Solution and Parameter Calculation Method of Steady-State Boundary-Driven Solute Transport Model Based on Laplace Transform

Abstract

This invention discloses a method for solving and calculating parameters of a steady-state boundary-driven solute transport model based on the Laplace transform, comprising the following steps: establishing a one-dimensional finite-scale mathematical model of groundwater solute transport based on the groundwater solute transport system; solving the mathematical model by combining groundwater dynamics, convection-dispersion equations, and the Laplace transform method, deriving the theoretical model of convection-dispersion experiments under different Pecklet numbers, and obtaining standard curves for determining the permeability coefficient and hydrodynamic dispersion parameters of the medium under different Pecklet numbers; conducting convection-dispersion experiments, and using the wiring method to determine the permeability coefficient and hydrodynamic dispersion parameters of the medium under different Pecklet numbers. This invention has a rigorous mathematical model and strict mathematical theoretical derivation, improves the relevant theories and methods for determining the permeability coefficient and hydrodynamic dispersion parameters of the medium under different Pecklet numbers, and solves the technical problems existing in the application of convection-dispersion experiments in one-dimensional finite-scale models.

Description

Technical Field

[0001] This invention relates to a method for determining the permeability coefficient and hydrodynamic dispersion parameters, specifically to a method for solving and calculating parameters of a steady-state boundary-driven solute transport model based on the Laplace transform. Background Technology

[0002] Permeability coefficient and hydrodynamic dispersion parameters are key parameters for groundwater resource assessment and environmental protection and management. Existing convection-dispersion test methods, when deriving analytical expressions and calculating parameters, typically assume that the test medium is approximately a semi-infinite space in the flow direction, with a constant source term at the inlet, and often assumes that the concentration tends to zero or there is no disturbance propagation at the outlet, simplifying the boundary conditions to a semi-infinite domain model. Based on this assumption, a classic one-dimensional convection-dispersion analytical solution under continuous injection can be obtained, but these solutions all rely on the semi-infinite domain assumption and cannot describe the actual behavior of solute convection-dispersion in a finite test section.

[0003] Furthermore, in indoor columnar or small-scale experiments, the test section length is limited, and the semi-infinite domain model cannot accurately characterize the process after the solute reaches the outlet. In addition, the commonly used "far-field constant" assumption does not match the actual closed or recirculation outlet conditions. If the semi-infinite solution is directly applied to a finite-scale one-dimensional experiment, deviations often occur in the later concentration change stage, thus affecting the fitting accuracy of the hydrodynamic dispersion parameters. At the same time, the semi-infinite domain model cannot effectively calculate the permeability coefficient.

[0004] Therefore, although traditional methods are mature in large-scale aquifer logging and tracer tests in the field, they lack effective means for finite-scale models and outlet boundary conditions of real indoor test scenarios. There is an urgent need to construct and solve a finite-scale model to improve the fitting accuracy and precision of test data. Summary of the Invention

[0005] Purpose of the invention: This invention aims to address the problem that the one-dimensional convection-dispersion analytical solution under the traditional semi-infinite domain assumption cannot accurately describe the outlet behavior of indoor finite-scale experiments. It creatively proposes a method for solving and calculating parameters of a steady-state boundary-driven solute transport model based on Laplace transform, which can quickly and accurately obtain key parameters such as the permeability coefficient and hydrodynamic dispersion parameters of the medium.

[0006] To solve the above-mentioned technical problems, the technical solution adopted by the present invention is as follows:

[0007] A method for solving and calculating parameters of a steady-state boundary-driven solute transport model based on Laplace transform includes the following steps:

[0008] S1: Based on the groundwater solute transport system, establish a one-dimensional finite-scale mathematical model of groundwater solute transport;

[0009] S2: Combining groundwater dynamics, convection-dispersion equations and the Laplace transform method to solve the mathematical model, the theoretical model of convection-dispersion test under different Peckle numbers is derived, and the standard curves for determining the permeability coefficient and hydrodynamic dispersion parameters of the medium under different Peckle numbers are obtained.

