A method for determining soil moisture content based on Maxwell-Wagner interfacial polarization theory

Abstract

The present invention discloses a method for determining soil moisture content based on Maxwell-Wagner interfacial polarization theory, which comprises the following steps: determining the volume fraction of each component material of natural soil; determining the relative dielectric constant of each component material of natural soil; determining the shape factor of each component material of natural soil; respectively determining the relationship between the apparent dielectric constant of soil and soil moisture content when the natural soil is all sand, silt or clay; determining the relative content of sand, silt and clay in natural soil; obtaining the relationship between the apparent dielectric constant of natural soil and water content, and obtaining the natural soil moisture content. The physical model involved in the present invention is close to the actual situation of soil, and the physical meaning of each parameter is clear and easy to obtain, which can provide theoretical support for the relevant research on the physical and mechanical properties of natural soil.

Description

Technical Field

[0001] The present invention relates to a soil moisture content determination method, in particular to a soil moisture content calculation method based on the Maxwell-Wagner interface polarization theory. Background Art

[0002] The moisture content in the soil has a significant impact on the smooth progress of agricultural production, engineering activities, etc. In agriculture, soil moisture content and salt ion concentration directly affect the water absorption of crop roots and the degree of surface salinization, determining the benefits of agricultural production. In engineering construction, soil moisture content affects the stability and service life of engineering facilities. In permafrost and seasonally frozen areas, which account for more than 70% of my country's land area, the interaction between moisture and heat in the soil causes frost heave and thaw settlement of the soil, leading to deformation and stability problems for engineering facilities above the soil, seriously restricting infrastructure construction and local economic development. Moisture is the material basis for water and salt migration in farmland and frost heave and thaw settlement of frozen soil. The moisture content directly affects the mechanical properties, moisture migration and deformation characteristics of the soil.

[0003] Accurately measuring soil moisture content is a prerequisite for determining the physical and mechanical properties of soil. Currently, the commonly used soil moisture content measurement methods can be roughly divided into five categories: the first category: destructive measurement methods, such as the drying method. This method is accurate for mineral soils, but it destroys the original structure of the soil and cannot measure the unfrozen water content in frozen soil; the second category: contact measurement methods, such as the neutron method. This method is easy to operate and can continuously monitor changes in soil moisture content, but the physical and chemical properties of the soil will affect the accuracy of the measurement results; the third category: non-contact measurement methods, such as infrared remote sensing The fourth category includes indirect measurement methods such as the tensiometer method and the gamma-ray method. These methods generally require expensive equipment, and some instruments pose health risks to test personnel during use. The fourth category includes indirect measurement methods using suction, humidity, or thermodynamic equilibrium, such as the tensiometer method and the adiabatic calorimetry method. These methods typically require instrument calibration and place high demands on the test environment and operation. The fifth category includes dielectric constant measurement methods, such as the TDR method. These methods invert soil moisture content based on the dielectric physical quantity measured by the measuring instrument and can continuously and non-destructively monitor changes in soil moisture content. The instruments are more affordable and portable. Therefore, the dielectric constant method for measuring soil moisture has been widely used in actual engineering activities.

[0004] After obtaining the apparent dielectric constant of the soil under test using a TDR or FDR instrument, the dielectric constant method simply establishes a relationship model between water content and the apparent dielectric constant to quickly and continuously determine the moisture content of the soil under test. Dielectric constant relationship models can be divided into three categories: The first category: empirical models. These methods use a large amount of experimental data to determine the relationship between water content and the apparent dielectric constant through fitting, such as the Topp model and the Alharthi model. The second category: semi-empirical and semi-physical models. Based on empirical models, these methods consider the physical and chemical properties of the soil to establish a relationship model between water content and the apparent dielectric constant. Examples include the Dobson model and the Malicki model. The third category: theoretical models. These methods consider the relative content and relative dielectric constant of each component in the soil, as well as the interfacial scattering of different particles in the electromagnetic field, to establish a relationship model between water content and the apparent dielectric constant. A large number of empirical and semi-empirical-semi-physical models proposed by scholars at home and abroad generally have specific scopes of application and often require calibration of the parameters in the models. Compared with empirical and semi-empirical-semi-physical models, the parameters in the physical models have clear physical meanings and the models are more applicable. Summary of the Invention

[0005] The purpose of the present invention is to provide a method for calculating the moisture content of soil based on the Maxwell-Wagner interfacial polarization theory, and at the same time provide theoretical formulas for calculating the moisture content of unfrozen soil and frozen soil. Compared with other measurement methods, the method of the present invention is simple to calculate, the physical model is close to the actual situation of the soil, and the physical meaning of each parameter is clear and easy to obtain.

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

[0007] A method for determining soil moisture content based on Maxwell-Wagner interfacial polarization theory includes the following steps:

[0008] S1. Determine the volume fraction of each component of natural soil;

[0009] The step S1 includes the following sub-steps:

[0010] S101. Consider the soil as a three-phase inhomogeneous medium consisting of air, water, and solid particles. When the soil temperature is less than or equal to the freezing temperature, that is, T≤T f When the temperature drops, ice crystals exist in the soil. As the temperature drops, a small amount of unfrozen water and ice crystals will coexist in the soil. According to the Maxwell-Wagner interfacial polarization theory, water or ice-water mixture is regarded as a continuous medium in the soil. In this case, the soil is considered to be composed of air, water or ice-water mixture and solid particles:

[0011] V0=V a +Vwi +V s (1)

[0013] In the formula, the total volume of the soil to be tested is V0, and the volume fraction of air in the soil is V a , the volume fraction of ice-water mixture in soil is V wi , the volume fraction of solid particles in the soil is V s , and it is believed that soil particles contain sand, silt and clay, among which:

[0014] V wi =V w +V i (2)

[0016] Where V w is the volume fraction of water in the soil, V i is the volume fraction of ice crystals in the soil. The ice-water mixture term is f When the soil contains only unfrozen water, V wi =V w T≤T f When the unfrozen water in the soil coexists with ice crystals, V wi =V w +V i . Introduce soil porosity n:

