A method for calculating effective thermal conductivity of unsaturated porous media considering thermal expansion

Abstract

The present invention provides a method for calculating the effective thermal conductivity of an unsaturated porous medium taking thermal expansion into consideration, comprising the following steps: S1, based on fractal theory and equivalent ideas, calculating the diameter λ of the equivalent pores and the maximum pore diameter λ of the pores after heating max and minimum pore diameter λ min ; S2, calculate the fractal dimension of pores after heating D f ; S3, calculate the porosity φ after heating; S4, calculate the fractal dimension D of tortuosity after heating T and the characteristic length L0 of the pore after heating; S5, calculate the average height h of the rough units in the pore av , and the relative roughness γ of the pores after heating; S6, calculate the total thermal resistance R of the gas phase in the porous medium after heating nwt , total thermal resistance of water phase R wt And the total solid thermal resistance R s ; S7, calculate the total effective thermal conductivity k of the unsaturated porous medium after heating e The present invention is based on fractal theory and considers factors such as temperature change, rough pores, water phase saturation and curved pores to solve the total effective thermal conductivity, providing an effective basis for geothermal mining research.

Description

Technical Field

[0001] The invention relates to the technical field of geothermal resource exploitation theory, and in particular to a method for calculating the effective thermal conductivity of an unsaturated porous medium taking thermal expansion into consideration. Background Art

[0002] In recent years, my country's social development has been extremely rapid, and the demand for energy has been unprecedentedly huge. It is particularly important to expand the diversified energy structure, which requires us to accelerate the research and development of deep energy. Geothermal energy, as an emerging renewable energy source, originates from underground radioactive substances or magma, comes from the interior of the earth's crust, and can be transmitted to the ground through natural cracks or artificial drilling, and can be used by people. Therefore, the research and development of geothermal energy has attracted widespread attention in recent years.

[0003] Geothermal energy exploitation involves heat-fluid-solid multi-field coupling. The complex mechanism of heat-fluid-solid multi-field coupling and the complexity of the pore distribution of porous media restrict the efficient exploitation of geothermal energy. Thermal conductivity is an important parameter for quantitatively characterizing the thermal conductivity of porous media. The accurate and efficient determination of thermal conductivity has a very important guiding significance for geothermal energy exploitation. At present, domestic and foreign scholars mainly use indoor experiments, numerical simulations and other means to carry out research on the thermal conductivity of porous media. These studies are of great significance to promoting the exploitation of geothermal resources, but these methods cannot efficiently predict the thermal conductivity of porous media during the heat-fluid-solid coupling process, and the main controlling factors of thermal conductivity during the heat-fluid-solid coupling process are not clear. For example: existing studies mostly test the thermal conductivity of porous media through experiments. Since the experimental test results are affected by the pore structure and representativeness of the samples, it is not clear whether the experimental test results can represent the parameters of geothermal energy reservoirs. In addition, experimental testing is time-consuming and labor-intensive, and the test results are affected by experimental methods and experimental errors. In order to solve the problem of thermal conductivity of porous media in geothermal energy mining, scholars at home and abroad use empirical equations to predict thermal conductivity of porous media. However, the empirical equations do not take into account the pore structure of porous media and contain empirical parameters with unclear physical meanings, which brings uncertainty to the prediction and exploitation of geothermal energy. Therefore, in order to solve the problem of unclear thermal conductivity of porous media in geothermal energy mining, there is an urgent need for a method to comprehensively consider factors such as temperature change of porous media, rough pores, water phase saturation and curved pores, and establish a thermal conductivity model of porous media under the influence of temperature, so as to reveal the thermal-fluid-solid coupling law of porous media in geothermal mining and guide efficient geothermal mining. Summary of the invention

[0004] In view of this, in order to solve the above-mentioned problem of unclear thermal conductivity of porous media, an embodiment of the present invention provides a method for calculating the effective thermal conductivity of unsaturated porous media taking thermal expansion into consideration.