[0010] S3: Conduct convection-dispersion tests and use the wiring method to determine the permeability coefficient and hydrodynamic dispersion parameters of the medium under different Peckle numbers.

[0011] The convection-diffusion experimental theoretical model in step S2 above is based on assumptions, specifically:

[0012] (1) Assume that the groundwater solute transport system is a one-dimensional soil column model;

[0013] (2) The hydrodynamic field in the model remains stable, and the groundwater flow velocity at each point in the soil column is the same and remains constant.

[0014] (3) Ignore the mixing effect of solutes inside the water column at the upper boundary of the soil column, that is, the concentration of each point in the upper water column remains stable and equal;

[0015] (4) Ignore the mechanical dispersion effect at the lower boundary exit of the soil column and only consider the convection effect;

[0016] Furthermore, in step S1, a one-dimensional finite-scale mathematical model of groundwater solute transport is established based on the groundwater continuity equation, the convection-dispersion equation, and the solute conservation law.

[0017] One-dimensional steady-flow groundwater continuity equation:

[0018]

[0019] Hydrodynamic field boundary conditions:

[0020] H(0,t)=H1(2)

[0021] H(L,t)=H2(3)

[0022] One-dimensional convection-diffusion equation:

[0023]

[0024] The convection-dispersion equation includes convection, molecular diffusion, and mechanical dispersion, and the parameter relationships among the three are as follows:

[0025] D L =D0+α L u(5)

[0026] Initial conditions of the chemical field:

[0027] C(z,0)=C0(6)

[0028] Boundary conditions at the entrance of the chemical field:

[0029] C(0,t)=C w0 +C0(7)

[0030] At the chemical field outlet, convection of water flow is allowed to transfer solutes, and dispersion at the outlet is ignored. This is the biggest difference from the semi-infinite boundary model. Therefore, the boundary conditions at the outlet under the finite scale model are established as follows:

[0031]

[0032] Integrating equation (1) twice and substituting it into equations (2) and (3), we can obtain the head distribution in the model:

[0033]

[0034] Based on Darcy's law, the permeability coefficient of the aquifer medium and the seepage velocity of the test section can be further determined:

[0035]

[0036] Where: H is the head at any point in the test section, in L; H1 is the boundary head at the inlet of the test section, in L; H2 is the boundary head at the outlet of the test section, in L; C is the concentration of the solute at any point in the test section, in mL. -3 ;D L L is the longitudinal hydrodynamic dispersion coefficient. 2 T -1 u is the average particle seepage velocity, LT -1 D0 is the molecular diffusion coefficient, L 2 T -1 ;α L L represents longitudinal dispersion; L represents the length of the model test section; z represents the spatial coordinates; C(z,0)=C0 represents the initial concentration; ML represents the vertical dispersion. -3 C w0 For the upper boundary concentration increment, ML -3 K is the permeability coefficient of the aquifer medium, LT -1 Q represents the seepage flow rate per unit time, and L represents the flow rate per unit time. 3 T -1 A is the cross-sectional area of ​​the water passage, L 2 n represents the porosity of the aquifer medium; t represents the measured time, T.

[0037] The units used in this application are expressed in international dimensions.

[0038] Furthermore, in step S2, the mathematical model of one-dimensional finite-scale groundwater solute transport needs to be dimensionless, and the selected dimensionless factor is:

[0039] Dimensionless distance:

[0040]

[0041] Dimensionless concentration:

[0042]

[0043] Dimensionless time:

[0044]

[0045] Peclet number (Peclet number):

[0046]

[0047] Dimensionless transformation of equations (4) to (7) yields the new dimensionless solution problem:

[0048]

[0049] Furthermore, in step S2, the new definite solution problem after dimensionless transformation is solved using the Laplace transform method;

[0050] For both sides of equation (16) with respect to t D Taking the Laplace transform, the convection-diffusion equation becomes the equation with respect to z. D Ordinary differential equations:

[0051]

[0052] The boundary conditions in the Lagrange space become:

[0053]

[0054] Solving the ordinary differential equation in the Laplace domain yields the general solution of equation (17):

[0055]

[0056] Where: λ 1,2 The characteristic equation λ 2 The two roots of -Peλ-s=0 A(s) and B(s) are the parameters to be determined.

