Aquifer hydraulic diffusion coefficient solving method based on semi-analytical collocation method

Abstract

The invention discloses an aquifer hydraulic diffusion coefficient solving method based on a semi-analytical collocation method, belongs to the technical field of groundwater seepage analysis, and aims to solve the problems that an existing finite element method is tedious and time-consuming in grid division when processing is complex, and a finite difference method depends on regular grids and is difficult to adapt to irregular boundaries and the like. The method comprises the following steps: constructing a mathematical model of a two-dimensional seepage problem under a polar coordinate system, solving a groundwater seepage control equation through a variable separation method under a space-time coordinate system to obtain a complete primary function, establishing a space-time collocation constraint system, forming a linear algebraic equation set about an unknown coefficient, and solving the linear algebraic equation set. Establishing an objective function taking a hydraulic diffusion coefficient as an unknown parameter, and combining boundary conditions to form a mathematical expression of a nonlinear optimization problem; according to the method, point collocation calculation is directly carried out in a time-space domain through a Trefftz time-space point collocation method, and the calculation process is remarkably simplified. The method is suitable for complex geometric boundary problems, and the application range of numerical simulation is expanded.

Description

Technical Field

[0001] The present invention belongs to the technical field of groundwater seepage analysis, and in particular relates to a method for solving the hydraulic diffusion coefficient of an aquifer based on a semi-analytical collocation method. Background Art

[0002] The hydraulic diffusivity of an aquifer is a key parameter characterizing the flow and solute transport characteristics of groundwater in porous media. Its accurate solution is of great significance to groundwater dynamics research, groundwater resource development and protection, and groundwater pollution prevention and control. For example, in groundwater remediation projects, the hydraulic diffusivity directly influences the prediction of pollutant diffusion range and the design of remediation plans. In oil and gas field development, this parameter is a key basis for optimizing injection and production plans and improving recovery rates. In the geological disposal of nuclear waste, its value is directly related to the risk assessment of radioactive material migration in groundwater. Therefore, efficiently and accurately determining the hydraulic diffusivity of an aquifer is a research hotspot and technical difficulty in the field of hydrogeology.

[0003] Solving the hydraulic diffusivity of an aquifer is essentially a problem involving nonlinear partial differential equations. The governing equations often involve nonlinear source and sink terms, variable permeability, and complex boundary conditions, making direct analytical solutions difficult to obtain using traditional analytical methods. Currently, engineering and scientific research fields primarily rely on traditional numerical methods such as the finite element method (FEM) and the finite difference method (FDM). However, these methods have limitations when dealing with nonlinear problems and complex geometric boundaries. The finite element method (FEM) requires the generation of high-quality meshes, making the meshing process cumbersome and time-consuming for complex geometric problems. Furthermore, in high-dimensional or transient problems, the computational complexity increases dramatically with the number of meshes. The finite difference method (FDM), on the other hand, typically relies on regular meshes and has difficulty handling irregular boundaries. This makes boundary conditions more complex when solving the hydraulic diffusivity of an aquifer. Furthermore, low-order difference schemes suffer from low accuracy, while high-order difference schemes, while improving accuracy, can lead to reduced stability and increased computational difficulty. To address the above problems, we propose a method for solving the hydraulic diffusion coefficient of aquifers based on the semi-analytical collocation method. Summary of the Invention

[0004] The purpose of the present invention is to address the shortcomings of the existing technology and provide a method for solving the hydraulic diffusion coefficient of an aquifer based on a semi-analytical collocation method. This method solves the problems in the existing methods: the finite element method is cumbersome and time-consuming in mesh division when facing complex geometric problems, and the computational complexity of high-dimensional or transient problems increases dramatically with the increase of the grid; the finite difference method relies on regular grids, has difficulty in handling irregular boundaries, and has complex boundary condition processing when solving the hydraulic diffusion coefficient of an aquifer. The present invention provides a method for solving the hydraulic diffusion coefficient of an aquifer based on a semi-analytical collocation method, and combines the Trefftz time-space collocation method with the FTIM iterative method to solve the hydraulic diffusion coefficient problem of an aquifer. This method does not require grid division, avoids the grid dependence of FEM and FDM when calculating complex flow fields, and has higher computational efficiency. In addition, this method shows good adaptability when dealing with irregular boundaries, and provides an efficient and stable calculation solution for solving nonlinear problems of porous media flow.

