Method and system for estimating large-pore reactive solute transport

Abstract

The present application relates to the method and system for estimating the migration of reactive solute in large pores, comprising the following steps: establishing the model of the migration of reactive solute in large pores considering the joint influence of movable zone, immovable zone and matrix zone, solving the semi-analytical solution of the model in Laplace domain under pulse boundary condition; obtaining the measured experimental data to get the breakthrough curve of organic compounds; fitting the semi-analytical solution and the breakthrough curve of organic compounds, selecting the parameter value corresponding to the fitting curve meeting the condition as the final value of parameter inversion; the present application is based on the solving method of analytical model, according to the existing measured experimental data, using the analytical model proposed in the present application for calculation, the results show that the solution of the model and the experimental data can be well matched, the measured results are related to the theoretical results, the utilization value of the existing data is fully tapped, and the model can accurately reflect the solute migration process in large pores.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of groundwater pollution prevention and control, more particularly to a method and system for estimating the transport of reactive solute in packed macropores. BACKGROUND

[0002] As the theoretical basis for groundwater pollution prevention and control, the research on solute transport mechanism has always been an important topic in the field of hydrogeology. However, due to the heterogeneity, anisotropy and scale uncertainty of porous media, the research on solute transport has been slow. Therefore, it is of great scientific value to construct a model that can comprehensively reflect the characteristics of reactive solute transport in macropores for precise prediction of pollutant migration paths and optimization of remediation schemes.

[0003] The present application is applicable to the prediction of pollutant migration and the optimization of remediation schemes. The packed macropores present dead-end pores, which are immobile regions, and significantly change the migration pattern of the solute plume through hysteresis effect. However, in the prior art, the traditional single-pore model cannot describe the reactive transport process of solute in packed pores, and there are problems such as ignoring the dynamic mass transfer between the mobile and immobile regions of the packed pores, leading to prediction bias of the tailing effect, and failing to reflect the solute retention mechanism. Moreover, the traditional research method only considers the solute transport in the mobile region, ignoring the influence of complex macropore structure on the reactive transport of solute. Therefore, there is a need for a method and system for estimating the reactive solute transport in packed macropores that more comprehensively considers the mass transfer process of solute transport, solves the problem of over-simplified structure representation of traditional single-pore models, and improves the accuracy of prediction due to incomplete mass transfer matrix. SUMMARY

[0004] The technical problem to be solved by the present application is to provide a method and system for estimating the reactive solute transport in packed macropores, which addresses the above-mentioned deficiencies of the prior art.

[0005] The technical solution adopted by the present application to solve the technical problem is:

[0006] A method for estimating the reactive solute transport in packed macropores is constructed, which includes the following steps:

[0007] A model for the reactive solute transport in packed macropores considering the joint influence of mobile, immobile and matrix regions is established, and a semi-analytical solution in the Laplace domain of the model under pulse boundary conditions is obtained;

[0008] Obtain the breakthrough curve of the organic compound based on the measured experimental data;

[0009] The semi-analytical solution obtained is fitted with the penetration curve of the organic compound, and the parameter value corresponding to the fitting curve meeting the condition is selected as the final value of the parameter inversion.

[0010] The filling macropore reactive solute transport estimation method provided by the application, wherein the filling macropore reactive solute transport model is established based on the following basis:

[0011] The filling macropore is generalized into movable areas and immovable areas and linear adsorption occurs;

[0012] The model water flow is steady flow and is Darcy water flow;

[0013] The pore radius does not change with the change of the pore length;

[0014] The matrix is semi-infinite length;

[0015] The pore and the matrix area are both first-order irreversible reactions;

[0016] The influence of convection and dispersion in the movable area is considered, and the radial diffusion of the solute in the matrix area is considered;

[0017] A geometric model is established with the pore center as the origin.