[0017]

[0018] Substituting (3) into (1) we get:

[0019] V s =V0(1-n) (4)

[0021] Considering the mass of ice crystals during the freezing process, it is the mass of water in the unfrozen state minus the mass of unfrozen water during the freezing process:

[0022] m i =ρ w V0(θ0-θ u (T))=ρ i V i (5)

[0024] Where m i is the mass of ice crystal, ρ w is the density of unfrozen water, θ0 is the initial volumetric moisture content of soil, θ u (T) is the unfrozen water content of the soil, expressed as volumetric water content, ρ i is the ice crystal density;

[0025] When T≤T f hour:

[0026] V w =V0θ u (T) (6)

[0028] When T>T f When there are no ice crystals in the soil:

[0029] V w =V0θ u (T)=V0θ0 (7)

[0031] Substituting equations (5) and (6) into equation (2), we obtain:

[0032]

[0033] Substituting equations (4) and (8) into equation (1), we obtain:

[0034]

[0035] S2. Determine the relative dielectric constant of each component of natural soil;

[0036] The step S2 includes the following sub-steps:

[0037] S201. Let the relative dielectric constant of air in soil be ε air The solid particles in the soil are divided into sand, silt and clay, and their relative dielectric constants are ε sand , ε silt and ε clay The relative dielectric constants of air and solid particles in the soil can be determined by referring to geological radar manuals or books;

[0038] S202. Let the relative dielectric constant of the ice-water mixture in the soil be ε wi (T), the relative dielectric constant of water in the soil is ε w (T), the relative dielectric constant of ice particles in the soil is ε ice The relative dielectric constant of water changes with temperature. According to the research of Cheng Xianjun, Zhuang Haijun, etc., ε is generally taken as w (T>T f )=81,ε w (T≤T f )=65;

[0039] Consider two cases: unfrozen soil and frozen soil:

[0040]

[0041] Where, T>T f When ε is the unfrozen soil wi (T) value. T≤T f When ε is the frozen soil wi (T) value. w (T),ε ice Consult a geological radar manual or book to be sure.

[0042] S3. Determine the shape factors of various components of natural soil;

[0043] The step S3 includes the following sub-steps:

[0044] S301. According to the Maxwell-Wagner interface polarization theory:

[0045]

[0046] Where K is the shape factor of substance i in the mixed system under the influence of the mixed system and the external electromagnetic field. a, b, and c are the three coordinate axes in the ellipsoidal coordinate system. ε i is the relative dielectric constant of substance i in the mixed system, ε a is the relative dielectric constant of the continuous medium in the mixed system. g j is the depolarization factor of the ellipsoid, and the Laplace equation of the potential is solved in the ellipsoidal surface coordinate system by the electrostatic method:

[0047]

[0048] And it is easy to prove:

[0049] g a +g b +g c =1 (15)

[0051] S302. Based on the Maxwell-Wagner interfacial polarization theory, the water in the soil and the ice-water mixture after the soil is frozen are considered as continuous media in the soil. According to the definition of the shape factor in this theory:

[0052]

[0053] Where K wi is the shape factor of water, or the shape factor of ice-water mixture during freezing. i It is the electric field excited by the substance i in the mixed system under the action of the external electromagnetic field. a It is the electric field excited by the continuous medium in the mixed system under the action of the external electromagnetic field;

[0054] S303. Consider the air in the soil as a spherical discontinuous medium, see Figure 2 Air model in soil. Assuming that air exists in the soil in the form of spherical bubbles, then:

[0055] a=b=c (17)

[0057] Substituting the limiting conditions of equation (17) into equations (12), (13), (14) and (15), we can obtain the depolarization factors of the air in the soil along the three axes in the ellipsoidal coordinate system:

[0058]

[0059] Substituting equation (18) into equation (11), we can obtain the shape factor of air in soil:

[0060]

[0061] Where K air is the shape factor of air in the soil-water system;

[0062] S304. Consider the clay particles in the soil as a disc-shaped discontinuous medium. Figure 2 Clay particle model. Assuming that clay particles exist in the soil in the shape of disks, then:

[0063]

[0064] Substituting the limiting conditions of formula (20) into formulas (12), (13), (14) and (15), we can obtain the depolarization factors of the three axes of the clay particles in the soil in the ellipsoidal coordinate system:

[0065] g a =1,g b =g c =0 (twenty one)

[0067] Substituting equation (21) into equation (11), we can obtain the shape factor of clay particles in the soil:

[0068]

[0069] Where K s，clay is the shape factor of clay particles in the soil-water system;

[0070] S305. Consider the silt particles in the soil as ellipsoidal discontinuous media, see Figure 2 Particle model. Assuming that the silt particles exist in the soil in the form of ellipsoids, taking a = b = lc, and fixing c = 0.075 mm as the upper limit of the particle size, then a, b∈(0.005 mm, 0.075 mm), we get

[0071] when When a=b=lc is substituted into formula (14), we get:

[0072]

[0073] make:

[0074] g c =1-2m (twenty four)

[0076] Substituting (24) into (15) we get:

[0077] g a =g b =m (25)

[0079] Substituting equations (24) and (25) into equation (11), we can obtain the shape factor of the silt particles in the soil:

[0080]

[0081] Where K s，silt is the shape factor of silt particles in the soil-water system, where:

[0082]

[0083] S306. Consider the sand particles in the soil as spherical discontinuous media. Figure 2 Sand particle model. Assuming that sand particles exist in the soil in a spherical shape, taking a=b=c, and substituting this constraint into equations (12), (13), (14), and (15), the depolarization factors of the three axes of the sand particles in the soil in the ellipsoidal coordinate system are obtained as follows: The g of sand a 、g b and g c Substituting into formula (11), we can obtain the shape factor of sand particles in the soil:

[0084]

[0085] Where K s，sand is the shape factor of sand particles in the soil-water system.