[0005] The present invention is achieved by providing a method for calculating the hydraulic diffusion coefficient of an aquifer based on a semi-analytical collocation method, the method comprising:

[0006] S10, constructing a mathematical model of a two-dimensional seepage problem in a polar coordinate system, wherein the mathematical model of the two-dimensional seepage problem includes a groundwater seepage control equation and boundary conditions;

[0007] S20, solve the groundwater seepage control equation by separation of variables method in polar coordinate system to obtain complete basis functions;

[0008] S30, the field solution of the two-dimensional seepage problem is approximated by the linear superposition of the complete basis functions, points are arranged on the boundary of the space-time domain, a space-time point allocation constraint system is established, and a linear algebraic equation system with unknown coefficients is formed;

[0009] S40, establishing an objective function with the hydraulic diffusion coefficient as an unknown parameter, and combining the boundary conditions to form a mathematical expression for the nonlinear optimization problem;

[0010] S50, using the fictitious time integration method (FTIM) to deal with the nonlinear optimization problem, iteratively updating the hydraulic diffusion coefficient, and terminating the iteration when the preset convergence criterion is met;

[0011] When constructing a mathematical model for two-dimensional groundwater seepage problems in a polar coordinate system, the groundwater seepage control equation is expressed as:

[0012]

[0013] Where: D is the hydraulic diffusion coefficient, h is the hydraulic head;

[0014] Among them, in the space-time coordinate system, the initial value and boundary value are both boundary conditions, and the initial value is expressed as follows:

[0015] h=h0(r,θ,t=0)(2)

[0016] Where: h0 is the initial water head in the target study area;

[0017] The boundary value expression is as follows:

[0018] h=B D (r,θ,t)(3)

[0019]

[0020] Where: B D (r,θ,t) is the water head boundary condition, B N (r,θ,t) is the flow boundary condition.

[0021] The groundwater seepage control equation is solved by the separation of variables method to obtain a complete basis function, and the linear superposition of the complete basis functions is performed to obtain the general solution of the groundwater seepage control equation, which is expressed as follows:

[0022]

[0023] Where: m, s are the order, A1, A 2q ,A 3q ,A 4p ,A 5pq ,A 6pq ,B1,B 2p ,B 3q ,B 4q ,B 5pq ,B 6pq , C 1p ,C 2p ,C 3pq ,C 4pq ,C 5pq ,C 6pq The unknown coefficients of the control equation are arbitrary constants, I0, K0 are 0-order second-kind Bessel functions, I q ,K q is the qth order second kind Bessel function, J0, Y0 are the 0th order first kind Bessel functions, J q ,Y q is a Bessel function of the first kind of order q. In a simply connected region, the head value is approximated by a positive basis function:

[0024]

[0025] The flow value is approximated by linear superposition of the positive basis function normal:

[0026]

[0027] Where: n x and n y It is the Dharma phase quantity;

[0028] In the space-time coordinate system, the initial value and boundary value are discretized into n1 points and n2 points to satisfy the initial conditions and boundary conditions of the study area as constraints, and the expression of the seepage problem in the target study area is obtained.

[0029] The arrangement of points under boundary conditions, discretization in the space-time domain, establishment of a space-time point constraint system, and formation of a linear algebraic equation system with unknown coefficients include:

[0030] Arrange points on the space-time boundary and bring the coordinates of the initial conditions and boundary conditions configuration points into the linear equation system composed of equations (6) and (7);

[0031]

[0032] Where: T 2(υ) ,T 3(υ) ,...,T 5(υ) is the T-type basis function corresponding to the flow boundary condition;

[0033] Formula (8) can be expressed in the form of the following matrix:

[0034] Aα=B(9)

[0035] Where A is a (n1+n2)×(2s+m+2ms+1) matrix composed of T-type basis, B is a (n1+n2)×1 matrix composed of boundary values ​​and initial values, and the undetermined coefficient α is a (2s+m+2ms+1)×1 matrix;

[0036] Divide the system matrix constructed by the T-type basis function by the left of equation (9) to obtain the coefficient matrix, which can be expressed as α = A -1 B, after finding the coefficient, substitute it into formula (6) to find the value of any internal point;

[0037] The objective function with the hydraulic diffusion coefficient as an unknown parameter is established, and the mathematical expression of the nonlinear optimization problem is formed in combination with the boundary conditions, including:

[0038] In the area where hydraulic diffusion is unknown, a set of linear algebraic equations is formed by the boundary conditions and initial conditions of the target study area. An initial guess value D0 is given and substituted into the linear algebraic equation (8) composed of the boundary conditions. A set of coefficient matrices can be obtained through the linear algebraic equation (8). Since the initial condition time term t = 0, t = 0 is substituted into equation (6) to obtain the linear algebraic equation of the initial condition. The obtained coefficient matrix is ​​substituted into the linear algebraic equation composed of the initial condition, which is expressed as follows:

[0039]

[0040] Where: R0 is the characteristic length, h(r i ,θ i , t=0) is the initial condition, is the D value of the κth iteration operation;

[0041] The method adopts the fictitious time integration method (FTIM) to process the nonlinear optimization problem, iteratively updates the hydraulic diffusion coefficient, and terminates the iteration when the preset convergence criterion is met, including:

[0042] The FTIM iterative calculation format is used to process the nonlinear algebraic equations generated by the spatial discretization of the Trefftz space-time collocation method. The iterative function used is:

[0043]

[0044] The method for iteratively solving the hydraulic diffusion coefficient by using the Trefftz time-space collocation method combined with the quasi-time integration method includes:

[0045] Given an initial guess value D0, a new hydraulic diffusion coefficient is obtained by formula (12): Will Substituting into formula (11), if The iteration ends when the following conditions are met:

[0046]

[0047] Where: ε represents the preset convergence tolerance.

[0048] Compared with the prior art, the embodiments of the present application have the following beneficial effects:

[0049] In the embodiments of the present invention, the Trefftz space-time collocation method is used to perform collocation calculations directly in the space-time domain, avoiding the grid division problem in traditional methods and significantly simplifying the calculation process. At the same time, the global characteristics of the Trefftz basis function are utilized to reduce the amount of calculation and significantly improve the calculation efficiency. The method is suitable for large-scale scientific computing and engineering simulation. The complete basis function accurately satisfies the control equations, and high-precision solutions can be obtained in both local areas and global scopes to meet the requirements of high-precision simulation. No grid division is required when solving the diffusion coefficient. The boundary conditions are directly applied through the collocation method, which simplifies the boundary condition processing and improves the accuracy and efficiency of the boundary condition processing. It is suitable for complex geometric boundary problems and expands the application range of numerical simulation. BRIEF DESCRIPTION OF THE DRAWINGS

[0050] Figure 1 A schematic diagram of space-time distribution in the space-time coordinate system is shown.

[0051] Figure 2 A schematic diagram of the process of iteratively solving the hydraulic diffusion coefficient using the Trefftz space-time collocation method combined with the virtual time integration method is shown.

[0052] Figure 3 The figure shows the relationship between the number of iterations and the hydraulic diffusion coefficient of Example 1 of the present invention.

[0053] Figure 4 The relationship between the number of iterations and the hydraulic diffusion coefficient of Example 2 of the present invention is shown.

[0054] Figure 5 The graph shows the relationship between the absolute error of the initial value analytical solution and the numerical solution and the number of iterations of Example 1 of the present invention.

[0055] Figure 6 The graph shows the relationship between the absolute error and the number of iterations of the initial value analytical solution and the numerical solution of Example 2 of the present invention.

[0056] Figure 7 The figure shows a flow chart of the method for solving the hydraulic diffusion coefficient of aquifers based on the semi-analytical collocation method. DETAILED DESCRIPTION

[0057] Unless otherwise defined, all technical and scientific terms used herein have the same meanings as commonly understood by those skilled in the art to which this application belongs. The terms used in the specification of the application are only for the purpose of describing specific embodiments and are not intended to limit this application. The terms "including" and "having" and any variations thereof in the specification and claims of this application and the above-mentioned drawings are intended to cover non-exclusive inclusions. The terms "first", "second", etc. in the specification and claims of this application or the above-mentioned drawings are used to distinguish different objects, not to describe a specific order.

[0058] References herein to "embodiments" mean that a particular feature, structure, or characteristic described in connection with the embodiments may be included in at least one embodiment of the present application. The appearance of this phrase in various places in the specification does not necessarily refer to the same embodiment, nor does it constitute an independent or alternative embodiment that is mutually exclusive of other embodiments. It is understood, both explicitly and implicitly, by those skilled in the art that the embodiments described herein may be combined with other embodiments.

