Steady-state boundary convection-dispersion model parameter acquisition method based on variable separation method

Abstract

The invention discloses a steady-state boundary convection-dispersion model parameter acquisition method based on a variable separation method, and the method comprises the steps: constructing a convection-dispersion mathematical model under a one-dimensional finite-scale steady-state boundary condition, and carrying out the dimensionless processing of an original mathematical model through a dimensionless factor; secondly, analyzing and solving the new mathematical model after nondimensionalization by adopting a separation variable method, systematically deriving an analytical solution expression of a one-dimensional steady-state boundary convection-dispersion model in combination with a groundwater dynamic theory and a solute transport mechanism, and constructing a standard curve capable of reflecting different permeability parameters and solute transport related parameters; an indoor one-dimensional convection-dispersion test is carried out, water level and solute concentration measured data at different observation points are obtained, and permeability parameters and solute transport related parameters of a test medium are efficiently obtained through wiring. According to the method, the accuracy and adaptability of the convection-dispersion model in the aspect of parameter identification under the limited scale condition are enhanced; and operation is simple, parameter identification quantity is large, and precision is high.

Description

Technical Field

[0001] The present invention relates to a method for determining aquifer permeability parameters and solute transport related parameters, and in particular to a method for obtaining parameters of a steady-state boundary convection-diffusion model based on a separation of variables method. Background Art

[0002] Permeability parameters and solute transport-related parameters are key parameters in hydrogeological and environmental studies. To accurately obtain these parameters, researchers established a one-dimensional convection-dispersion model and combined it with indoor experiments for data fitting and parameter inversion. In traditional modeling, infinite or semi-infinite domain assumptions are often used to simplify boundary conditions, setting the inlet as a constant source term and assuming infinite concentration dilution or undisturbed diffusion at the outlet. However, in actual indoor finite-scale experiments, the above assumptions are often difficult to meet, resulting in the actual physical behavior not being accurately reflected by traditional models. In addition, the convection-dispersion behavior under finite closed boundary conditions has obvious spatiotemporal coupling characteristics. It is necessary to introduce a solution method with clear physical meaning and good mathematical adaptability to accurately simulate the concentration evolution law during the experiment and further use it for the inversion calculation of permeability coefficient and diffusion parameters.

[0003] This application introduces the separation of variables method into the modeling of convection-diffusion problems in a specific way, establishes a solution method for the convection-diffusion model suitable for finite closed boundary conditions, constructs an accurate analytical solution through the separation of variables method, and based on this, realizes the efficient and accurate acquisition of aquifer permeability parameters and solute transport parameters to meet the needs of actual indoor experiments and engineering applications. Summary of the Invention

[0004] Purpose of the invention: The present invention aims to solve the modeling and parameter identification problems of solute migration processes in limited spaces, and proposes a convection-diffusion model and parameter acquisition method suitable for finite-scale test systems. This method constructs an analytical solution framework based on the separation of variables method. By introducing reasonable steady-state boundary conditions and variable separation solution strategies, it realizes the calculation of core hydrogeological parameters such as hydrodynamic diffusion coefficient, permeability coefficient, and particle flow rate. It breaks through the traditional model's dependence on the semi-infinite domain assumption and solves the problem of difficulty in accurately calculating concentration evolution under finite boundary conditions. This method has the advantages of clear mathematical structure, strong boundary adaptability, and high computational stability. It can be widely used in scenarios such as groundwater pollution simulation, aquifer characteristic assessment, and hydrogeological experiment design.

[0005] In order to solve the above technical problems, the technical solutions adopted by the present invention are as follows:

[0006] A method for obtaining parameters of a steady-state boundary convection-dispersion model based on a separation of variables method comprises the following steps:

[0007] 1) Construct a one-dimensional convection-dispersion mathematical model under finite-scale steady-state boundary conditions, and non-dimensionalize the original mathematical model through dimensionless factors to obtain a new mathematical model after non-dimensionalization;

[0008] 2) The dimensionless mathematical model was analytically solved using the separation of variables method. Combining groundwater dynamics theory and solute transport mechanisms, an analytical expression for the one-dimensional steady-state boundary convection-diffusion model was systematically derived. From this, a standard curve reflecting the correlation between different permeability parameters and solute transport parameters was constructed.