[0005] An embodiment of the present invention provides a method for calculating the effective thermal conductivity of an unsaturated porous medium taking into account thermal expansion, comprising the following steps:

[0006] S1. Equivalent the cross section of the irregular pores in the unsaturated porous medium after the temperature rises to the cross section of a cylinder with the same area, and calculate the diameter of the equivalent pores , and according to the pore diameter Get the maximum pore diameter after heating and minimum pore diameter ,

[0007] Wherein, the pore diameter The calculation formula is:

[0008]

[0009] In the formula, is the thermal expansion coefficient of the solid particles, is the temperature change, is the initial pore diameter of the cylindrical cross section equivalent to the cross section of the irregular pore before heating;

[0010] S2. According to fractal theory, calculate the fractal dimension of the pores in porous media after heating , the pore fractal dimension after heating The calculation formula is:

[0011]

[0012] In the formula, is the initial fractal dimension of the porous medium before heating, The maximum initial pore diameter of a circular cross section equivalent to the cross section of a single irregular pore before heating;

[0013] S3. Calculate the porosity of porous media after heating based on fractal theory , the porosity of the porous medium after heating The calculation formula is:

[0014]

[0015] S4. According to fractal theory, calculate the fractal dimension of the tortuosity of the porous medium after heating and the characteristic length of the pores in the porous medium after heating ;

[0016] Among them, the tortuosity fractal dimension of the porous medium after heating is The calculation formula is:

[0017]

[0018] The characteristic length of the pores of the porous medium after heating The calculation formula is:

[0019]

[0020] S5. According to the principle of pore roughness solution, the cone height of the rough unit in the pores of the porous medium is The base diameter of the cone is ,and The ratio of is constant and satisfies the fractal theory, and the average height of the rough units in the pores of the porous medium is calculated. , and according to the average height of the rough unit Calculate the relative roughness of porous media after heating ;

[0021] The average height of the rough unit The calculation formula is:

[0022]

[0023] In the formula, is the ratio of the total base area of ​​the roughness unit to the pore surface area, is the maximum roughness unit base diameter, is the basis fractal dimension of the rough unit, is the ratio of the minimum to maximum base diameter of the roughness unit;

[0024] The relative roughness of the porous medium pores after heating The calculation formula is:

[0025]

[0026] S6. Calculate the total thermal resistance of the gas phase in the porous medium after heating according to the thermoelectric analogy theory. , total thermal resistance of water phase , and calculate the total solid thermal resistance according to the definition of thermal resistance , the calculation formulas for the three are:

[0027]

[0028]

[0029] In the formula, is the gas phase thermal conductivity, is the thermal conductivity of water phase, is the amplitude of the radius when the pore changes sinusoidally, is the critical pore diameter, is the cross-sectional area of ​​the porous medium characterization unit cell, and , is the solid phase thermal conductivity;

[0030] S7. Calculate the total effective thermal conductivity of unsaturated porous media according to the definition of thermal conductivity , and its calculation formula is:

[0031]

[0032] In the formula, is the total thermal resistance of each phase.

[0033] Furthermore, in step S1, the maximum pore diameter of the pores after heating is and minimum pore diameter The expression is:

[0034]

[0035] In the formula, It is the minimum initial pore diameter of a circular cross section equivalent to the cross section of a single irregular pore before heating.

[0036] Further, before step S2, according to fractal theory, the initial fractal dimension of the porous medium before heating is calculated: , and its calculation formula is:

[0037]

[0038] In the formula, is the initial porosity of the porous medium before heating.

[0039] Further in step S7, according to the thermoelectric analogy theory, the total thermal resistance of each phase The expression is:

[0040]

[0041] In the formula, is the total thermal resistance of the gas phase, is the total thermal resistance of water phase, is the solid phase thermal resistance.

[0042] The beneficial effects brought about by the technical solution provided by the embodiments of the present invention are: a method for calculating the effective thermal conductivity of unsaturated porous media taking into account thermal expansion of the present invention fully considers the influence of factors such as temperature, rough pores, water vapor solid phase saturation and curved pores when calculating the effective thermal conductivity of porous media, so that the obtained thermal conductivity is more in line with the actual working conditions, so as to reveal the heat-fluid-solid coupling law of porous media in the geothermal exploitation process and guide the efficient geothermal exploitation. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] Figure 1 It is a flow chart of a method for calculating effective thermal conductivity of unsaturated porous media taking thermal expansion into consideration according to the present invention;

[0044] Figure 2 It is a schematic diagram of the change of a single pore before and after heating in step S1 of the method of the present invention;

[0045] Figure 3 It is a simplified diagram of heat transfer in porous media. DETAILED DESCRIPTION

[0046] To make the objectives, technical solutions and advantages of the present invention more clear, the embodiments of the present invention will be further described below with reference to the accompanying drawings.