[0086] S4. Determine the relationship between the apparent dielectric constant and soil moisture content when the natural soil is composed entirely of sand, silt, or clay.

[0087] The step S4 includes the following sub-steps:

[0088] S401. The relationship between the apparent dielectric constant of a mixed system and the relative dielectric constants of its components is determined by the Maxwell-Wagner interfacial polarization theory:

[0089]

[0090] Where K * is the apparent dielectric constant of the mixed system, ε j 、V j and K j are the relative dielectric constant, volume fraction and shape factor of substance j in the mixed system, respectively.

[0091] S402. Consider the case where all the solid particles in the soil are clay particles, see Figure 3 Clay soil-water system model. In this case, the soil-water system is a non-uniform mixed system composed of air, ice-water mixture, and clay particles. Substituting the relative dielectric constants of the above components and equations (4), (8), (9), (10), (16), (19), and (22) into equation (29), we can obtain the relationship between the apparent dielectric constant and water content of the soil when all solid particles are clay particles:

[0092]

[0093] make:

[0094] ε air +2ε wi (T)=C (31)

[0096] Substituting (31) into (30) we get:

[0097]

[0098] Where K a，clay is the apparent dielectric constant of the soil-water system when all soil particles are clay;

[0099] S403. Consider the case where all the solid particles in the soil are powder particles, see Figure 3 Silt-water system model. Given equation (26), let:

[0100] F=(2-3m)ε silt +(1+3m)ε wi (T) (33)

[0102] D=3[mε silt +(1-m)ε wi (T)][2mε wi (T)+(1-2m)ε silt] (34)

[0104] Substituting equations (33) and (34) into equation (26), we obtain:

[0105]

[0106] At this time, the soil-water system is a non-uniform mixed system composed of air, ice-water mixture and silt particles. Substituting the relative dielectric constants of the above components and equations (4), (8), (9), (10), (16), (19), (31) and (35) into equation (29), we can obtain the relationship between the apparent dielectric constant and water content of the soil when all solid particles are silt particles:

[0107]

[0108] Where K a，silt is the apparent dielectric constant of the soil-water system when all soil particles are silt;

[0109] S404. Consider the case where all the solid particles in the soil are sand particles, see Figure 3 Sand-soil-water system model. In this case, the soil-water system is a non-uniform mixed system consisting of air, ice-water mixture, and sand particles. Substituting the relative dielectric constants of the above components and equations (4), (8), (9), (10), (16), (19), (28), and (31) into equation (29), we can obtain the relationship between the apparent dielectric constant and water content of the soil when all solid particles are sand particles:

[0110]

[0111] Where K a，sand is the apparent dielectric constant of the soil-water system when all soil particles are sand particles.

[0112] S5. Determine the relative amounts of sand, silt, and clay in natural soils;

[0113] The step S5 includes the following sub-steps:

[0114] S501. Collect soil samples for particle analysis to determine the volume ratio of sand, silt and clay in the soil, and record the relative content of sand, silt and clay as W. sand 、W silt and W clay ,have:

[0115]

[0116] Where V sand 、V silt and V clayare the volume proportions of sand, silt and clay in the soil samples, respectively.

[0117] S6. Determine the relative contents of sand, silt, and clay in the natural soil according to step S5. Substitute these values ​​into step S4 and multiply them by the corresponding relationships between the apparent dielectric constant and moisture content of the soil obtained when the soil is entirely sand, silt, and clay, respectively, to obtain the relationship between the apparent dielectric constant and moisture content of the natural soil. Measure the apparent dielectric constant of the natural soil under any operating condition using a TDR instrument and substitute this value into the theoretical relationship obtained in step S6 to obtain the moisture content of the natural soil under the corresponding operating condition.

[0118] The step S6 comprises:

[0119] S601. The relative content W of sand, silt and clay in the soil to be tested determined by step S5 sand 、W silt and W clay , multiplied by the relationships between the apparent dielectric constant and water content of the soil when all the solid particles obtained in step S4 are sand, silt and clay (32), (36) and (37), respectively, we get:

[0120] K a =W sand K a，sand +W silt K a，silt +W clay K a，clay (39)

[0122] Where K a is the apparent dielectric constant of natural soil containing sand, silt and clay;

[0123] S602. Using TDR to measure the apparent dielectric constant K of natural soil a , and substituting the measured value into formula (39) to obtain the moisture content in the natural soil under the corresponding working conditions.

[0124] Beneficial effects of the present invention:

[0125] Compared to other methods for determining soil moisture content, this method takes into account the shape of soil particles and the composition of each phase from a microscopic perspective. It explores the influence of more physical soil properties on soil moisture content, including porosity, particle size composition, and initial soil moisture content. Furthermore, it can measure the moisture content of both unfrozen and frozen soil, thus having a wide range of applications. The physical model closely resembles the actual soil conditions, and the physical meaning of each parameter is clear and easily accessible. BRIEF DESCRIPTION OF THE DRAWINGS

[0126] The specific embodiments of the present invention will be further described in detail below with reference to the accompanying drawings.

[0127] Figure 1 This is a schematic diagram of the physical model of the present invention;

[0128] Figure 2 Schematic diagram of the air and solid particle model in soil;

[0129] Figure 3 Schematic diagram of the soil-water system model with solid particles of clay, silt and sand respectively;

[0130] Figure 4 This is the curve of the apparent dielectric constant of unfrozen soil changing with water content;

[0131] Figure 5 This is a graph showing the prediction error of the theoretical formula of the present invention for the moisture content of unfrozen soil;

[0132] Figure 6 This is the curve of the apparent dielectric constant of frozen soil changing with water content;

[0133] Figure 7 This is a diagram showing the prediction error of the theoretical formula of the present invention for the moisture content of frozen soil.