[0009] 3) Conduct indoor one-dimensional convection-diffusion experiments to obtain measured data of water level and solute concentration at different observation points, and use wiring to efficiently obtain the permeability parameters and solute transport-related parameters of the test medium.

[0010] This application can effectively realize the coupled identification of aquifer permeability parameters and solute transport parameters, and improve the stability and calculation accuracy of parameter fitting.

[0011] In the above step 1), a one-dimensional convection-dispersion mathematical model under finite-scale steady-state boundary conditions is constructed, and the original mathematical model is dimensionlessly processed by a dimensionless factor:

[0012] The one-dimensional convection-dispersion model with finite boundaries in the experiment can be expressed as the following well-defined problem:

[0013]

[0014] Choose an appropriate dimensionless factor:

[0015]

[0016] Substituting the dimensionless factor into equation (1), we can obtain the new solution to the problem after dimensionless transformation:

[0017]

[0018] Where: C is the concentration of solute at any point in the test section, ML -3 ;D L is the longitudinal hydrodynamic dispersion coefficient, L 2 T -1 ; u is the average particle seepage velocity, LT -1 ; D0 is the molecular diffusion coefficient, L 2 T -1 ; α L is the longitudinal dispersivity, L; L is the length of the model test section, L; z is the spatial position coordinate, L; C(z,0)=C0 is the initial concentration, ML -3 ; C w0 is the upper boundary concentration increment, ML -3 ;zD is the dimensionless distance; C D is the dimensionless concentration; t D is the dimensionless time; Pe is the Peclet number; t is the measured time, T.

[0019] The units used in this application are expressed in international dimensions.

[0020] Furthermore, in step 2), the new dimensionless mathematical model is analytically solved using the separation of variables method:

[0021] Due to the inhomogeneous boundary conditions, the solution is decomposed into steady-state and transient parts:

[0022] C D (z D ,t D )=C D,s (z D )+C D,t (z D ,t D ) (4)

[0023] The steady-state part satisfies the following steady-state equation:

[0024]

[0025] Solving the ordinary differential equation (5) yields the steady-state solution:

[0026] C D,s (z D )=1 (6)

[0027] Substituting equation (4) and steady-state solution equation (6) into equation (3) yields the equation satisfied by the transient solution:

[0028]

[0029] There is a first-order derivative term in the equation. To facilitate the solution, transform equation (7) and let:

[0030]

[0031] Substituting equation (8) into equation (7) yields a new solution to the steady-state problem:

[0032]

[0033] Separate the variables and solve:

[0034] W(z D ,t D )=Z(z D )T(t D ) (10)

[0035] Substitute equation (10) into the control equation in equation (9),

[0036]

[0037] Transformation by simplification:

[0038]

[0039] The dimensionality reduction from partial differential equations to ordinary differential equations is completed, and two ordinary differential equations about time and space are obtained:

[0040]

[0041] Combine the boundary conditions and initial conditions to solve Equation (13). First, solve the spatial equation and assume:

[0042]

[0043] Then the spatial equation in formula (13) becomes:

[0044] Z″+γ 2 Z=0 (15)

[0045] The boundary conditions are:

[0046]

[0047] The solution of the spatial equation obtained by combining equations (15) and (16) can be expressed by the characteristic function:

[0048] Z n (z D )=sin(γ n z D ) (17)

[0049] The solution to the time equation is:

[0050]

[0051] Use series expansion to find the general solution for W:

[0052]

[0053] The coefficient A is obtained by using the initial conditions and the orthogonality of the trigonometric function system n :

[0054]

[0055] Substituting equations (19) and (20) into equation (8) yields the steady-state solution:

[0056]

[0057] Substituting equations (6) and (21) into equation (4), we can obtain the final solution expression of the problem equation (3):

[0058]

[0059] Eigenvalue γ n It is defined by the positive roots of the following transcendental equation:

[0060]

[0061] Among them, C D,s (z D ) is the steady-state portion of the dimensionless concentration; C D,t (z D ,t D ) is the transient part of the dimensionless concentration; W is the transformation function (used to simplify the equation); λ is the separation constant; γ is the eigenvalue of the characteristic equation; γ n is the nth characteristic root; Z(z D ) is the spatial characteristic function; T(t D ) is a time function; A n is the series expansion coefficient, and the other symbols have the same meanings as above.