[0057] A(s) and B(s) are obtained by applying boundary conditions:

[0058]

[0059] Substituting equations (21) and (22) into (20) yields the frequency domain solution of the convection-dispersion equation in the Laplace domain:

[0060]

[0061] The Stehfest algorithm can be used to directly perform a numerical inverse transformation on equation (23). Its core idea is to approximate the Bromwich integral as a linear combination of a set of discrete points, thereby obtaining a semi-analytical solution to the new boundary value problem after dimensionless transformation.

[0062]

[0063] Where: N is an even number, usually ranging from 8 to 16; This represents the points in the Laplace variable s, used to approximate the Bromwich integral; This represents the weight of each value point in the inverse transformation result.

[0064] Furthermore, in step S2, Python code is used to plot the z-axis at different dimensionless positions. D and dimensionless concentration C under Pekley number Pe. D Standard curve as a dimensionless time-varying curve.

[0065] Furthermore, in step S3, based on the dimensionless concentration change value C at the collected observation point... D Using the observed data over time t, plot the concentration change C with the same modulus as the standard curve. D The measured curve is plotted against time t, with all measured times t taken logarithmically. The origins of the measured curve and the standard curve are positioned at the same height on the coordinate layer. The measured curve layer is then fitted to the standard curve by translation, and the optimal fitting position is selected. The dimensionless position z of the fitted standard curve is recorded. D Pekley number (Pe value), dimensionless time (t) D Value and measured time t.

[0066] Furthermore, in step S3, the specific process of calculating the permeability coefficient and hydrodynamic dispersion parameters based on the experimental data is as follows:

[0067] A1: Conduct the experiment and record all raw data required during the experiment;

[0068] A2: Calculate the longitudinal hydrodynamic dispersion coefficient D of the aquifer medium according to equations (14) and (15). L1 and particle velocity u1;

[0069] A3: The permeability coefficient K1 and porosity n1 of the aquifer medium are calculated according to equations (10) and (11);

[0070] A4: By changing the hydrodynamic field conditions and conducting the experiment again, repeating the process A1-A3 above, the permeability coefficient K2, particle velocity u2, and longitudinal hydrodynamic dispersion coefficient D can be obtained. L2 ;

[0071] A5: The permeability coefficient and porosity are obtained by calculating the average value of the permeability coefficient from the two experiments. The molecular diffusion coefficient D0 and longitudinal dispersion α are calculated according to equation (5). L .

[0072] The longitudinal hydrodynamic dispersion coefficient D of the medium can be calculated through a single test by following the above steps. L The molecular diffusion coefficient D0 and longitudinal dispersion α can be calculated from the particle velocity u, permeability coefficient K, and porosity n through at least two experiments. L .

[0073] The above method has significant advantages in terms of rigorous theoretical derivation, convenient operation, high computational efficiency, and accurate results, and has good engineering application and promotion value.

[0074] This invention features a rigorous mathematical model and strict theoretical derivation, improving the relevant theories and methods for determining the permeability coefficient and hydrodynamic dispersion parameters of a medium under different Pecklet numbers in convection-dispersion experiments. It also solves the problems existing in the application of convection-dispersion experiments in one-dimensional finite-scale models. Furthermore, the invention employs a wiring method to calculate the permeability and hydrodynamic dispersion parameters of the medium under different Pecklet numbers, which is simple, efficient, and yields a large number of parameters with high accuracy, making it highly valuable for application and promotion.

[0075] Any techniques not mentioned in this invention are based on existing technologies.