[0134] Icons: 1-sand; 2-silt; 3-clay; 4-water or ice-water mixture; 5-air. DETAILED DESCRIPTION

[0135] In order to make the purpose, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention, but the scope of protection of the present invention is not limited to the following.

[0136] A method for determining soil moisture content based on Maxwell-Wagner interfacial polarization theory includes the following steps:

[0137] S1. Determine the volume fraction of each component of natural soil;

[0138] The step S1 includes the following sub-steps:

[0139] S101. Consider soil as a three-phase heterogeneous medium consisting of air, water, and solid particles, such as Figure 1 As shown, when the soil temperature is less than or equal to the freezing temperature, that is, T≤T f When the temperature drops, ice crystals are generated in the soil. As the temperature drops, a small amount of unfrozen water and ice crystals will coexist. According to the Maxwell-Wagner interfacial polarization theory, water or ice-water mixture is regarded as a continuous medium in the soil. In this case, the soil is considered to be composed of air, water or ice-water mixture and solid particles:

[0140] V0=Va +V wi +V s (1)

[0142] In the formula, the total volume of the soil to be tested is V0, and the volume fraction of air in the soil is V a , the volume fraction of ice-water mixture in soil is V wi , the volume fraction of solid particles in the soil is V s , and it is believed that soil particles contain sand, silt and clay, among which:

[0143] V wi =V w +V i (2)

[0145] Where V w is the volume fraction of water in the soil, V i is the volume fraction of ice crystals in the soil. The ice-water mixture term is f When the soil contains only unfrozen water, V wi =V w T≤T f When the unfrozen water in the soil coexists with ice crystals, V wi =V w +V i . Introduce soil porosity n:

[0146]

[0147] Substituting (3) into (1) we get:

[0148] V s =V0(1-n) (4)

[0150] Considering the mass of ice crystals during the freezing process, it is the mass of water in the unfrozen state minus the mass of unfrozen water during the freezing process:

[0151] m i =ρ w V0(θ0-θ u (T))=ρ i V i (5)

[0153] Where m i is the mass of ice crystal, ρ w is the density of unfrozen water, usually 1g / cm 3 , θ0 is the initial volumetric moisture content of the soil, θ u (T) is the unfrozen water content, expressed as volumetric water content, ρi is the density of ice crystals, usually 0.9 g / cm 3 ;

[0154] When T≤T f hour:

[0155] V w =V0θ u (T) (6)

[0157] When T>T f When there are no ice crystals in the soil:

[0158] V w =V0θ u (T)=V0θ0 (7)

[0160] Substituting equations (5) and (6) into equation (2), we obtain:

[0161] V wi =V0{θ u (T)+[1.11(θ0-θ u (T))]} (8)

[0163] Substituting equations (4) and (8) into equation (1), we obtain:

[0164] V a =V0{n-[θ u (T)+[1.11(θ0-θ u (T))]]} (9)

[0166] S102. The V0 term will be eliminated in the subsequent derivation process and will not affect the theoretical formula. Let V0 = 1, and we have:

[0167] Volume fraction of air in soil:

[0168] V a =n-{θ u (T)+[1.11(θ0-θ u (T))]} (10)

[0170] Volume fraction of ice-water mixture in soil:

[0171] V wi =θ u (T)+[1.11(θ0-θ u (T))] (11)

[0173] Volume fraction of solid particles in soil:

[0174] V s =(1-n) (12)

[0176] S2. Determine the relative dielectric constant of each component of natural soil;

[0177] The step S2 includes the following sub-steps:

[0178] S201. Let the relative dielectric constant of air in soil be ε air Soil particles are divided into sand, silt and clay, and their relative dielectric constants are ε sand , ε silt and ε clay The relative dielectric constants of air and solid particles in the soil can be determined by referring to geological radar manuals or books;

[0179] Since the relative dielectric constant of air and soil particles changes very little with temperature, ε is usually taken air =1,ε clay =4,ε silt =5,ε sand =6;

[0180] S202. Let the relative dielectric constant of the ice-water mixture in the soil be ε wi (T), the relative dielectric constant of water in the soil is ε w (T), the relative dielectric constant of ice particles in the soil is ε ice The relative dielectric constant of water changes with temperature. According to the research of Cheng Xianjun, Zhuang Haijun, etc., ε is generally taken as w (T>T f )=81,ε w (T≤T f )=65;

[0181] Consider two cases: unfrozen soil and frozen soil:

[0182]

[0183] Where, T>T f When ε is the unfrozen soil wi (T) value. T≤T f When ε is the frozen soil wi (T) value. w (T),ε ice Refer to the geological radar manual or book to determine. At room temperature, ε is usually taken w =81. When the temperature is below 0℃, ε is usually used. ice =3.

[0184] S3. Determine the shape factors of various components of natural soil;

[0185] The step S3 includes the following sub-steps:

[0186] S301. According to the Maxwell-Wagner interface polarization theory:

[0187]

[0188] Where K is the shape factor of substance i in the mixed system under the influence of the mixed system and the external electromagnetic field. a, b, and c are the three coordinate axes in the ellipsoidal coordinate system. ε i is the relative dielectric constant of substance i in the mixed system, ε a is the relative dielectric constant of the continuous medium in the mixed system. j is the depolarization factor of the ellipsoid, and the Laplace equation of the potential is solved in the ellipsoidal surface coordinate system by the electrostatic method:

[0189]

[0190] And it is easy to prove:

[0191] g a +g b +g c =1 (18)

[0193] S302. Based on the Maxwell-Wagner interfacial polarization theory, the water in the soil and the ice-water mixture after the soil is frozen are considered as continuous media in the soil. According to the definition of the shape factor in this theory:

[0194]

[0195] Where K wi is the shape factor of water, or the shape factor of ice-water mixture during freezing. i It is the electric field excited by the substance i in the mixed system under the action of the external electromagnetic field. a It is the electric field excited by the continuous medium in the mixed system under the action of the external electromagnetic field;