[0062] By using Equations (22) and (23), we can obtain the standard curve data corresponding to different Peclet numbers and different observation point positions using Python programming. Then, we can obtain the standard curve by plotting the standard curve data.

[0063] Furthermore, in step 3), the measured water level and conductivity data at different observation points are obtained by using an LTC three-parameter high-frequency and high-precision probe, and the data are recorded and transmitted remotely in real time in combination with a wireless transmission module.

[0064] Furthermore, in step 3), the water level and conductivity data during the test are dimensionlessly processed, and a measured curve is drawn. The longitudinal hydrodynamic diffusion coefficient D of the test medium is efficiently obtained using the wiring method. L and particle velocity u, record the [Pe] value and [Z] value corresponding to the corresponding cone curve D ] value, and at the same time, select any matching point and write down the corresponding coordinate value [t D ] and [t], and then calculate the longitudinal hydrodynamic diffusion coefficient D of the aquifer L and particle velocity u, where:

[0065]

[0066] Furthermore, in step 3), the permeability coefficient and porosity of the aquifer medium are calculated using the water level and flow rate data from the one-dimensional convection-diffusion test;

[0067] Hydraulic head distribution in the one-dimensional convection-dispersion experimental model:

[0068]

[0069] Darcy's law can be expressed as:

[0070] Q (t) =-KAJ (27)

[0071]

[0072] Where: H is the hydraulic head at any point in the test section, L; J is the hydraulic gradient; ΔH is the hydraulic head difference between two points in the aquifer, L; ΔL is the infiltration path between two points in the aquifer, L; K is the permeability coefficient of the aquifer medium, LT -1 ;Q (t) is the seepage rate per unit time, L 3 T -1 , A is the cross-sectional area of ​​water flow, L 2 ; n is the porosity of the aquifer medium.

[0073] According to the location and water level data of different observation points, the hydraulic gradient of the aquifer can be calculated by formula (26), and then the permeability coefficient and porosity of the aquifer medium can be calculated according to formulas (27) and (28).

[0074] Theoretically, this method features clear boundary control conditions and a rigorous analytical derivation process, enhancing the accuracy and adaptability of parameter identification for convection-dispersion models under finite-scale conditions. Furthermore, the wiring method employed offers advantages such as ease of operation, a large number of parameter identifications, and high accuracy, providing reliable support for the promotion and engineering application of convection-dispersion testing in groundwater quality and quantity issues.

[0075] Beneficial effects: Compared with the prior art, the present invention has the following advantages:

[0076] 1. The present invention no longer relies on the semi-infinite domain assumption, but explicitly introduces finite space boundary conditions through the separation of variables method, which is closer to the actual boundary behavior in indoor experiments and small-scale engineering environments, improving the physical authenticity and applicability of the model.

[0077] 2. By separating variables, a well-defined spatiotemporal analytical expression is constructed without the need for numerical inversion, thus avoiding computational errors caused by numerical instability or initial value sensitivity.

[0078] 3. Synchronous processing of water level data and concentration data can realize the joint calculation of multiple field parameters, improving the integrity and accuracy of parameter calculation. BRIEF DESCRIPTION OF THE DRAWINGS

[0079] Figure 1 is an operational flow chart of the method of the present invention;

[0080] Figure 2 Schematic diagram of the test device of the present invention;

[0081] Figure 3 is the experimental standard curve based on the analytical solution;

[0082] Figure 4 is a measured curve based on test data;

[0083] Figure 5 is the curve fitting diagram of the convection-diffusion test; DETAILED DESCRIPTION

[0084] The present invention is further illustrated below with reference to the accompanying drawings and specific embodiments. It should be understood that these embodiments are only used to illustrate the present invention and are not used to limit the scope of the present invention. After reading the present invention, modifications of various equivalent forms of the present invention made by those skilled in the art all fall within the scope defined by the claims attached to this application.