[0076] Beneficial effects: Compared with the prior art, the present invention has the following advantages:

[0077] 1. A finite-scale convection dispersion model was constructed, and the boundary condition of zero diffusion flux at the outlet was adopted to eliminate the process error of concentration change in the later stage of the semi-infinite model in the indoor experiment.

[0078] 2. By combining dimensionless transformation, Laplace transform, and Stehfest algorithm, high-precision standard curves can be generated quickly.

[0079] 3. It can simultaneously calculate the permeability coefficient K and the longitudinal hydrodynamic dispersion coefficient D. L Molecular diffusion coefficient D0 and longitudinal dispersion α L It can handle multiple key parameters and has high experimental efficiency.

[0080] 4. The experiment is simple and convenient, highly versatile, and highly accurate. Attached Figure Description

[0081] Figure 1 This is a flowchart illustrating the operation of the method of the present invention;

[0082] Figure 2 To determine the standard curve of convection-dispersion test for the hydrodynamic dispersion parameters of the aquifer;

[0083] Figure 3 The first set of measured curves from the convection-diffusion experiment;

[0084] Figure 4 The second set of measured curves from the convection-diffusion experiment;

[0085] Figure 5 The fitted curves for the first group of convection-diffusion experiments;

[0086] Figure 6 The fitting curves for the second group of convection-diffusion experiments; Detailed Implementation

[0087] The present invention will be further illustrated below with reference to the accompanying drawings and specific embodiments. It should be understood that these embodiments are for illustrative purposes only and are not intended to limit the scope of the invention. After reading this invention, any modifications of the invention in various equivalent forms by those skilled in the art will fall within the scope defined by the appended claims.

[0088] like Figure 1 As shown, this invention provides a method for solving and calculating parameters of a steady-state boundary-driven solute transport model based on Laplace transform, comprising the following steps:

[0089] 1) Construct a finite-scale one-dimensional convection-diffusion model, select an appropriate dimensionless factor for normalization, obtain the frequency domain solution through Laplace transform, and use the Stehfest algorithm for inverse transformation to obtain dimensionless concentration-time standard curves under different dimensionless distances and Peckley numbers.

[0090] A mathematical model for solute transport in groundwater at a one-dimensional finite scale is established based on the groundwater continuity equation, the convection-dispersion equation, and the solute conservation law.

[0091] One-dimensional steady-flow groundwater continuity equation:

[0092]

[0093] Hydrodynamic field boundary conditions:

[0094] H(0,t)=H1(2)

[0095] H(L,t)=H2(3)

[0096] One-dimensional convection-diffusion equation:

[0097]

[0098] The convection-dispersion equation includes convection, molecular diffusion, and mechanical dispersion, and the parameter relationships among the three are as follows:

[0099] D L =D0+α L u(5)

[0100] Initial conditions of the chemical field:

[0101] C(z,0)=C0(6)

[0102] Boundary conditions at the entrance of the chemical field:

[0103] C(0,t)=C w0 +C0(7)

[0104] At the chemical field outlet, convection of water flow is allowed to transfer solutes, and dispersion at the outlet is ignored. This is the biggest difference from the semi-infinite boundary model. Therefore, the boundary conditions at the outlet under the finite scale model are established as follows:

[0105]

[0106] Integrating equation (1) twice and substituting it into equations (2) and (3), we can obtain the head distribution in the model:

[0107]

[0108] Based on Darcy's law, the permeability coefficient of the aquifer medium and the seepage velocity of the test section can be further determined:

[0109]

[0110] Where: H is the head at any point in the test section, in L; H1 is the boundary head at the inlet of the test section, in L; H2 is the boundary head at the outlet of the test section, in L; C is the concentration of the solute at any point in the test section, in mL. -3 ;D L L is the longitudinal hydrodynamic dispersion coefficient. 2 T -1 u is the average particle seepage velocity, LT -1 D0 is the molecular diffusion coefficient, L 2 T -1 ;α L L represents longitudinal dispersion; L represents the length of the model test section; z represents the spatial coordinates; C(z,0)=C0 represents the initial concentration; ML represents the vertical dispersion. -3 C w0 For the upper boundary concentration increment, ML -3K is the permeability coefficient of the aquifer medium, LT -1 Q represents the seepage flow rate per unit time, and L represents the flow rate per unit time. 3 T -1 A is the cross-sectional area of ​​the water passage, L 2 ; n is the porosity of the aquifer medium.