[0047] Please refer to Figures 1 to 3 The embodiment of the present invention provides a method for calculating the effective thermal conductivity of an unsaturated porous medium taking into account thermal expansion. Based on the fractal theory, the influence of temperature on pores is fully considered to calculate the thermal conductivity of the unsaturated porous medium under temperature changes. The method specifically includes the following steps:

[0048] S1. Equivalent the cross section of the irregular pores in the unsaturated porous medium after the temperature rises to the cross section of a cylinder with the same area, and calculate the diameter of the equivalent pores , and according to the pore diameter Get the maximum pore diameter after heating and minimum pore diameter ,

[0049] Wherein, the pore diameter The calculation formula is:

[0050] (1)

[0051] In the formula, is the thermal expansion coefficient of the solid particles, is the temperature change, It is the initial pore diameter of the cylindrical cross section equivalent to the cross section of the irregular pore before heating.

[0052] Specifically, this embodiment calculates the thermal conductivity of an unsaturated porous medium. Therefore, when calculating the pore diameter after heating, a single pore is used as an example for ease of understanding. Figure 2 The pores selected in this embodiment are the special-shaped pores (i.e., attached pores) formed by the contact between three solid phases (i.e., medium particles) Figure 2 In the middle space formed by the three solid phases in the porous medium), when the temperature of the porous medium increases, the solid phase will expand due to heat, resulting in a smaller pore size; according to fractal theory, it is assumed that Figure 2The area of ​​the triangle formed by the centers of the solid phase cross sections connected to each other remains unchanged before and after heating, that is, the area of ​​the triangle formed by each solid phase is controlled to remain unchanged, and the overlapping or extrusion deformation of each solid phase particle is ignored. According to the area of ​​the triangle before and after heating and the thermal expansion formula of the solid particles, the following relationship can be obtained:

[0053] (2)

[0054] In the formula, is the radius of the solid particles before heating, is the radius of the solid particles after heating, is the pore area after heating.

[0055] Eliminate the above formula (2) And simplify, get the pore area after heating The calculation expression is:

[0056] (3)

[0057] Since the pore area obtained at this time The area of ​​the cross section of the irregular pore after heating is not conducive to calculating the diameter of the irregular pore. Therefore, the equivalent idea is adopted to equate the cross section of the irregular pore to the cross section of a cylinder with the same area, and the relationship can be obtained:

[0058] (4)

[0059] In the formula, is the equivalent pore diameter after heating, that is, .

[0060] Combining formula (3) and formula (4), we get the expression:

[0061] (5)

[0062] The initial diameter of the pore before heating is calculated using the same equivalent method: , its calculation expression is:

[0063] (6)

[0064] Combining formula (5) and formula (6), the expression of the pore diameter after heating is calculated, namely formula (1).

[0065] Thus, according to the commonly used similar deduction in mathematics, the maximum pore diameter after heating is obtained: and the minimum pore diameter The calculation expression is:

[0066] (7)

[0067] In the formula, is the maximum initial pore diameter of the circular cross section equivalent to the cross section of the irregular pore before heating, It is the minimum initial pore diameter of a circular cross section equivalent to the cross section of the irregular pore before heating.

[0068] S2. According to fractal theory, calculate the fractal dimension of the pores in porous media after heating , the pore fractal dimension after heating The calculation formula is:

[0069] (8)

[0070] In the formula, is the initial fractal dimension of the porous medium before heating, It is the maximum initial pore diameter of a circular cross section equivalent to the cross section of a single irregular pore before heating.

[0071] Specifically, according to the fractal theory, the initial pore fractal dimension of the unsaturated porous medium before heating is obtained: , whose expression is:

[0072] (9)

[0073] In the formula, is the initial porosity of the porous medium before heating.