[0018] The filling macropore reactive solute transport estimation method provided by the application, wherein the filling macropore reactive solute transport model considering the joint influence of the movable area, the immovable area and the matrix area is established, and the semi-analytical solution of the filling macropore reactive solute transport model in the Laplace domain under the pulse boundary condition is solved, and the semi-analytical solution of the filling macropore reactive solute transport model in the Laplace domain under the pulse boundary condition includes:

[0019] The solute transport process is described by the convection-dispersion equation, and the mathematical equation of the reactive solute transport is expressed as:

[0020] The solute transport process in the movable area is:

[0021] (1)

[0022] The solute transport process in the immovable area is:

[0023] (2)

[0024] The solute transport process in the matrix is:

[0025] (3)

[0026] In the formula, C m and C im respectively represent the solute concentration in the movable area and the immovable area of the pore; C a represents the solute concentration in the matrix area; t represents time; vm where λ is the first order reaction rate constant; θ is the porosity; z is the vertical distance; r is the radial distance from the center of the pore; ω is the first order mass transfer coefficient between the mobile region and the immobile region; and R is the retardation factor, which is defined as: m where r is the radial distance from the center of the pore; ω is the first order mass transfer coefficient between the mobile region and the immobile region; and R is the retardation factor, which is defined as:

[0027] R is the retardation factor, which is defined as:

[0028] ;

[0029] ;

[0030] where m, im, and a are the mobile region of the pore, the immobile region of the pore, and the matrix, respectively; D is the longitudinal dispersion coefficient of the pore, which is defined as:

[0031] (4)

[0032] where α is the first order mass transfer coefficient between the mobile region and the immobile region; and D0 is the effective molecular diffusion coefficient in the pore; m where α is the first order mass transfer coefficient between the mobile region and the immobile region; and D0 is the effective molecular diffusion coefficient in the pore;

[0033] For the model of reactive solute transport in a large pore filled with matrix, the concentration is continuous between the pore and the matrix, which is defined as:

[0034] (5)

[0035] The initial concentration of the pore-matrix is defined as:

[0036] (6)

[0037] The initial condition is defined as:

[0038] (7)

[0039] (8)

[0040] The boundary condition of the solute concentration is considered as a pulse injection boundary condition:

[0041] (9)

[0042] where t0 is the pulse injection time;

[0043] The solution of the model is obtained by taking the Laplace transform of both sides of equations (1), (2), and (3) with respect to t, where is a function of z, and s is the variable, which is defined as:

[0044] (10)

[0045] (11)

[0046] (12)

[0047] where s is the Laplace transform variable; , and denote the Laplace transform of , and ; the boundary condition (9) is transformed into Laplace space to give

[0048] (13)

[0049] (14)

[0050] (15)

[0051] According to equation (11), and the relationship is expressed as:

[0052] (16)

[0053] Substituting equation (16) into equation (10) gives the following equation:

[0054] (17)

[0055] where . The general solution of equation (12) is:

[0056] (18)

[0057] where K0() is the second kind of zero-order modified Bessel function, and the coefficients A and B can be determined from the boundary condition equation (15):

[0058] (19)

[0059] Therefore, equation (18) is:

[0060] (20)

[0061] Substituting the boundary condition equation (14) into equation (20) gives:

[0062] (21)

[0063] Substituting equation (21) into equation (20) gives:

[0064] (22)

[0065] Substituting equation (22) into equation (17) gives:

[0066] (23)

[0067] (24)

[0068] (25)

[0069] Taking Laplace transform of the pulse injection boundary condition (9) gives:

[0070] (26)

[0071] The solution of equation (24) is:

[0072] (27)

[0073] (28).

[0074] The method for estimating the transport of reactive solute in large pores filled by the present application, wherein the measured experimental data is selected from soil column experimental data.

[0075] A system for estimating the transport of reactive solute in large pores filled, for realizing the method for estimating the transport of reactive solute in large pores filled as described above, wherein the system comprises a model construction unit, a data acquisition unit and a fitting wiring unit.