[0111] The mathematical model for one-dimensional finite-scale groundwater solute transport needs to be dimensionless. The dimensionless factor selected is:

[0112] Dimensionless distance:

[0113]

[0114] Dimensionless concentration:

[0115]

[0116] Dimensionless time:

[0117]

[0118] Peclet number (Peclet number):

[0119]

[0120] Dimensionless transformation of equations (4) to (7) yields the new dimensionless solution problem:

[0121]

[0122] The new boundary value problem after dimensionless transformation is solved using the Laplace transform method; the two sides of equation (16) with respect to t D Taking the Laplace transform, the convection-diffusion equation becomes the equation with respect to z. D Ordinary differential equations:

[0123]

[0124] The boundary conditions in the Lagrange space become:

[0125]

[0126] Solving the ordinary differential equation in the Laplace domain yields the general solution of equation (17):

[0127]

[0128] Where: λ 1,2 The characteristic equation λ 2 The two roots of -Peλ-s=0 A(s) and B(s) are the parameters to be determined.

[0129] A(s) and B(s) are obtained by applying boundary conditions:

[0130]

[0131] Substituting equations (21) and (22) into (20) yields the frequency domain solution of the convection-dispersion equation in the Laplace domain:

[0132]

[0133] The Stehfest algorithm can be used to directly perform a numerical inverse transformation on equation (23). Its core idea is to approximate the Bromwich integral as a linear combination of a set of discrete points, thereby obtaining a semi-analytical solution to the new boundary value problem after dimensionless transformation.

[0134]

[0135] Where: N is an even number, usually ranging from 8 to 16; This represents the points in the Laplace variable s, used to approximate the Bromwich integral; This represents the weight of each value point in the inverse transformation result.

[0136] Write Python code to plot the z-axis at different dimensionless positions. D and dimensionless concentration C under Pekley number Pe. D Standard curve as a dimensionless time-varying curve;

[0137] 2) Conduct convection-dispersion tests in a one-dimensional soil column indoors, record the solute concentration changes over time at each observation point, and plot the measured curves on a semi-logarithmic coordinate system. Align the measured curves with the standard curve on semi-logarithmic paper with the same modulus, and fit the measured curve layer to the standard curve by translation, selecting the optimal fitting position. Record the dimensionless position, Peckley number, dimensionless time, and measured time of the fitted standard curve, substitute them into the (semi-)analytical formula, and calculate the permeability coefficient, particle velocity, and longitudinal hydrodynamic dispersion coefficient. If further determination of the molecular diffusion coefficient and longitudinal dispersion is required, the test can be repeated under different flow rates, and the two sets of fitting parameters can be combined to calculate the key parameters such as permeability coefficient, longitudinal hydrodynamic dispersion coefficient, molecular diffusion coefficient, and longitudinal dispersion coefficient in one go through a closed-form relationship.

[0138] Based on the dimensionless concentration change value C collected at the observation point D Using the observed data over time t, plot the concentration change C with the same modulus as the standard curve. DThe measured curve is plotted against time t, where all measured times t are logarithmic. The origins of the measured curve and the standard curve are aligned at the same height on the coordinate layer. The measured curve layer is then fitted to the standard curve by translation. The optimal fitting position is selected, and the dimensionless position z of the fitted standard curve is recorded. D Pekley number (Pe value), dimensionless time (t) D Value and measured time t.