[0074] According to fractal theory, the probability density function of pore size distribution in porous media before heating is obtained And the total number of pores in the unit cell The expression is:

[0075] (10)

[0076] In the formula, is the pore radius before heating, and , is the maximum initial pore radius before heating, is the minimum initial pore radius before heating.

[0077] Similarly, the cross-section of the irregular pores before heating is equivalent to the cross-section of the cylindrical pores, and combined with formula (10), the total cross-sectional area of ​​the pores before heating can be calculated: and the total volume of the pores , the calculation formulas for the two are:

[0078] (11)

[0079] Similarly, according to the similar deduction commonly used in mathematics, the total cross-sectional area of ​​the pores after heating can be obtained: and the total volume of the pores , the calculation formulas for the two are:

[0080] (12)

[0081] In the formula, is the maximum pore radius after heating, is the minimum pore radius after heating, is the fractal dimension of the pores after heating.

[0082] According to fractal theory, assuming that the ratio of the total cross-sectional area to the total volume of the pores remains unchanged before and after heating, the following expression can be obtained according to formulas (11) and (12):

[0083] (13)

[0084] According to fractal theory and empirical summary, in porous media, it can be considered that ,and Based on this, equation (13) is simplified to obtain the pore fractal dimension after heating: The expression of , that is, formula (8) is obtained.

[0085] S3. Calculate the porosity of porous media after heating based on fractal theory , the porosity of the porous medium after heating The calculation formula is:

[0086] (14)

[0087] Specifically, according to the fractal theory and combined with formula (8), the porosity of the porous medium after heating is calculated as: , that is, formula (14).

[0088] S4. According to fractal theory, calculate the fractal dimension of the tortuosity of the porous medium after heating and the characteristic length of the pores in the porous medium after heating .

[0089] Specifically, according to the fractal theory, the initial tortuosity fractal dimension of the porous medium before heating is obtained: , whose expression is:

[0090] (15)

[0091] Similarly, according to the commonly used similarity deduction in mathematics and combined with fractal theory, the fractal dimension of the tortuosity of the porous medium after heating can be obtained through formula (15): The calculation formula is:

[0092] (16)

[0093] Furthermore, according to the fractal theory, the characteristic length of the pores in the porous medium after heating is calculated: , and its calculation formula is:

[0094] (17)

[0095] S5. According to the principle of pore roughness solution, the cone height of the rough unit in the pores of the porous medium is The base diameter of the cone is ,and The ratio of is constant and satisfies the fractal theory, and the average height of the rough units in the pores of the porous medium is calculated. , and according to the average height of the rough unit Calculate the relative roughness of porous media after heating .

[0096] Specifically, based on fractal theory, the pores in porous media are rough, and it is assumed that the rough units in the pores are cone-shaped, and the height of the cone is The base diameter of the cone is ,and If the ratio of is constant, then the cumulative number equation of the particles characterizing the base diameter can be obtained according to the fractal theory:

[0097] (18)

[0098] In the formula, The base diameter is equal to or greater than The cumulative number of coarse units; is the maximum roughness unit base diameter; is the base fractal dimension.

[0099] By taking the derivative of formula (18), we can get the following expression:

[0100] (19)

[0101] At the same time, based on fractal theory, in two-dimensional space, the base fractal dimension of the rough unit It can be expressed as:

[0102] (20)

[0103] In the formula, is the ratio of the total base area of ​​the roughness unit to the pore surface area; is the ratio of the minimum to maximum base diameter of the roughness unit; is the minimum roughness unit base diameter.

[0104] Based on the geometric characteristics of the conical rough unit and the average height formula, and combined with formula (19) and formula (20), the average height of the rough unit is calculated as , and its calculation formula is:

[0105] (twenty one)

[0106] In the formula, is the total volume of all coarse elements, For the The base diameter of the roughness unit; is the total pore surface area.

[0107] Based on fractal theory, it can be considered that the diameter of each pore is equivalent to the ratio of the maximum rough unit base diameter in the pore, so:

[0108] (twenty two)

[0109] In the formula, is the base diameter of the largest roughness unit in the pore with the smallest diameter.

[0110] Simplifying formula (22), we get formula (23):

[0111] (twenty three)

[0112] Based on the definition of relative roughness and combined with formula (21) and formula (23), the relative roughness of the porous medium pores after heating is It can be expressed as:

[0113] (twenty four)

[0114] In the formula, is the height of the largest coarse element in the pore with the smallest diameter.