[0196] S303. Consider the air in the soil as a spherical discontinuous medium, see Figure 2 Air model in soil. Assuming that air exists in the soil in the form of spherical bubbles, then:

[0197] a=b=c (20)

[0199] Substituting the limiting conditions of equation (20) into equations (15), (16), (17), and (18), we can obtain the depolarization factors of the air in the soil along the three axes in the ellipsoidal coordinate system:

[0200]

[0201] Substituting equation (21) into equation (14), we can obtain the shape factor of air in soil:

[0202]

[0203] Where K air is the shape factor of air in the soil-water system;

[0204] S304. Consider the clay particles in the soil as a disc-shaped discontinuous medium. Figure 2 Clay particle model. Assuming that clay particles exist in the soil in the shape of disks, then:

[0205]

[0206] Substituting the limiting conditions of formula (23) into formulas (15), (16), (17) and (18), we can obtain the depolarization factors of the three axes of the clay particles in the soil in the ellipsoidal coordinate system:

[0207] g a =1,g b =g c =0 (twenty four)

[0209] Substituting equation (24) into equation (14), we can obtain the shape factor of clay particles in the soil:

[0210]

[0211] Where K s,clay is the shape factor of clay particles in the soil-water system;

[0212] S305. Consider the silt particles in the soil as ellipsoidal discontinuous media, see Figure 2 Particle model. Assuming that the silt particles exist in the soil in the form of ellipsoids, taking a=b=lc, and fixing c=0.075mm as the upper limit of the particle size, then a,b∈(0.005mm,0.075mm), we get l represents the flattening of the ellipsoid. When l approaches 1, the ellipsoid becomes closer and closer to a sphere. As l = 0.5, the ellipsoid becomes more and more like a long rod.

[0213] When l = 0.5, substituting the constraint of a = b = 0.5c into equation (17) yields:

[0214]

[0215] make:

[0216]

[0217] Substituting (27) into (18) we have:

[0218]

[0219] Substituting equations (27) and (28) into equation (14), we can obtain the shape factor of the silt particles in the soil:

[0220]

[0221] Where K s,silt is the shape factor of silt particles in the soil-water system, where:

[0222]

[0223] S306. Consider the sand particles in the soil as spherical discontinuous media. Figure 2 Sand particle model. Assuming that sand particles exist in the soil in a spherical shape, taking a=b=c, and substituting this constraint into equations (15), (16), (17), and (18), the depolarization factors of the three axes of the sand particles in the soil in the ellipsoidal coordinate system are obtained as follows: The g of sand a 、g b and g c Substituting into formula (14), we can obtain the shape factor of sand particles in the soil:

[0224]

[0225] Where K s，sand is the shape factor of sand particles in the soil-water system.

[0226] S4. Determine the relationship between the apparent dielectric constant and soil moisture content when the natural soil is composed entirely of sand, silt, or clay.

[0227] The step S4 includes the following sub-steps:

[0228] S401. The relationship between the apparent dielectric constant of a mixed system and the relative dielectric constants of its components is determined by the Maxwell-Wagner interfacial polarization theory:

[0229]

[0230] Where K *is the apparent dielectric constant of the mixed system, ε j 、V j and K j are the relative dielectric constant, volume fraction and shape factor of substance j in the mixed system, respectively.

[0231] S402. Consider the case where all the solid particles in the soil are clay particles, see Figure 3 Clay soil-water system model. In this case, the soil-water system is a non-uniform mixed system composed of air, ice-water mixture, and clay particles. Substituting the relative dielectric constants of the above components and equations (10), (11), (12), (13), (19), (22), and (25) into equation (32), we can obtain the relationship between the apparent dielectric constant and water content of the soil when all solid particles are clay particles:

[0232]

[0233] make:

[0234] 1+2ε wi (T))=C (34)

[0236] Substituting (34) into (33) we get:

[0237]

[0238] Where K a，clay is the apparent dielectric constant of the soil-water system when all soil particles are clay;

[0239] S403. Consider the case where all the solid particles in the soil are powder particles, see Figure 3 Silt-water system model. Given equation (29), let:

[0240] F=18(3ε wi (T)+5) (36)

[0242] D=(5ε wi (T)+5)(7ε wi (T)+25) (37)

[0244] Substituting equations (36) and (37) into equation (29), we obtain:

[0245]

[0246] At this time, the soil-water system is a non-uniform mixed system composed of air, ice-water mixture and silt particles. Substituting the relative dielectric constants of the above components and equations (10), (11), (12), (13), (19), (22), (34) and (38) into equation (32), we can obtain the relationship between the apparent dielectric constant and water content of the soil when all solid particles are silt particles:

[0247]

[0248] Where K a，suit is the apparent dielectric constant of the soil-water system when all soil particles are silt;

[0249] S404. Consider the case where all the solid particles in the soil are sand particles, see Figure 3 Sand-soil-water system model. In this case, the soil-water system is a non-uniform mixed system consisting of air, ice-water mixture, and sand particles. Substituting the relative dielectric constants of the above components and equations (10), (11), (12), (13), (19), (22), (31), and (34) into equation (32), we can obtain the relationship between the apparent dielectric constant and water content of the soil when all the solid particles are sand particles:

[0250]

[0251] Where K a，sand is the apparent dielectric constant of the soil-water system when all soil particles are sand particles.

[0252] S5. Determine the relative amounts of sand, silt, and clay in natural soils;

[0253] The step S5 includes the following sub-steps:

[0254] S501. Collect soil samples for particle analysis to determine the volume ratio of sand, silt and clay in the soil, and record the relative content of sand, silt and clay as W. sand 、W silt and W clay ,have:

[0255]

[0256] Where V sand 、V silt and V clay are the volume proportions of sand, silt and clay in the soil samples, respectively.