[0139] The specific process for calculating the permeability coefficient and hydrodynamic dispersion parameters based on experimental data is as follows:

[0140] A1: Conduct the experiment and record all raw data required during the experiment;

[0141] A2: Calculate the longitudinal hydrodynamic dispersion coefficient D of the aquifer medium according to equations (14) and (15). L1 and particle velocity u1;

[0142] A3: The permeability coefficient K1 and porosity n1 of the aquifer medium are calculated according to equations (10) and (11);

[0143] A4: By changing the hydrodynamic field conditions and conducting the experiment again, repeating the process A1-A3 above, the permeability coefficient K2, particle velocity u2, and longitudinal hydrodynamic dispersion coefficient D can be obtained. L2 ;

[0144] A5: The permeability coefficient and porosity are obtained by calculating the average value of the permeability coefficient from the two experiments. The molecular diffusion coefficient D0 and longitudinal dispersion α are calculated according to equation (5). L .

[0145] The longitudinal hydrodynamic dispersion coefficient D of the medium can be calculated through a single test by following the above steps. L The molecular diffusion coefficient D0 and longitudinal dispersion α can be calculated from the particle velocity u, permeability coefficient K, and porosity n through at least two experiments. L .

[0146] To verify the actual effect of the method of the present invention, the above scheme is applied in this embodiment, and the specific experimental process is as follows:

[0147] 1. Apparatus preparation and test medium selection

[0148] Assemble a one-dimensional finite-scale convection-dispersion experimental setup with an inner diameter of 125 mm (cross-sectional area of ​​water passage A = 0.0123 m²). 2 The total height is 350 mm. The test medium is silica fine sand with a particle size range of 0.3-0.6 mm. The material is stable and does not easily undergo chemical reactions. The particle size distribution is uniform, simulating a homogeneous and isotropic aquifer medium, and serving as a representative aquifer material.

[0149] 2. Observation point layout and medium filling

[0150] The upper surface of the sand layer was set as z=0, and a coordinate axis z was established with the downward direction as positive. Four observation points were set along the z-axis at distances of -21.5 mm, 27 mm, 132 mm, and 232 mm from the upper surface of the sand layer. The probe at -21.5 mm was located in the water column region and was used to collect data on water level and concentration changes at the upper boundary. The other three probes were buried inside the sand layer to collect data on water level, temperature, and conductivity during solute transport at different depths. Sand sample filling and probe placement were carried out from bottom to top to ensure uniform sample loading and accurate probe placement throughout the device. The final uniform filling height of the sand sample in the device was 282 mm.

[0151] 3. Saturation treatment of sand samples and preparation of solutions

[0152] To eliminate the influence of gases in the aquifer medium and ensure consistent experimental conditions, the filled sand samples were repeatedly subjected to water saturation treatment until the aquifer medium was completely saturated. A sufficient amount of 1 g / L NaCl solution was prepared with deionized water and placed in the same constant temperature environment as the experimental apparatus for 24 hours to ensure that the temperatures of both were consistent.

[0153] 4. Conduct convection-diffusion experiments

[0154] Four LTC high-precision probes were activated, sampling at a frequency of 10 s / time. A 1 g / L NaCl solution was continuously injected into the upper part of the apparatus using a peristaltic pump to maintain a constant water head at the upper and lower boundaries of the experimental setup. Water level, temperature, and conductivity data were automatically collected at each observation point using the probes to obtain the solute concentration changes at each depth. The experiment duration should cover the entire process of concentration increase, peak appearance, and concentration stabilization at all observation points.

[0155] 5. Data Analysis and Parameter Calculation

[0156] Concentration-time data at different observation points were dimensionlessized, and the measured curves were plotted on a semi-logarithmic coordinate graph (e.g., Figure 3-4 As shown), and the corresponding analytical derivation yields the obtained standard curve (as shown). Figure 2 As shown) perform wiring fitting (e.g. Figure 5-6 (As shown). The longitudinal hydrodynamic dispersion coefficient, particle velocity, permeability coefficient, and porosity of the medium are calculated using formulas.