[0076] The model construction unit is configured to establish a model for the transport of reactive solute in large pores filled, which takes into account the combined effects of mobile zone, immobile zone and matrix zone, and to solve the semi-analytical solution of the model for the transport of reactive solute in large pores filled in the Laplace domain under pulse boundary conditions.

[0077] The data acquisition unit is configured to obtain measured experimental data, and to obtain the breakthrough curve of the organic compound based on the measured experimental data.

[0078] The fitting wiring unit is configured to fit the obtained semi-analytical solution with the breakthrough curve of the organic compound, and to select the parameter values corresponding to the fitting curve that meets the conditions as the final values of the parameter inversion.

[0079] The beneficial effect of the present application is that: the present application is based on the solving method of the analytical model, according to the existing measured experimental data, the analytical model is calculated, the results show that the solution of the model can be well matched with the experimental data, the measured results are related to the theoretical results, the utilization value of the existing data is fully tapped, and the model can accurately reflect the large-pore solute transport process. BRIEF DESCRIPTION OF DRAWINGS

[0080] In order to more clearly illustrate the technical solutions of the embodiments of the present application or the prior art, the present application will be further described below in conjunction with the drawings and embodiments. The drawings in the following description are only some embodiments of the present application, and for those skilled in the art, other drawings can be obtained without creative labor on the basis of these drawings:

[0081] Figure 1 is a filling large-pore reactive solute transport estimation method flow chart of the preferred embodiment of the present application;

[0082] Figure 2 is a filling large-pore reactive solute transport estimation method model schematic diagram of the preferred embodiment of the present application;

[0083] Figure 3 is a fitting wiring schematic diagram of the filling large-pore reactive solute transport estimation method of the preferred embodiment of the present application;

[0084] Figure 4 is a filling large-pore reactive solute transport estimation system principle block diagram of the preferred embodiment of the present application. DETAILED DESCRIPTION

[0085] In order to make the purpose, technical scheme and advantages of the embodiments of the present application more clear, the technical solutions in the embodiments of the present application will be described clearly and completely below. Obviously, the described embodiments are some embodiments of the present application, not all embodiments. Based on the embodiments of the present application, all other embodiments obtained by those skilled in the art without creative labor are within the protection scope of the present application.

[0086] The filling large-pore reactive solute transport estimation method of the preferred embodiment of the present application, as shown in Figure 1 , and referring to Figure 2 and Figure 3 , comprises the following steps:

[0087] S01: A filling large-pore reactive solute transport model considering the joint influence of movable zone, immovable zone and matrix zone is established, and a semi-analytical solution of the filling large-pore reactive solute transport model in Laplace domain under pulse boundary condition is solved;

[0088] S02: obtaining measured experimental data, and obtaining a breakthrough curve of the organic compound based on the measured experimental data;

[0089] S03: fitting the obtained semi-analytical solution with the breakthrough curve of the organic compound, and selecting the parameter value corresponding to the fitting curve meeting the condition as the final value of parameter inversion;

[0090] Based on the solving method of the analytical model, the existing measured experimental data are used for calculation by using the analytical model proposed in the application, and the results show that the solution of the model can well agree with the experimental data, the measured results are associated with the theoretical results, the utilization value of the existing data is fully tapped, and the model can accurately reflect the large-pore solute transport process.

[0091] More specifically, the application considers a filling large-pore reactive solute transport model (Mobile-Immobile Model, MIM) affected by a mobile zone and an immobile zone, wherein the process is to establish a two-zone mathematical model, and to obtain a semi-analytical solution of the model in the Laplace domain under pulse boundary conditions, since the semi-analytical solution obtained in the Laplace domain contains special functions such as Bessel functions, the method of Laplace numerical inverse transformation is adopted to obtain the solution of the MIM model in the real domain space; the analytical solution obtained by the analytical model is fitted with the soil column experimental data of the organic compound at fixed observation points in a set time range, and the parameter value corresponding to the optimal fitting curve is selected as the final value of parameter inversion, which provides a scientific basis for the application of groundwater pollution remediation in actual engineering, and has important significance.