[0085] like Figure 1 As shown, the present invention provides a method for obtaining parameters of a steady-state boundary convection-diffusion model based on the separation of variables method, comprising the following steps: constructing a one-dimensional convection-diffusion mathematical model under finite-scale steady-state boundary conditions, and non-dimensionalizing the original mathematical model through a dimensionless factor; using the separation of variables method to analytically solve the new dimensionless mathematical model, combining groundwater dynamics theory and solute transport mechanism, systematically deriving an analytical solution expression of the one-dimensional steady-state boundary convection-diffusion model, and thereby constructing a standard curve that can reflect different permeability parameters and solute transport-related parameters; further, designing and conducting an indoor one-dimensional convection-diffusion test, obtaining measured data of water level and solute concentration at different observation points, and using wiring to efficiently obtain the permeability parameters and solute transport-related parameters of the test medium.

[0086] The experimental one-dimensional convection-dispersion model with finite boundaries is formulated as the following well-defined problem:

[0087]

[0088] Choose an appropriate dimensionless factor:

[0089]

[0090] Substituting the dimensionless factor into equation (1), we can obtain the new solution to the problem after dimensionless transformation:

[0091]

[0092] Where: C is the concentration of solute at any point in the test section, ML -3 ;D L is the longitudinal hydrodynamic dispersion coefficient, L 2 T -1 ; u is the average particle seepage velocity, LT -1 ; D0 is the molecular diffusion coefficient, L 2 T -1 ; α L is the longitudinal dispersivity, L; L is the length of the model test section, L; z is the spatial position coordinate, L; C(z,0)=C0 is the initial concentration, ML -3 ; C w0 is the upper boundary concentration increment, ML -3 ;z D is the dimensionless distance; C D is the dimensionless concentration; t D is the dimensionless time; Pe is the Peclet number.

[0093] The new mathematical model after dimensionless transformation is solved analytically using the separation of variables method. Due to the nonhomogeneous boundary conditions, the solution is decomposed into steady-state and transient parts:

[0094] C D (z D ,t D )=C D,s (z D )+C D,t (z D ,t D ) (4)

[0095] The steady-state part satisfies the following steady-state equation:

[0096]

[0097] Solving the ordinary differential equation (5) yields the steady-state solution:

[0098] C D,s (z D )=1 (6)

[0099] Substituting equation (4) and steady-state solution equation (6) into equation (3) yields the equation satisfied by the transient solution:

[0100]

[0101] There is a first-order derivative term in the equation. To facilitate the solution, transform equation (7) and let:

[0102]

[0103] Substituting equation (8) into equation (7) yields a new solution to the steady-state problem:

[0104]

[0105] Separate the variables and solve:

[0106] W(z D ,t D )=Z(z D )T(t D ) (10)

[0107] Substitute equation (10) into the control equation in equation (9),

[0108]

[0109] Transformation by simplification:

[0110]

[0111] The dimensionality reduction from partial differential equations to ordinary differential equations is completed, and two ordinary differential equations about time and space are obtained:

[0112]

[0113] Combine the boundary conditions and initial conditions to solve Equation (13). First, solve the spatial equation and assume:

[0114]

[0115] Then the spatial equation in formula (13) becomes:

[0116] Z″+γ 2 Z=0 (15)

[0117] The boundary conditions are:

[0118]

[0119] The solution of the spatial equation obtained by combining equations (15) and (16) can be expressed by the characteristic function:

[0120] Z n (z D )=sin(γ n z D ) (17)

[0121] The solution to the time equation is:

[0122]

[0123] Use series expansion to find the general solution for W:

[0124]

[0125] The coefficient A is obtained by using the initial conditions and the orthogonality of the trigonometric function system n :

[0126]

[0127] Substituting equations (19) and (20) into equation (8) yields the steady-state solution:

[0128]