[0157]

[0158]

[0159] By continuing the experiment and repeating the above steps, the longitudinal hydrodynamic dispersion coefficient, particle velocity, permeability coefficient and porosity under different flow velocity conditions can be obtained.

[0160]

[0161] According to equation (5), the following relationship can be obtained:

[0162] D L1 =D0+α L u1

[0163] D L2 =D0+α L u2

[0164] By combining the two equations, the molecular diffusion coefficient and longitudinal dispersion can be obtained:

[0165] D0 = 4.87 × 10 -7 m 2 / s

[0166] α L =1.03×10 -3 m

[0167] Thus, the longitudinal hydrodynamic dispersion coefficient, particle velocity, permeability coefficient, and porosity of the medium can be calculated through at least one set of convection-dispersion tests, and the molecular diffusion coefficient and longitudinal dispersion can be calculated through at least two sets of tests.

[0168] Furthermore, to verify the accuracy of the parameter calculation method proposed in this invention, the basic parameters of the same test medium were tested using constant head tests, porosity measurement tests, and conventional convection-dispersion tests. The parameter values ​​measured by the conventional method were compared with those measured by the new method (Table 1). The comparison results show that the relevant parameters measured by the new method proposed in this patent are consistent with the relevant parameters measured by conventional tests, further verifying the accuracy of the method proposed in this invention.

[0169] Table 1 Comparison of test results between traditional and new methods

[0170]

Claims

1. A method for solving and calculating parameters of a steady-state boundary-driven solute transport model based on Laplace transform, characterized in that, Includes the following steps: S1: Based on the groundwater solute transport system, establish a one-dimensional finite-scale mathematical model of groundwater solute transport; S2: Combining groundwater dynamics, convection-dispersion equations and the Laplace transform method, a mathematical model for solute transport in one-dimensional finite-scale groundwater is solved, and theoretical models for convection-dispersion experiments under different Peckle numbers are derived, resulting in standard curves for determining the permeability coefficient and hydrodynamic dispersion parameters of the medium under different Peckle numbers. S3: Conduct convection-dispersion tests and use the wiring method to determine the permeability coefficient and hydrodynamic dispersion parameters of the medium under different Peckle numbers; The convection-diffusion experimental theoretical model in step S2 is based on assumptions, specifically: (1) Assume that the groundwater solute transport system is a one-dimensional soil column model; (2) The hydrodynamic field in the model remains stable, and the groundwater flow velocity at each point in the soil column is the same and remains constant; (3) Ignore the mixing effect of solute inside the water column at the upper boundary of the soil column, that is, the concentration at each point in the upper water column remains stable and equal; (4) Ignore the mechanical dispersion effect at the lower boundary exit of the soil column and only consider the convection effect; Step S2 specifically includes the following steps: S21: Dimensionless processing of the mathematical model for one-dimensional finite-scale groundwater solute transport: The selected dimensionless factor is: Dimensionless distance: (12) Dimensionless concentration: (13) Dimensionless time: (14) Peclet number: (15) Dimensionless transformation of equations (4) to (7) yields the new dimensionless solution problem: (16) S22: The new boundary value problem after dimensionless transformation is solved using the Laplace transform method: Regarding both sides of equation (16) Taking the Laplace transform, the convection-diffusion equation becomes a parallel equation. Ordinary differential equations: (17) The boundary conditions in the Lagrange space become: (18) (19) Solving the ordinary differential equation in the Laplace domain yields the general solution of equation (17): (20) in: Characteristic equation The two roots, ; , These are the parameters to be determined; Obtained by applying boundary conditions and : (21) (22) Substituting equations (21) and (22) into (20) yields the frequency domain solution of the convection-diffusion equation in the Laplace domain: (23) The Stehfest algorithm is used to directly perform a numerical inverse transformation on equation (23). The core idea is to approximate the Bromwich integral as a linear combination of a set of discrete points, thereby obtaining a semi-analytical solution to the new boundary value problem after dimensionless transformation. (24) in: It is an even number, and its value ranges from 8 to 16; , representing the Laplace variable The points where are taken are used to approximate the Bromwich integral; , representing the weight of each value point in the inverse transform result; S23: Write Python code to plot different dimensionless positions. and Pecley number dimensionless concentration Standard curve as a dimensionless time-varying curve; in, Spatial location coordinates; The length of the model test section; C0 represents the concentration of the solute at any point in the test section; C0 represents the initial concentration. This represents the upper boundary concentration increment; The longitudinal hydrodynamic dispersion coefficient; This is the actual measured time; The average particle seepage velocity; Dimensionless position; Dimensionless concentration; For Laplace variables; Dimensionless time.