[0092] The filling large-pore reactive solute transport model is established based on the following basis:

[0093] 1. The filling pore is generalized into a mobile zone and an immobile zone and linear adsorption occurs;

[0094] 2. The model water flow is steady flow, and is Darcy water flow;

[0095] 3. The pore radius does not change with the change of the pore length;

[0096] 4. The matrix is semi-infinite long;

[0097] 5. The pore and the matrix region are first-order irreversible reactions;

[0098] 6. The influence of convection and dispersion is considered in the mobile zone, and the radial diffusion of solute is considered in the matrix region;

[0099] Finally, a geometric model is established with the center of the pore as the origin.

[0100] Reference Figure 2, a reactive solute transport model considering the influence of mobile zone, immobile zone and matrix zone together was established, and the semi-analytical solution of the model in Laplace domain under pulse boundary condition was solved, the model establishment and calculation specifically adopted:

[0101] The solute transport process was described by convection-dispersion equation, and the mathematical equation of reactive solute transport was expressed as:

[0102] The solute transport process in mobile zone:

[0103] (1)

[0104] The solute transport process in immobile zone:

[0105] (2)

[0106] The solute transport process in matrix:

[0107] (3)

[0108] In the formula, C m and C im respectively represent the solute concentration in the mobile zone and the immobile zone of the pore; C a represents the solute concentration in the matrix zone; t represents time; v m represents the groundwater flow velocity in the mobile zone; λ represents the first-order reaction rate constant; θ represents porosity; z represents vertical distance; r m represents the pore radius; r represents the radial distance from the center of the pore to a certain point; ω represents the first-order mass transfer coefficient between the mobile zone and the immobile zone;

[0109] R is the retardation factor, which meets the condition:

[0110] ;

[0111] ;

[0112] The subscripts m, im and a are the mobile zone of the pore, the immobile zone of the pore and the matrix respectively; D is the longitudinal dispersion coefficient of the pore, which is expressed as:

[0113] (4)

[0114] Where, α m represents the longitudinal dispersion of the pore; D0 is the effective molecular diffusion coefficient in the pore;

[0115] For the reactive solute transport model of filled large pore, the concentration between the pore and the matrix is continuous, which is expressed as:

[0116] (5)

[0117] The initial concentration of the pore-matrix is expressed as:

[0118] (6)

[0119] The initial condition is expressed as:

[0120] (7)

[0121] (8)

[0122] The boundary condition for the solute concentration is considered to be a pulse injection boundary condition:

[0123] (9)

[0124] where t0is the pulse injection time;

[0125] The solution of the model is found by taking the Laplace transform of equations (1), (2), and (3) with respect to t, where is a function of z and s is the argument, to obtain:

[0126] (10)

[0127] (11)

[0128] (12)

[0129] where s is the Laplace transform argument; , and denote the Laplace transform of , and respectively; the boundary condition (9) is Laplace transformed to obtain:

[0130] (13)

[0131] (14)

[0132] (15)

[0133] From equation (11), and the relationship is expressed as:

[0134] (16)

[0135] Substituting equation (16) into equation (10) gives the following equation:

[0136] (17)

[0137] where, The general solution of equation (12) is:

[0138] (18)

[0139] where: K0() is the second kind zero-order modified Bessel function, the coefficients A, B can be determined by the boundary condition equation (15):

[0140] (19)

[0141] Therefore, equation (18) is:

[0142] (20)

[0143] Substituting the boundary condition equation (14) into equation (20) gives:

[0144] (21)

[0145] Substituting equation (21) into equation (20) gives:

[0146] (22)

[0147] Substituting equation (22) into equation (17) gives:

[0148] (23)