[0257] S6. Determine the relative contents of sand, silt, and clay in the natural soil according to step S5. Substitute these values ​​into step S4 and multiply them by the corresponding relationships between the apparent dielectric constant and moisture content of the soil obtained when the soil is entirely sand, silt, and clay, respectively, to obtain the relationship between the apparent dielectric constant and moisture content of the natural soil. Measure the apparent dielectric constant of the natural soil under any operating condition using a TDR instrument and substitute this value into the theoretical relationship obtained in step S6 to obtain the moisture content of the natural soil under the corresponding operating condition.

[0258] The step S6 comprises:

[0259] S601. The relative content W of sand, silt and clay in the soil to be tested determined by step S5 sand 、W silt and W clay , multiplied by the relationships between the apparent dielectric constant and water content of the soil when all the solid particles obtained in step S4 are sand, silt and clay (35), (39) and (40), respectively, we get:

[0260] K a =W sand K a，sand +W silt K a，silt +W clay K a，clay (42)

[0262] Where K a is the apparent dielectric constant of natural soil containing sand, silt and clay;

[0263] S602. Using TDR to measure the apparent dielectric constant K of natural soil a , and put the measured value into formula (42) to obtain the moisture content in the natural soil under the corresponding working conditions.

[0264] The foregoing description is merely a preferred embodiment of the present invention. It should be understood that the present invention is not limited to the form disclosed herein and should not be construed as excluding other embodiments. Rather, the present invention may be used in various other combinations, modifications, and environments, and may be modified within the scope of the concepts described herein by utilizing the above teachings or techniques or knowledge in the relevant field. Modifications and variations made by those skilled in the art without departing from the spirit and scope of the present invention are intended to be within the scope of the appended claims. Specific embodiments

[0266] Unfrozen soil model verification:

[0267] This paper examines the theoretical formula using two commonly used empirical soil dielectric models: the Topp formula and the Alharthi formula. The theoretical formula is evaluated using measured data from unfrozen Lanzhou loess soil under different working conditions to determine its moisture content.

[0268] Topp's empirical formula is:

[0269] K a =3.03+9.3θ+146θ 2 -76.7θ 3 (43)

[0271] Alharthi's empirical formula is:

[0272] K a =2.54+24.902θ+61.035θ 2 (44)

[0274] Where K a is the apparent dielectric constant of the soil to be tested, and θ is the volumetric water content.

[0275] The test soil is a disturbed sample. The relative contents of clay, silt and sand in the soil are determined by the particle distribution curve, see Table 1. The water content test of unfrozen soil is carried out at room temperature. w =81, the soil dry density is controlled to be ρ d =1.35g / cm 3 The porosity of the soil is n=49.81%. Five soil samples with different moisture contents were prepared and the test data were recorded as shown in Table 2. The volumetric moisture content is obtained by multiplying the mass moisture content measured by the drying method by the dry density of the soil. The apparent dielectric constant of the soil is measured by the Hydra-Probe probe.

[0276] Table 1 Relative contents of clay, silt and sand in the test soil

[0277]

[0278] Table 2 Unfrozen soil test data

[0279]

[0280] Depend on Figure 4It can be seen that when the soil moisture content is zero, the apparent dielectric constant values ​​corresponding to the Topp formula, Alharthi formula and the theoretical formula of this paper are not 0, and are all around 4, indicating that the apparent dielectric constant of the soil at this time is determined by the relative dielectric constant of the soil particles. As the soil moisture content increases, the apparent dielectric constant also increases. The changing trends of the theoretical formula of the present invention are completely consistent with those of the Topp formula and the Alharthi formula curves. However, when the apparent dielectric constant of the Alharthi formula is close to 80, the corresponding soil moisture content is only about 90%, which underestimates the soil moisture content. The moisture content corresponding to the theoretical formula of the present invention and the Topp formula is close to 100%, which is more in line with the actual situation. The theoretical formula of the present invention is in good agreement with the measured data of Lanzhou loess, with an average error of less than 3%. Therefore, it is feasible to apply the theoretical formula derived by the present invention to the calculation of the moisture content of unfrozen soil.

[0281] Depend on Figure 5 It can be seen that when the soil moisture content is between 0% and 15%, the soil moisture content predicted by the theoretical formula is very close to the actual soil moisture content, with an error of about 1%. When the moisture content is between 15% and 30%, the predicted value of the theoretical formula is also in good agreement with the measured data, with an error of about 4%. When the soil moisture content is greater than 30%, the predicted value of the theoretical formula is in good agreement with the measured data, with an error of about 1%. The prediction results of the theoretical formula are generally good. Through statistical analysis of the predicted results of the theoretical formula and the actual measured values, the following determination coefficient R is obtained: 2 =0.985, and Nash efficiency coefficient (NSE) =0.939. These two indicators indicate that the model predictions match the measured values ​​well and with high accuracy. The root mean square error (RMSE) =0.028 indicates that the theoretical formula predictions differ from the measured data by an average of 2.8%, demonstrating that the model predictions are reliable and have a low error. Therefore, the theoretical formula of the present invention provides reliable predictions and can be applied to the calculation of the moisture content of unfrozen soil.

[0282] Frozen soil model verification:

[0283] The present invention uses the widely recognized Topp formula for frozen soil to test the theoretical formula of this invention. The unfrozen water content of the frozen soil was measured by taking disturbed samples of Lanzhou loess to evaluate the theoretical formula of this invention.

[0284] The empirical formula for Topp in frozen soil is:

[0285]

[0286] Where θ u is the unfrozen water content, given as volumetric moisture content.

[0287] The test soil is a disturbed sample, and the particle composition of the soil is shown in Table 1. For frozen soil, take εw (T≤T f )=65,ε ice =3, the soil dry density is controlled to be ρ d =1.35g / cm 3 The initial moisture content θ0 = 30%, and the porosity n = 49.81%. The soil column was placed in a Meiling freezer and frozen at -21°C for 72 hours. After freezing, the column was removed and thawed at room temperature (25°C) for 8 hours. This test involved one freeze-thaw cycle. The test data are shown in Table 3. The unfrozen water content was obtained using a calibrated Hydra-Probe and is expressed as volumetric water content. The apparent dielectric constant of the soil was also measured using the Hydra-Probe.