2. The method for solving and calculating parameters of a steady-state boundary-driven solute transport model based on Laplace transform according to claim 1, characterized in that, In step S1, a one-dimensional finite-scale mathematical model of groundwater solute transport is established based on the groundwater continuity equation, the convection-dispersion equation, and the solute conservation law. One-dimensional steady-flow groundwater continuity equation: (1) Hydrodynamic field boundary conditions: (2) (3) One-dimensional convection-diffusion equation: (4) The convection-dispersion equation includes convection, molecular diffusion, and mechanical dispersion, and the parameter relationships among the three are as follows: (5) Initial conditions of the chemical field: (6) Boundary conditions at the entrance of the chemical field: (7) At the chemical field outlet, convection of water flow is allowed to transfer solutes. Ignoring dispersion at the outlet, the boundary conditions at the outlet under the finite-scale model are as follows: (8) Integrating equation (1) twice and substituting it into equations (2) and (3), we can obtain the head distribution in the model: (9) Based on Darcy's law, the permeability coefficient of the aquifer medium and the seepage velocity of the test section can be further determined: (10) (11) in: The water head at any point in the test section; The boundary head at the entrance of the test section; The boundary head at the outlet of the test section; The concentration of the solute at any point in the test section; The longitudinal hydrodynamic dispersion coefficient; The average particle seepage velocity; The molecular diffusion coefficient; Longitudinal dispersion; The length of the model test section; Spatial location coordinates; This is the initial concentration; This represents the upper boundary concentration increment; The permeability coefficient of the aquifer medium; The seepage flow rate per unit time. This refers to the cross-sectional area of ​​the water passage. Porosity of the aquifer medium; This is the actual measured time.

3. The method for solving and calculating parameters of a steady-state boundary-driven solute transport model based on Laplace transform according to claim 1 or 2, characterized in that, In step S3, the dimensionless concentration change value at the collected observation point is used as the basis for further analysis. With time Based on the observed data, concentration changes with the same modulus as the standard curve were plotted. With time The measured curve, the measured time Take the logarithm of each coordinate to ensure that the origin of the measured curve and the origin of the standard curve are at the same height on the coordinate layer. Fit the measured curve layer to the standard curve by translation, select the optimal fitting position, and record the dimensionless position of the fitted standard curve. Value, Pecley number Value, dimensionless time Value and measured time value.

4. The method for solving and calculating parameters of a steady-state boundary-driven solute transport model based on Laplace transform according to claim 1 or 2, characterized in that, In step S3, the specific process of calculating the permeability coefficient and hydrodynamic dispersion parameters based on the experimental data is as follows: A1: Conduct the experiment and record all raw data required during the experiment; A2: Calculate the longitudinal hydrodynamic dispersion coefficient of the aquifer medium according to equations (14) and (15). and particle velocity ; A3: The permeability coefficient of the aquifer medium is calculated according to equations (10) and (11). and porosity ; A4: By changing the hydrodynamic field conditions and conducting the experiment again, repeating the process from A1 to A3 above, the permeability coefficient can be obtained. particle velocity and longitudinal hydrodynamic dispersion coefficient ; A5: The permeability coefficient and porosity are obtained by calculating the average value of the permeability coefficient from the two experiments. The molecular diffusion coefficient is calculated according to equation (5). and longitudinal diffusion .

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