[0149] (24)

[0150] (25)

[0151] Substituting the pulse injection boundary condition (9) into the Laplace transform gives:

[0152] (26)

[0153] The solution of equation (24) is:

[0154] (27)

[0155] (28);

[0156] The formula (24) obtained by the above derivation is an analytical solution of the large-pore reactive solute transport model in the Laplace domain. Since the analytical solution contains special functions such as Bessel functions, the Laplace numerical inverse transformation method is used to solve it. To this end, the MIM model in the real domain space can be obtained by using the numerical inverse transformation method of F.R.D. Hoog et al.

[0157] According to the measured experimental data, the soil column experimental data are selected here. The large-pore soil column used in the experiment is 12.1 cm long, the large-pore radius is 0.422 cm, and the matrix region radius is 1.05 cm. The analytical model proposed in step 2 is used for calculation, that is, the model input is initialized.

[0158] As shown in Figure 3 The analytical solution of the large-pore reactive solute transport obtained by the analytical model is fitted and wired with the breakthrough curve of the organic compound of the experimental data. The solution of the model is well matched with the solution obtained from the measured experimental data, and can accurately reflect the solute transport process in the large-pore.

[0159] The theoretical breakthrough curve generated by the pulse injection boundary condition is least squares fitted with the experimental data TCE organic compound, and the pore region dispersion coefficient Dm = 0.05 cm²·min-1, the matrix molecular diffusion coefficient Da = 1.092×10-5 cm²·min-1 and the mass transfer coefficient 0.001 1 / mins are inversely calculated.

[0160] A large-pore reactive solute transport estimation system for implementing the large-pore reactive solute transport estimation method as described above, as shown in Figure 4 The system comprises a model construction unit 10, a data acquisition unit 11 and a fitting and wiring unit 12.

[0161] The model construction unit is used to establish a large-pore reactive solute transport model considering the joint influence of movable area, immovable area and matrix area, and to solve the semi-analytical solution of the large-pore reactive solute transport model in the Laplace domain under the pulse boundary condition.

[0162] The data acquisition unit is used to obtain measured experimental data and to obtain the breakthrough curve of the organic compound based on the measured experimental data.

[0163] The fitting and wiring unit is used to fit and wire the obtained semi-analytical solution with the breakthrough curve of the organic compound, and to select the parameter value corresponding to the fitting curve meeting the condition as the final value of parameter inversion.

[0164] The application is based on a solving method of an analytical model, according to existing measured experimental data, the analytical model is used to calculate, the result shows that the solution of the model can be well matched with the experimental data, the measured result is connected with the theoretical result, the utilization value of the existing data is fully mined, and the model can accurately reflect the large-pore solute migration process.

[0165] It should be understood that, for those skilled in the art, all the improvements and changes made according to the above description should belong to the protection scope of the appended claims of the application.