[0288] Table 3 Frozen soil test data

[0289]

[0290]

[0291] Depend on Figure 6 It can be seen that the dielectric theoretical model derived by the present invention and the Topp model in frozen soil have the same curve trend. When the moisture content of the soil is low, the apparent dielectric constant of the soil is mainly determined by the ice and soil particles in the soil, which is about 4 to 5. As the temperature rises, the ice melts and turns into water, the unfrozen water content in the soil increases, and the apparent dielectric constant of the soil also increases. It can be seen from the measured data points that under freezing conditions, the accuracy of the prediction results of this model is higher than that of the Topp model in frozen soil. Therefore, this model can be applied to the calculation of the moisture content of frozen soil. Figure 7 It can be seen that the unfrozen water content predicted by the theoretical formula in this paper is close to the measured value, and most of the measured points are within and around the ±10% error line. The model established by this invention has a good overall prediction result. Through statistical analysis of the theoretical formula prediction results and the actual measured values, the determination coefficient R 2 =0.984, and Nash efficiency coefficient (NSE) =0.955. These two indicators indicate that the model predictions are close to the measured values, and the two data sets have a strong correlation. The root mean square error (RMSE) =0.0207 indicates that the theoretical formula predictions differ from the measured data by an average of 2.07%. Therefore, the theoretical formula predictions have a small error and are reliable. This shows that the proposed model's predictions of the unfrozen water content in frozen soil are reliable.

[0292] In summary, the theoretical formula for soil moisture content obtained in the present invention can reliably predict the soil moisture content of natural soils with different particle size compositions. It is feasible to apply the soil moisture content calculation method of the present invention to obtain actual soil moisture content.