[0059] Among existing methods, the finite element method (FEM) is cumbersome and time-consuming when faced with complex geometric problems, and the computational complexity of high-dimensional or transient problems increases dramatically with the number of meshes. The finite difference method (FDM) relies on regular meshes, making it difficult to handle irregular boundaries, and the boundary conditions involved in solving the hydraulic diffusion coefficient of an aquifer are complex. To address these issues, we propose a method for solving the hydraulic diffusion coefficient of an aquifer based on a semi-analytical collocation method. Briefly, the method first constructs a mathematical model of a two-dimensional groundwater flow problem in a polar coordinate system. The governing groundwater flow equations are solved using the separation of variables method in this polar coordinate system to obtain complete basis functions. The hydraulic head distribution is represented by a linear combination of these complete basis functions. Collocation points are then placed at the boundary conditions, discretized in the spatiotemporal domain, and a spatiotemporal collocation constraint system is established, forming a system of linear algebraic equations involving unknown coefficients. An objective function is then established with the hydraulic diffusion coefficient as the unknown parameter. Combined with the boundary conditions, this forms a mathematical expression for the nonlinear optimization problem. Finally, the fictitious time integration method (FTIM) is used to solve the nonlinear optimization problem, iteratively updating the hydraulic diffusion coefficient. The iterations terminate when a preset convergence criterion is met. In the embodiments of the present invention, the Trefftz space-time collocation method is used to perform collocation calculations directly in the space-time domain, avoiding the grid division problem in traditional methods and significantly simplifying the calculation process. At the same time, the global characteristics of the Trefftz basis function are utilized to reduce the amount of calculation and significantly improve the calculation efficiency. The method is suitable for large-scale scientific computing and engineering simulation. The complete basis function accurately satisfies the control equations, and high-precision solutions can be obtained in both local areas and global scopes to meet the requirements of high-precision simulation. No grid division is required when solving the diffusion coefficient. The boundary conditions are directly applied through the collocation method, which simplifies the boundary condition processing and improves the accuracy and efficiency of the boundary condition processing. It is suitable for complex geometric boundary problems and expands the application range of numerical simulation.

[0060] The embodiment of the present invention provides a method for solving the hydraulic diffusion coefficient of an aquifer based on a semi-analytical collocation method. Figure 7 A schematic diagram of the implementation process of a method for solving the hydraulic diffusion coefficient of an aquifer based on a semi-analytical collocation method is shown. The method for solving the hydraulic diffusion coefficient of an aquifer based on a semi-analytical collocation method specifically includes:

[0061] S10, constructing a mathematical model of a two-dimensional seepage problem in a polar coordinate system, wherein the mathematical model of the two-dimensional seepage problem includes a groundwater seepage control equation and boundary conditions;

[0062] S20, solve the groundwater seepage control equation by separation of variables method in polar coordinate system to obtain complete basis functions;

[0063] S30, the field solution of the two-dimensional seepage problem is approximated by the linear superposition of the complete basis functions, points are arranged on the boundary of the space-time domain, a space-time point allocation constraint system is established, and a linear algebraic equation system with unknown coefficients is formed;

[0064] S40, a target function with the hydraulic diffusion coefficient as an unknown parameter is established, and a mathematical expression of a nonlinear optimization problem is constituted in combination with a boundary condition;

[0065] S50, a fictitious time integration method (FTIM) is used to process the nonlinear optimization problem, the hydraulic diffusion coefficient is iteratively updated, and the iteration is terminated when a preset convergence criterion is met;

[0066] In the embodiment, the Trefftz space-time collocation method is used to directly perform collocation calculation in the time-space domain, the grid division problem in the traditional method is avoided, the calculation process is significantly simplified, the calculation amount is reduced by using the global characteristics of the complete basis function, the calculation efficiency is significantly improved, the method is suitable for large-scale scientific calculation and engineering simulation, the complete basis function is used to accurately meet the control equation, high-precision solutions can be obtained in local areas and global ranges, the high-precision simulation demand is met, and the grid division is not required when the diffusion coefficient is solved, the boundary condition is directly applied by using the collocation method, the boundary condition processing is simplified, and the accuracy and efficiency of the boundary condition processing are improved. It is suitable for complex geometric boundary problems, and expands the application range of numerical simulation.