Claims

1. A method for estimating the transport of a reactive solute in a large pore medium, the method comprising: The method comprises the following steps: A filling macropore reactive solute transport model considering the joint influence of movable area, immovable area and matrix area is established, and a semi-analytical solution of the filling macropore reactive solute transport model in Laplace domain under pulse boundary conditions is solved; Obtaining measured experimental data, and obtaining the breakthrough curve of the organic compound based on the measured experimental data; The semi-analytical solution obtained is fitted and wired with the breakthrough curve of the organic compound, and the parameter value corresponding to the optimal fitting curve is selected as the final value of parameter inversion; The selected parameters are: the pore region diffusion coefficient D m , the matrix molecule diffusion coefficient D a , and the first order mass transfer coefficient ω between the mobile and immobile regions; The establishment of the filling macropore reactive solute transport model considering the joint influence of movable area, immovable area and matrix area, and the solving of the semi-analytical solution of the filling macropore reactive solute transport model in Laplace domain under pulse boundary conditions comprise: The solute transport process is described by the convection-dispersion equation, and the mathematical equation of the reactive solute transport is expressed as: The solute transport process in the movable area: （1) The solute transport process in the immovable area: （2) The solute transport process in the matrix: （3) where C m and C im represent the solute concentrations in the mobile and immobile regions of the pore, respectively; C a represents the solute concentration in the matrix region; t represents time; v m represents the groundwater flow velocity in the mobile region; λ represents the first-order reaction rate constant; θ represents the porosity; z represents the vertical distance; r m represents the pore radius; r represents the radial distance from the center of the pore to a point; and ω represents the first-order mass transfer coefficient between the mobile and immobile regions. R is a retardation factor, and meets the condition: ； ； The subscripts m, im and a are the movable area of the pore, the immovable area of the pore and the matrix respectively; （4) where a m represents the longitudinal diffusivity of the pores; D0is the effective molecular diffusion coefficient in the pores; For the filling macropore reactive solute transport model, the concentration between the pore and the matrix is continuous, and is expressed as: （5) The initial concentration of the pore-matrix is expressed as: （6) The initial condition is expressed as: （7) （8) The solute concentration boundary condition is considered as a pulse injection boundary condition: （9) In the formula, t0 is the pulse injection time; Solve the model: take the Laplace transform of both sides of equations (1), (2) and (3) with respect to t, where is a function of z, to obtain: （10) （11) （12) where s is the Laplace transform variable; , and denotes the Laplace transform of , and ; the boundary condition equation (9) is transformed into （13) The boundary condition formula (5) and the formula (6) are subjected to Laplace transformation to obtain: （14) （15) According to the relationship of formula (11), and is expressed as: （16) The formula (16) is substituted into the formula (10) to obtain the following equation: （17) In the formula, The general solution of equation (12) is: （18) In the formula, K0() is a second-order modified Bessel function of the first kind, and the coefficients A and B are determined by the boundary condition formula (15): （19) Therefore, the formula (18) is: （20) The boundary condition formula (14) is substituted into the formula (20) to obtain: （21) The formula (21) is substituted into the formula (20) to obtain: （22) The formula (22) is substituted into the formula (17) to obtain: （23) K1() is a second-order modified Bessel function of the first kind; （24) （25) The pulse injection boundary condition formula (9) is subjected to Laplace transformation to obtain: （26) When the initial condition and the boundary condition are pulse injection solutes, the solution of the formula (24) is: （27) （28)。 2. The method of estimating large-pore reactive solute transport by filling according to claim 1, wherein, The filling macropore reactive solute transport model is established based on the following basis: The filling pore is generalized as a movable area and an immovable area and linear adsorption occurs; The model water flow is stable flow, and is Darcy water flow; The pore radius does not change with the change of the pore length; The matrix is semi-infinite in length; The pore and the matrix area are both first-order irreversible reactions; The influence of the convection-dispersion effect is considered in the movable area, and the radial diffusion effect of the solute is considered in the matrix area; A geometric model is established with the pore center as the origin.

3. The method of estimating large-pore reactive solute transport by filling according to claim 1, wherein, The measured experimental data are selected from soil column experimental data.

4. A system for estimating large-pore reactive solute transport, for implementing the method for estimating large-pore reactive solute transport according to any one of claims 1 to 3, characterized in that, The system comprises a model construction unit, a data acquisition unit and a fitting and wiring unit; The model construction unit is used to establish a filling macropore reactive solute transport model considering the joint influence of movable area, immovable area and matrix area, and to solve a semi-analytical solution of the filling macropore reactive solute transport model in Laplace domain under pulse boundary conditions. The data acquisition unit is configured to acquire measured experimental data and obtain a breakthrough curve of the organic compound based on the measured experimental data. The fitting wiring unit is configured to fit and wire the obtained semi-analytical solution with the breakthrough curve of the organic compound, and select a parameter value corresponding to an optimal fitting curve as a final value of parameter inversion.

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