[0115] In formula (7), Substituting into formula (24), we can get the relative roughness after temperature rise: The representation relationship:

[0116] (25)

[0117] S6. Calculate the total thermal resistance of the gas phase in the porous medium after heating based on the thermoelectric analogy theory. , total thermal resistance of water phase , and calculate the total solid thermal resistance according to the definition of thermal resistance .

[0118] Specifically, please refer to Figure 3 Based on fractal theory, the pore boundary can be regarded as a parallel, curved, sinusoidal periodic contraction boundary along the coordinate axis X direction. Therefore, the boundary of the pore along the X direction is It can be characterized as:

[0119] (26)

[0120] In the formula, is the effective radius of the pore and is the effective diameter of the pore, and , , is the amplitude of the sinusoidally varying pore radius, is the wavelength.

[0121] Based on fractal theory, for a single diameter The actual length of the pore can be expressed as:

[0122] (27)

[0123] Then the volume of a single pore can be calculated for:

[0124] (28)

[0125] In fractal theory, for unsaturated porous media, it is usually assumed that there is an initial critical pore diameter before heating. , pores with a pore diameter larger than the critical pore diameter contain only gas phase, on the contrary, pores with a diameter smaller than the critical pore diameter contain only water phase. Based on the definition of saturation, saturation is the ratio of the volume of water phase to the total pore volume. The initial saturation of the porous medium before heating is for:

[0126] (29)

[0127] Combined with formula (9), the initial critical pore diameter before heating can be calculated as The expression is converted into the initial pore fractal dimension of the unsaturated porous medium before heating , the maximum initial pore diameter of a circular cross section equivalent to the cross section of a single irregular pore before heating , the initial saturation of the porous medium before heating , and the initial porosity of the porous medium before heating The formula is:

[0128] (30)

[0129] Similarly, according to the similar derivation commonly used in mathematics, the critical pore diameter after heating can be obtained according to formula (7): The expression is:

[0130] (31)

[0131] In the process of heat transfer, based on the thermoelectric analogy theory, heat flow can be analogized to electric current, and thermal resistance can be analogized to resistance. According to Fourier's law, the differential of the pore thermal resistance after heating can be obtained: , the expression is:

[0132] (32)

[0133] In the formula, is the cross-sectional area of ​​a single pore, is the pore thermal conductivity.

[0134] Based on formula (32) and combined with fractal theory, the pore length direction is integrated and mathematically transformed to obtain the diameter after heating: Thermal resistance of gas or water phase in the pores:

[0135] (33)

[0136] In the formula, and After heating, the diameter is Thermal resistance of the gas or water phase in the pores; and are the thermal conductivities of the gas phase and the water phase, respectively.

[0137] Combining fractal theory and based on the meaning of critical pore diameter, integrating formula (33) yields the following relationship:

[0138] (34)

[0139] In the formula, and are the total thermal resistances of the gas phase and water phase in the porous medium after heating, respectively.

[0140] Based on formula (34), the total thermal resistance of the gas phase and the water phase in the porous medium is derived as and The expression is:

[0141] (35)

[0142] Furthermore, according to the physical meaning of thermal resistance, the thermal resistance of the solid phase in the porous medium after heating is It can be expressed as:

[0143] (36)

[0144] In the formula, is the cross-sectional area of ​​the solid phase, is the cross-sectional area of ​​the porous medium characterization unit cell, and , is the solid phase thermal conductivity.

[0145] S7. Calculate the total effective thermal conductivity of unsaturated porous media according to the definition of thermal conductivity .

[0146] Specifically, according to the definition of thermal conductivity, the total effective thermal conductivity of the porous medium after heating is It can be expressed as:

[0147] (37)

[0148] In the formula, in the formula, is the total thermal resistance of each phase, and .

[0149] The above formula (37) is converted to:

[0150] (38)

[0151] Substituting formula (35) and formula (36) into formula (38), the total effective thermal conductivity of the porous medium after heating is obtained: The final expression of the total effective thermal conductivity is Taking into account temperature changes, rough pores, water saturation and tortuous pores, the total effective thermal conductivity is The final expression of is:

[0152] (39).