Claims

1. A method for determining soil moisture content based on Maxwell-Wagner interfacial polarization theory, characterized in that: For the following steps: S1. Determine the volume fraction of each component of natural soil; It includes: S101. Consider the soil as a three-phase inhomogeneous medium consisting of air, water, and solid particles. When the soil temperature is less than or equal to the freezing temperature, that is, T≤T f When the temperature drops, ice crystals exist in the soil. As the temperature drops, a small amount of unfrozen water and ice crystals will coexist in the soil. According to the Maxwell-Wagner interfacial polarization theory, water or ice-water mixture is regarded as a continuous medium in the soil. At this time, the soil is considered to be composed of air, water or ice-water mixture and solid particles: V0=V a +V wi +V s (1) In the formula, the total volume of the soil to be tested is V0, and the volume fraction of air in the soil is V a , the volume fraction of ice-water mixture in soil is V wi , the volume fraction of solid particles in the soil is V s , and it is believed that soil particles contain sand, silt and clay, among which: V wi =V w +V i (2) Where V w is the volume fraction of water in the soil, V i is the volume fraction of ice crystals in the soil; the ice-water mixture term is when T>T f When the soil contains only unfrozen water, V wi =V w ; T≤T f When the unfrozen water in the soil coexists with ice crystals, V wi =V w +V i , introducing the soil porosity n: Substituting (3) into (1) we get: V s =V0(1-n) (4) Considering the mass of ice crystals during the freezing process, it is the mass of water in the unfrozen state minus the mass of unfrozen water during the freezing process: m i =ρ w V0(θ0-θ u (T))=ρ i V i (5) Where m i is the mass of ice crystal, ρ w is the density of unfrozen water, θ0 is the initial volumetric moisture content of soil, θ u (T) is the unfrozen water content of the soil, expressed as volumetric water content, ρ i is the ice crystal density; When T≤T f hour: V w =V0θ u (T) (6) When T>T f When there are no ice crystals in the soil: V w =V0θ u (T)=V0θ0 (7) Substituting equations (5) and (6) into equation (2), we obtain: Substituting equations (4) and (8) into equation (1), we obtain: S2. Determine the relative dielectric constant of each component of the natural soil; this includes: S201. Let the relative dielectric constant of air in soil be ε air , the solid particles in the soil are divided into sand, silt and clay, and their relative dielectric constants are ε sand , ε silt and ε clay , the relative dielectric constants of air and solid particles in the soil can be determined by referring to geological radar manuals or books; S202. Let the relative dielectric constant of the ice-water mixture in the soil be ε wi (T), the relative dielectric constant of water in the soil is ε w (T), the relative dielectric constant of ice particles in the soil is ε ice , the relative dielectric constant of water changes with temperature, take ε w (T>T f )=81,ε w (T≤T f )=65; Consider two cases: unfrozen soil and frozen soil: Where, T>T f When ε is the unfrozen soil wi (T) value; T≤T f When ε is the frozen soil wi (T) value; S3. Determine the shape factor of each component of the natural soil; Step S3 comprises: S301. According to the Maxwell-Wagner interface polarization theory: Where K is the shape factor of substance i in the mixed system under the action of the mixed system and the external electromagnetic field, a, b, and c are the three coordinate axes in the ellipsoidal coordinate system, and ε i is the relative dielectric constant of substance i in the mixed system, ε a is the relative dielectric constant of the continuous medium in the mixed system, g j is the depolarization factor of the ellipsoid, and the Laplace equation of the potential is solved in the ellipsoidal surface coordinate system by the electrostatic method: Attainment: g a +g b +g c =1 (15) S302. Based on the Maxwell-Wagner interfacial polarization theory, the water in the soil and the ice-water mixture after the soil is frozen are considered as continuous media in the soil. According to the definition of the shape factor in this theory: Where K wi is the shape factor of water, or the shape factor of ice-water mixture during freezing, E i It is the electric field excited by the substance i in the mixed system under the action of the external electromagnetic field, E a It is the electric field excited by the continuous medium in the mixed system under the action of the external electromagnetic field; S303. Consider the air in the soil as a spherical discontinuous medium. Assuming that the air exists in the soil in the form of spherical bubbles, then: a=b=c (17) Substituting the limiting conditions of equation (17) into equations (12), (13), (14) and (15), we can obtain the depolarization factors of the air in the soil along the three axes in the ellipsoidal coordinate system: Substituting equation (18) into equation (11), we can obtain the shape factor of air in soil: Where K air is the shape factor of air in the soil-water system; S304. Consider the clay particles in the soil as a disc-shaped discontinuous medium. Assuming that the clay particles exist in the soil in a disc-shaped manner, then: Substituting the limiting conditions of formula (20) into formulas (12), (13), (14) and (15), we can obtain the depolarization factors of the three axes of the clay particles in the soil in the ellipsoidal coordinate system: g a =1,g b =g c =0 (21) Substituting equation (21) into equation (11), we can obtain the shape factor of clay particles in the soil: Where K s，clay is the shape factor of clay particles in the soil-water system; S305. Consider the silt particles in the soil as an ellipsoidal discontinuous medium. Assuming that the silt particles exist in the soil in an ellipsoidal shape, take a = b = lc, and fix c = 0.075 mm as the upper bound of the silt particle size. Then a, b ∈ (0.005 mm, 0.075 mm), we get when When a=b=lc is substituted into formula (14), we get: make: g c =1-2m (24) Substituting (24) into (15) we get: g a =g b =m (25) Substituting equations (24) and (25) into equation (11), we can obtain the shape factor of the silt particles in the soil: Where K s，silt is the shape factor of silt particles in the soil-water system, where: S306. Consider the sand particles in the soil as spherical discontinuous media. Assume that the sand particles exist in the soil in a spherical shape, take a = b = c, and substitute this constraint into equations (12), (13), (14), and (15). The depolarization factors of the three axes of the sand particles in the soil in the ellipsoidal coordinate system are obtained as follows: The g of sand a 、g b and g c Substituting into formula (11), we can obtain the shape factor of sand particles in the soil: Where K s，sand is the shape factor of sand particles in the soil-water system; S4. Determine the relationship between the apparent dielectric constant and soil moisture content when the natural soil is composed entirely of sand, silt, or clay. S5. Determine the relative amounts of sand, silt, and clay in natural soils; S6. Determine the relative contents of sand, silt, and clay in the natural soil according to step S5, substitute these values ​​into step S4, and multiply them by the corresponding relationships between the apparent dielectric constant and moisture content of the soil obtained when the soil is entirely sand, silt, and clay, respectively, to obtain the relationship between the apparent dielectric constant and moisture content of the natural soil; measure the apparent dielectric constant of the natural soil under any working condition using time domain reflectometry, and substitute this value into the theoretical relationship obtained in step S6 to obtain the moisture content of the natural soil under the corresponding working condition; Step S4 is specifically as follows: S401. The relationship between the apparent dielectric constant of a mixed system and the relative dielectric constants of its components is determined by the Maxwell-Wagner interfacial polarization theory: Where K * is the apparent dielectric constant of the mixed system, ε j 、V j and K j are the relative dielectric constant, volume fraction and shape factor of substance j in the mixed system respectively; S402. Consider the case where all the solid particles in the soil are clay particles. In this case, the soil-water system is a non-uniform mixed system consisting of air, ice-water mixture, and clay particles. Substituting the relative dielectric constants of the above components and equations (4), (8), (9), (10), (16), (19), and (22) into equation (29), we can obtain the relationship between the apparent dielectric constant and water content of the soil when all the solid particles are clay particles: make: e air +2e wi (T)=C (31) Substituting (31) into (30) we get: Where K a，clay is the apparent dielectric constant of the soil-water system when all soil particles are clay; S403. Consider the case where all solid particles in the soil are powder particles. Given equation (26), let: F=(2-3m)ε silt +(1+3m)ε wi (T) (33) D=3[mε silt +(1-m)e wi (T)][2mε wi (T)+(1-2m)e silt ] (34) Substituting equations (33) and (34) into equation (26), we obtain: At this time, the soil-water system is a non-uniform mixed system composed of air, ice-water mixture and silt particles. Substituting the relative dielectric constants of the above components and equations (4), (8), (9), (10), (16), (19), (31) and (35) into equation (29), we can obtain the relationship between the apparent dielectric constant and water content of the soil when all solid particles are silt particles: Where K a，silt is the apparent dielectric constant of the soil-water system when all soil particles are silt; S404. Consider the case where all the solid particles in the soil are sand particles. In this case, the soil-water system is a non-uniform mixed system consisting of air, ice-water mixture, and sand particles. Substituting the relative dielectric constants of the above components and equations (4), (8), (9), (10), (16), (19), (28), and (31) into equation (29), we can obtain the relationship between the apparent dielectric constant and water content of the soil when all the solid particles are sand particles: Where K a，sand is the apparent dielectric constant of the soil-water system when all soil particles are sand particles; The step S6 comprises: S601. The relative content W of sand, silt and clay in the soil to be tested determined by step S5 sand 、W silt and W clay , multiplied by the relationships between the apparent dielectric constant and water content of the soil when all the solid particles obtained in step S4 are sand, silt and clay (32), (36) and (37), respectively, to obtain: K a =W sand K a,sand +W silt K a,stilt +W clay K a,clay (39) Where K a is the apparent dielectric constant of natural soil containing sand, silt and clay; S602. Using TDR to measure the apparent dielectric constant K of natural soil a , and substituting the measured value into formula (39) to obtain the moisture content in the natural soil under the corresponding working conditions.

2. The method for determining soil moisture content based on the Maxwell-Wagner interfacial polarization theory according to claim 1, characterized in that: The step S5 comprises: S501. Collect soil samples for particle analysis to determine the volume ratio of sand, silt and clay in the soil, and record the relative content of sand, silt and clay as W. sand 、W silt and W clay ,have: Where V sand 、V silt and V clay are the volume proportions of sand, silt and clay in the soil samples, respectively.

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