the pore size range of nanoscale pores in the coal seam matrix is ​​2nm to 100nm, and the gas is methane. The parameter values ​​required for calculating the critical temperature of the gas are obtained by chemical testing, as shown in Table 1 below.

[0106]

[0107] Table 1. Parameter values.

[0108] Substituting the data into equation (16), the density distribution of methane in the nanopores under in-situ conditions of the coal seam can be plotted, as follows: Figure 2 As shown in the figure, the ordinate represents ρ / ρ0.

[0109] Substituting the data into formula (24), the critical temperature shift factor of gas in nanoscale pores of deep coal seams under different gas-solid interaction strength conditions can be calculated, and the variation trend of the shift factor due to changes in pore size and gas-solid interaction strength can be plotted, as shown in Figure 3 shown.

[0110] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or replace some or all of the technical features therein with equivalents. However, these modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for constructing a gas state equation suitable for nanoscale pores in deep coal seams, characterized by: The following steps are involved: S1: Determine the assumptions; S2: Establish the total potential energy function of gas molecules in nanopores: Use the van der Waals intermolecular interaction theory to establish the interaction potential energy between gas molecules and the interaction potential energy between gas and solid molecules, and superimpose them to obtain the total potential energy function of gas molecules in nanopores; S3: Establish the non-uniform distribution function of gas density in nanopores: Combined with the Boltzmann distribution theory, the non-uniform distribution function of gas density in nanopores is derived based on function replacement and simplification; S4: Construct the gas state equation for nanopores: Based on the thermodynamic relationship between pressure and gas density, derive the gas state equation applicable to nanoscale pores in deep coal seams.

2. The method for constructing a gas state equation applicable to nanoscale pores in deep coal seams according to claim 1, characterized in that: Assumptions include: 1) Ignore the polarization phenomenon of gas molecules caused by gas-solid interaction; 2) The translational and rotational kinetic energy of gas molecules themselves is only affected by temperature; 3) Ignore the influence of short-range forces between molecules; 4) The solid surface is an ideal surface, that is, the surface roughness is ignored.

3. The method for constructing a gas state equation applicable to nanoscale pores in deep coal seams according to claim 2, characterized in that: Interaction potential energy μ between gas molecules g-g The expression is as follows: In the formula: C represents the intermolecular van der Waals interaction constant, σ represents the gas molecule size, ρ represents the gas molecule number density, and A represents the intermolecular interaction parameter; A gas molecule at a distance Z from the left wall of the pore is subject to the additional potential energy μ generated by the interaction with the pore wall. g-s (Z) is expressed as: Where: C' represents the van der Waals interaction constant between gas molecules and solid molecules, D represents the characteristic size of nanoscale pores, ρ s It represents the number density of solid molecules per unit volume; The total potential energy μ(Z) of gas molecules in the nanopore at a distance Z from the left wall of the pore is:

4. The method for constructing a gas state equation applicable to nanoscale pores in deep coal seams according to claim 3, characterized in that: The expression of the non-uniform distribution function of gas density in nanopores is: Where: Z represents the number density of gas molecules at position Z from the left wall of the pore, ρ0 represents the number density of bulk gas molecules not affected by gas-solid interaction, μ0 represents the potential energy of bulk gas molecules not affected by gas-solid interaction, k represents the Boltzmann constant, and T represents the temperature.

5. The method for constructing a gas state equation applicable to nanoscale pores in deep coal seams according to claim 4, characterized in that: The expression of the gas state equation applicable to nanoscale pores in deep coal seams is: Where: P represents pressure, v represents gas molar volume, A represents intermolecular interaction parameter, B represents parameter in the derivation process, a represents internal pressure correction due to intermolecular interaction of gas, b represents gas molecular volume correction, c represents internal pressure correction caused by the action of solid wall on gas molecules, d represents external pressure correction caused by nanospace of nanoscale pores.

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