[0153] In this document, the directional words such as front, back, top, and bottom are defined by the positions of the components in the drawings and the positions of the components relative to each other, and are only for the sake of clarity and convenience in expressing the technical solution. It should be understood that the use of the directional words should not limit the scope of protection claimed in this application.

[0154] In the absence of conflict, the above embodiments and features in the embodiments may be combined with each other.

[0155] 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, improvements, etc. made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims

1. A method for calculating the effective thermal conductivity of unsaturated porous media considering thermal expansion, characterized in that: The following steps are involved: S1. Equivalent the cross section of the irregular pores in the unsaturated porous medium after the temperature rises to the cross section of a cylinder with the same area, and calculate the diameter of the equivalent pores , and according to the pore diameter Get the maximum pore diameter after heating and minimum pore diameter , Wherein, the pore diameter The calculation formula is: In the formula, is the thermal expansion coefficient of the solid particles, is the temperature change, is the initial pore diameter of the cylindrical cross section equivalent to the cross section of the irregular pore before heating; S2. According to fractal theory, calculate the fractal dimension of the pores in porous media after heating , the pore fractal dimension after heating The calculation formula is: In the formula, is the initial fractal dimension of the porous medium before heating, The maximum initial pore diameter of a circular cross section equivalent to the cross section of a single irregular pore before heating; S3. Calculate the porosity of porous media after heating based on fractal theory , the porosity of the porous medium after heating The calculation formula is: ； S4. According to fractal theory, calculate the fractal dimension of the tortuosity of the porous medium after heating and the characteristic length of the pores in the porous medium after heating ; Among them, the tortuosity fractal dimension of the porous medium after heating is The calculation formula is: The characteristic length of the pores of the porous medium after heating The calculation formula is: ； S5. According to the principle of pore roughness solution, the cone height of the rough unit in the pores of the porous medium is The base diameter of the cone is ,and The ratio of is constant and satisfies the fractal theory, and the average height of the rough units in the pores of the porous medium is calculated. , and according to the average height of the rough unit Calculate the relative roughness of porous media after heating ; The average height of the rough unit The calculation formula is: In the formula, is the ratio of the total base area of ​​the roughness unit to the pore surface area, is the maximum roughness unit base diameter, is the basis fractal dimension of the rough unit, is the ratio of the minimum to maximum base diameter of the roughness unit; The relative roughness of the porous medium pores after heating The calculation formula is: ； S6. Calculate the total thermal resistance of the gas phase in the porous medium after heating based on the thermoelectric analogy theory. , total thermal resistance of water phase , and calculate the total solid thermal resistance according to the definition of thermal resistance , the calculation formulas for the three are: In the formula, is the gas phase thermal conductivity, is the thermal conductivity of water phase, is the amplitude of the radius when the pore changes sinusoidally, is the critical pore diameter, is the cross-sectional area of ​​the porous medium characterization unit cell, and , is the solid phase thermal conductivity; S7. According to the definition of thermal conductivity, calculate the total effective thermal conductivity of the unsaturated porous medium after heating , and its calculation formula is: In the formula, is the total thermal resistance of each phase.

2. A method for calculating effective thermal conductivity of unsaturated porous media taking thermal expansion into consideration according to claim 1, characterized in that: In step S1, the maximum pore diameter of the pores after heating is and minimum pore diameter The expression is: In the formula, It is the minimum initial pore diameter of a circular cross section equivalent to the cross section of a single irregular pore before heating.

3. A method for calculating effective thermal conductivity of unsaturated porous media taking thermal expansion into consideration according to claim 2, characterized in that: Before step S2, the initial fractal dimension of the porous medium before heating is calculated according to fractal theory. , and its calculation formula is: In the formula, is the initial porosity of the porous medium before heating.

4. A method for calculating effective thermal conductivity of unsaturated porous media taking thermal expansion into consideration according to claim 1, characterized in that: In step S7, according to the thermoelectric analogy theory, the total thermal resistance of each phase The expression is: In the formula, is the total thermal resistance of the gas phase, is the total thermal resistance of water phase, is the solid phase thermal resistance.

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