[0129] Substituting equations (6) and (21) into equation (4), we can obtain the final solution expression of the problem equation (3):

[0130]

[0131] Eigenvalue γ n It is defined by the positive roots of the following transcendental equation:

[0132]

[0133] Among them, C D,s (z D ) is the steady-state portion of the dimensionless concentration; C D,t (z D ,t D ) is the transient part of the dimensionless concentration; W is the transformation function (used to simplify the equation); λ is the separation constant; γ is the eigenvalue of the characteristic equation; γ n is the nth characteristic root; Z(z D ) is the spatial characteristic function; T(t D ) is a time function; A n is the series expansion coefficient, and the other symbols have the same meanings as above.

[0134] By using formulas (22) and (23), Python programming is used to obtain the standard curve data corresponding to different Peclet numbers and different observation point locations. The standard curve can then be obtained by plotting the standard curve data. The water level and conductivity measured data at different observation points are obtained through the LTC three-parameter high-frequency and high-precision probe, and the data is recorded in real time and remotely transmitted in combination with the wireless transmission module. The water level and conductivity data during the test are dimensionlessly processed, and the measured curve is drawn. The longitudinal hydrodynamic diffusion coefficient D of the test medium is efficiently obtained using the wiring method. L and particle velocity u, record the [Pe] value and [Z] value corresponding to the corresponding cone curve D] value, and at the same time, select any matching point and write down the corresponding coordinate value [t D ] and [t], and then calculate the longitudinal hydrodynamic diffusion coefficient D of the aquifer L and particle velocity u, where:

[0135]

[0136] The permeability coefficient and porosity of the aquifer medium are calculated using the water level and flow data from the one-dimensional convection-diffusion experiment; the hydraulic head distribution in the one-dimensional convection-diffusion experiment model:

[0137]

[0138] Darcy's law can be expressed as:

[0139] Q (t) =-KAJ (27)

[0140]

[0141] Where: H is the hydraulic head at any point in the test section, L; J is the hydraulic gradient; ΔH is the hydraulic head difference between two points in the aquifer, L; ΔL is the infiltration path between two points in the aquifer, L; K is the permeability coefficient of the aquifer medium, LT -1 ;Q (t) is the seepage rate per unit time, L 3 T -1 , A is the cross-sectional area of ​​water flow, L 2 ; n is the porosity of the aquifer medium.

[0142] According to the location and water level data of different observation points, the hydraulic gradient of the aquifer can be calculated by formula (26), and then the permeability coefficient and porosity of the aquifer medium can be calculated according to formulas (27) and (28).

[0143] The specific steps of one-dimensional convection-diffusion experiment and parameter calculation are as follows:

[0144] (1) Build an indoor one-dimensional cylindrical test device (such as Figure 2) to ensure that the device has controllable injection and discharge systems as well as temperature, pressure, and concentration monitoring points. The device has an inner diameter of 125mm and a height of 400mm. The top of the device is the inlet and the bottom is equipped with an outlet. From bottom to top, the device includes a lower boundary solution layer (height 50mm), an aqueous medium layer (height 288mm; sand column, the sand used is silica fine sand with a particle size of 0.3-0.6mm), and an upper boundary solution layer (height 62mm). Three observation points Z1, Z2, and Z3 are set in the aqueous medium layer. Three parameter probes are installed at these three observation points to monitor and record the changes in pressure, temperature, and conductivity at each observation point in real time. First, saturate the sand column with deionized water, maintaining the water head height at the upper surface of the sand column. Then, quickly fill the upper boundary solution layer with 1g / L sodium chloride solution and continue to inject 1g / L sodium chloride solution into the upper boundary to maintain the constant water head boundary and constant concentration boundary at the upper boundary.

[0145] (2) During the test, the solute concentration change data of each monitoring point was recorded regularly, and all test condition parameters were fully recorded, including the injection rate (0.812 cm 3 / s), physical properties of the water-containing medium (silica fine sand with a particle size of 0.3-0.6 mm), and the changes in pressure, temperature, and conductivity over time during the test were recorded by a three-parameter probe (see Table 1), providing basic data for subsequent parameter calculations.

[0146] Table 1 Changes of pressure, temperature and conductivity over time during the test

[0147]

[0148]

[0149]

[0150]

[0151]

[0152]

[0153]

[0154]

[0155]

[0156]

[0157]

[0158]

[0159]

[0160]

[0161] (3) Based on the established finite boundary one-dimensional convection-diffusion analytical model, a standard curve is drawn (e.g. Figure 3 ), the concentration data in the test process were dimensionless processed to obtain the measured curve (such as Figure 4 ), and perform line fitting with the standard curve (such as Figure 5 ), and the longitudinal hydrodynamic diffusion coefficient D of the aquifer medium is obtained by using equations (24) and (25): L and particle flow velocity u.

[0162] (4) Based on the obtained particle velocity u, combined with the water level data of each observation point and the pressure tube during the test, the hydraulic gradient of the aquifer is calculated using formula (26), and then the permeability coefficient and porosity of the aquifer medium are calculated according to formulas (27) and (28).

[0163] Therefore, key hydrogeological parameters such as the permeability coefficient, hydrodynamic diffusion coefficient, porosity, and particle velocity of the aquifer medium can be obtained through a single convection-diffusion test. A comparative analysis with the results calculated from traditional hydrogeological tests shows that the differences in permeability coefficient and porosity are very small. The nearly doubled difference in the hydrodynamic diffusion coefficient is due to the difference in infiltration velocity under different test conditions, which significantly affects the value of the hydrodynamic diffusion coefficient. However, the difference between the two is within a controllable range (see Table 2).

[0164] Table 2 Calculation and comparison of key hydrogeological parameters

[0165]

Claims

1. A method for obtaining parameters of a steady-state boundary convection-diffusion model based on the separation of variables method, characterized in that: The steps include: 1) Construct a one-dimensional convection-dispersion mathematical model under finite-scale steady-state boundary conditions and perform dimensionless processing through dimensionless factors to obtain a new mathematical model after dimensionless processing; 2) The dimensionless mathematical model was analytically solved using the separation of variables method. Combining groundwater dynamics theory and solute transport mechanisms, an analytical expression for the one-dimensional steady-state boundary convection-diffusion model was systematically derived. From this, a standard curve reflecting the correlation between different permeability parameters and solute transport parameters was constructed. 3) Conduct indoor one-dimensional convection-diffusion experiments to obtain measured data of water level and solute concentration at different observation points, and use wiring to efficiently obtain the permeability parameters and solute transport-related parameters of the test medium.

2. A method for obtaining parameters of a steady-state boundary convection-diffusion model based on the separation of variables method according to claim 1, characterized in that: In the step 1), a one-dimensional convection-diffusion mathematical model under finite-scale steady-state boundary conditions is constructed and dimensionless processing is performed using dimensionless factors, including: The experimental one-dimensional convection-dispersion model with finite boundaries is formulated as the following well-defined problem: Select the dimensionless factor: Substituting the dimensionless factor into equation (1), we can obtain the new solution to the problem after dimensionless transformation: Where: C is the concentration of solute at any point in the test section, ML -3 ;D L is the longitudinal hydrodynamic dispersion coefficient, L 2 T -1 ; u is the average particle seepage velocity, LT -1 ; D0 is the molecular diffusion coefficient, L 2 T -1 ; α L is the longitudinal dispersivity, L; L is the length of the model test section, L; z is the spatial position coordinate, L; C(z,0)=C0 is the initial concentration, ML -3 ; C w0 is the upper boundary concentration increment, ML -3 ;z D is the dimensionless distance; C D is the dimensionless concentration; t D is the dimensionless time; Pe is the Peclet number; t is the measured time, T.

3. A method for obtaining parameters of a steady-state boundary convection-diffusion model based on the separation of variables method according to claim 2, characterized in that: In step 2), the new dimensionless mathematical model is analytically solved using the separation of variables method, including: Due to the inhomogeneous boundary conditions, the solution is decomposed into steady-state and transient parts: C D (z D ,t D )=C D,s (z D )+C D,t (z D ,t D ) (4) The steady-state part satisfies the following steady-state equation: Solving the ordinary differential equation (5) yields the steady-state solution: C D,s (z D )=1 (6) Substituting equation (4) and steady-state solution equation (6) into equation (3) yields the equation satisfied by the transient solution: There is a first-order derivative term in the equation. To facilitate the solution, transform equation (7) and let: Substituting equation (8) into equation (7) yields a new solution to the steady-state problem: Separate the variables and solve: W(z D ,t D )=Z(z D )T(t D ) (10) Substitute equation (10) into the control equation in equation (9), Transformation by simplification: The dimensionality reduction from partial differential equations to ordinary differential equations is completed, and two ordinary differential equations about time and space are obtained: Combine the boundary conditions and initial conditions to solve Equation (13). First, solve the spatial equation and assume: Then the spatial equation in formula (13) becomes: Z″+γ 2 Z=0 (15) The boundary conditions are: The solution of the spatial equation obtained by combining equations (15) and (16) can be expressed by the characteristic function: WITH n (With D )=sin(γ n With D ) (17) The solution to the time equation is: Use series expansion to find the general solution for W: The coefficient A is obtained by using the initial conditions and the orthogonality of the trigonometric function system n : Substituting equations (19) and (20) into equation (8) yields the steady-state solution: Substituting equations (6) and (21) into equation (4), we can obtain the final solution expression of the problem equation (3): Eigenvalue γ n It is defined by the positive roots of the following transcendental equation: Among them, C D,s (z D ) is the steady-state portion of the dimensionless concentration; C D,t (z D ,t D ) is the transient part of the dimensionless concentration; W is the transformation function; λ is the separation constant; γ is the eigenvalue of the characteristic equation; γ n is the nth characteristic root; Z(z D ) is the spatial characteristic function; T(t D ) is a time function; A n is the series expansion coefficient; By using Equations (22) and (23), we can obtain the standard curve data corresponding to different Peclet numbers and different observation point positions using Python programming. Then, we can obtain the standard curve by plotting the standard curve data.

4. The method for obtaining parameters of a steady-state boundary convection-diffusion model based on the separation of variables method according to claim 3, characterized in that: In step 3), the water level and conductivity measured data at different observation points are obtained by using an LTC three-parameter high-frequency and high-precision probe, and the data are recorded in real time and transmitted remotely in combination with a wireless transmission module.

5. The method for obtaining parameters of a steady-state boundary convection-diffusion model based on the separation of variables method according to claim 4, characterized in that: In step 3), the water level and conductivity data during the test are dimensionlessly processed, and the measured curve is drawn. The longitudinal hydrodynamic diffusion coefficient D of the test medium is efficiently obtained using the wiring method. L and particle velocity u, record the [Pe] value and [Z] value corresponding to the corresponding cone curve D ] value, and at the same time, select any matching point and write down the corresponding coordinate value [t D ] and [t], and then calculate the longitudinal hydrodynamic diffusion coefficient D of the aquifer L and particle velocity u, where:

6. The method for obtaining parameters of a steady-state boundary convection-diffusion model based on the separation of variables method according to claim 5, characterized in that: In step 3), the permeability coefficient and porosity of the aquifer medium are calculated using the water level and flow rate data from the one-dimensional convection-diffusion experiment; Hydraulic head distribution in the one-dimensional convection-dispersion experimental model: Darcy's law can be expressed as: Q (t) =-AND (27) Where: H is the hydraulic head at any point in the test section, L; J is the hydraulic gradient; ΔH is the hydraulic head difference between two points in the aquifer, L; ΔL is the infiltration path between two points in the aquifer, L; K is the permeability coefficient of the aquifer medium, LT -1 ;Q (t) is the seepage rate per unit time, L 3 T -1 , A is the cross-sectional area of ​​water flow, L 2 ; n is the porosity of the aquifer medium; According to the location and water level data of different observation points, the hydraulic gradient of the aquifer can be calculated by formula (26), and then the permeability coefficient and porosity of the aquifer medium can be calculated according to formulas (27) and (28).

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