[0067] In a further preferred embodiment of the application, when a mathematical model of a two-dimensional seepage problem is constructed in a polar coordinate system, the groundwater seepage control equation is expressed as:

[0068]

[0069] In the formula: D is the hydraulic diffusion coefficient h is the water head;

[0070] In the space-time coordinate system, the initial value and the boundary value are both boundary conditions, and the initial value is expressed as follows:

[0071] h=h0(r,θ,t=0)(2)

[0072] In the formula: h0 is the initial water head in the target research area;

[0073] The boundary value is expressed as follows:

[0074] h=B D (r,θ,t)(3)

[0075]

[0076] In the formula: B D (r,θ,t) is a water head boundary condition, B N (r,θ,t) is a flow boundary condition.

[0077] In this embodiment, the groundwater seepage control equation is solved by the separation of variables method in a polar coordinate system to obtain a complete basis function. The complete basis function is shown in Table 1.

[0078] Table 1

[0079]

[0080]

[0081] The linear combination of the complete basis functions is used to obtain the general solution of the groundwater seepage control equation, which is expressed as follows:

[0082]

[0083] Where: m, s are the order, A1, A 2q ,A 3q ,A 4p ,A 5pq ,A 6pq ,B1,B 2p ,B 3q ,B 4q ,B 5pq ,B 6pq , C 1p ,C 2p ,C 3pq ,C 4pq ,C 5pq ,C 6pq The unknown coefficients of the control equation are arbitrary constants, I0, K0 are 0-order second-kind Bessel functions, I q ,K q is the qth order second kind Bessel function, J0, Y0 are the 0th order first kind Bessel functions, J q ,Y q is a Bessel function of the first kind of order q. In a simply connected region, the head value is approximated by a positive basis function:

[0084]

[0085] The flow value is approximated by linear superposition of the positive basis function normal:

[0086]

[0087] Where: n x and n y It is the Dharma phase quantity;

[0088] In the space-time coordinate system, the initial value and boundary value are discretized into n1 points and n2 points. Figure 1A schematic diagram of the study area and the time-space distribution in the time-space coordinate system is shown. The expression of the seepage problem in the target study area is obtained by satisfying the initial conditions and boundary conditions of the study area.

[0089] The method for obtaining the expression of the seepage problem in the target study area under the constraints of satisfying the initial conditions and boundary conditions of the study area includes:

[0090] First, points are arranged on the boundary of space-time, and the coordinates of the initial conditions and boundary conditions are substituted into the linear equation system composed of equations (6) and (7);

[0091]

[0092] Where: T 2(υ) ,T 3(υ) ,...,T 5(υ) is the T-type basis function corresponding to the flow boundary condition (is the normal derivative of the T-type basis function corresponding to the head boundary condition);

[0093] Formula (8) can be expressed in the form of the following matrix:

[0094] Aα=B(9)

[0095] Where A is a (n1+n2)×(2s+m+2ms+1) matrix composed of T-type basis, B is a (n1+n2)×1 matrix composed of boundary values ​​and initial values, and the undetermined coefficient α is a (2s+m+2ms+1)×1 matrix;

[0096] Divide the system matrix constructed by the T-type basis function by the left of equation (9) to obtain the coefficient matrix, which can be expressed as α = A -1 B, after finding the coefficient, substitute it into formula (6) to find the value of any internal point;

[0097] In the area where hydraulic diffusion is unknown, a set of linear algebraic equations is formed by the boundary conditions and initial conditions of the target study area. Given an initial guess value D0, it is substituted into the linear algebraic equation (8) composed of the boundary conditions. A set of coefficient matrices can be obtained through the linear algebraic equation (8). Since the initial condition time term t = 0, t = 0 is substituted into the linear algebraic equation obtained by equation (6), and the obtained coefficient matrix is ​​substituted into the linear algebraic equation composed of the initial conditions, which is expressed as follows:

[0098]

[0099] Where: R0 is the characteristic length, h(r i ,θ i , t=0) is the initial condition, is the D value of the κth iteration operation;

[0100] The FTIM iterative calculation format is used to process the nonlinear algebraic equations generated by the spatial discretization of the Trefftz space-time collocation method. The iterative function used is:

[0101]

[0102] In a further preferred embodiment of the present invention, Figure 2 , which shows a schematic flow chart of iteratively solving the hydraulic diffusion coefficient by the Trefftz space-time collocation method combined with the quasi-time integration method. The method of iteratively solving the hydraulic diffusion coefficient by the Trefftz space-time collocation method combined with the quasi-time integration method includes:

[0103] First, an initial guess value D0 is given, and then a new hydraulic diffusion coefficient is obtained by formula (12): Will Substituting into formula (11), if The iteration ends when the following conditions are met:

[0104]

[0105] Where: ε represents the preset convergence tolerance;

[0106] In this embodiment, for the nonlinear algebraic equations in the inversion of hydraulic diffusion coefficient, the FTIM iterative method combined with the pseudo-time integration technology is used to transform the nonlinear problem into a linearized iterative process. Compared with traditional iterative strategies such as the Newton method, the number of computational iterations and storage overhead are significantly reduced.

[0107] Example 1

[0108] In a further preferred embodiment of the present invention, the initial conditions and boundary conditions are known, and the hydraulic diffusion coefficient D is solved. The control equation is the groundwater seepage control equation (1), and the target research area diagram and the point distribution method are as follows: Figure 1 As shown, the expression is as follows:

[0109]

[0110] The initial condition expressions are as follows:

[0111] h=sin(rcosθ)sin(rsinθ)(15)

[0112] The boundary conditions are expressed as follows:

[0113] h(r,θ,t)=sin(rcosθ)sin(rsinθ)e -2Dt (16)

[0114]

[0115] Where D = 0.1

[0116] The analytical solution is shown as follows:

[0117] h=sin(rcosθ)sin(rsinθ)e -2Dt (18)

[0118] Example 2

[0119] In a further preferred embodiment of the present invention, the initial conditions and boundary conditions are known, and the hydraulic diffusion coefficient D is solved. The control equation is the groundwater seepage control equation (1), and the target research area diagram and the point distribution method are as follows: Figure 1 As shown, the expression is as follows:

[0120]

[0121] The initial condition expressions are as follows:

[0122] h=sin(rcosθ)sin(rsinθ)(20)

[0123] The boundary conditions are expressed as follows:

[0124] h(r,θ,t)=sin(rcosθ)sin(rsinθ)e -2Dt (twenty one)

[0125]

[0126] Where D = 0.01;

[0127] The analytical solution is shown as follows:

[0128] h(r,θ,t)=sin(rcosθ)sin(rsinθ)e -2Dt (twenty three)

[0129] In this embodiment, the initial hydraulic diffusion coefficient is given as D0=10 5 , the number of boundary points is 980, the order is 15, and the hydraulic diffusion coefficient D of the study area is solved by knowing the initial value and boundary value. The present invention solves the nonlinear algebraic problem by the FTIM method, and the solution process of calculating the hydraulic diffusion coefficient is as follows: Figure 1 As the number of iterative operations increases, D0 gradually approaches the correct hydraulic diffusion coefficient D. The relationship between the hydraulic diffusion coefficient and the number of iterative operations is as follows: Figure 3 and Figure 4 As shown, Figure 3 The relationship between the number of iterations and the hydraulic diffusion coefficient in Example 1 of the present invention is shown. Figure 4 The relationship between the number of iterations and the hydraulic diffusion coefficient of Example 2 of the present invention is shown. Figure 3 and Figure 4 It can be seen that when Example 1 is iterated about 115 times and Example 2 is iterated about 30 times, D0 is approximately equal to D. The absolute error between the analytical solution and the numerical solution of the initial value of the target research area and the number of iterations is as follows: Figure 5 and Figure 6 As shown, Figure 5 The graph showing the relationship between the absolute error and the number of iterations of the initial value analytical solution and the numerical solution of Example 1 of the present invention is shown. Figure 6 The absolute error and the number of iterations of the analytical solution and the numerical solution of the initial value of Example 2 of the present invention are shown. Figure 5 and Figure 6 It can be seen that the D0 obtained by FTIM is brought into the Trefftz time-space collocation method to solve the initial value of the study area. As D0 gradually approaches the correct hydraulic diffusion coefficient D, the absolute error between the analytical solution and the numerical solution reaches 10 in Example 1 and Example 2. -10 and 10 -8 This shows that the numerical method proposed in this invention has high accuracy in estimating seepage hydraulic diffusion.

[0130] In summary, the present invention provides a method for solving the hydraulic diffusion coefficient of an aquifer based on a semi-analytical point matching method. In an embodiment of the present invention, the Trefftz time-space point matching method is used to directly perform point matching calculations in the time-space domain, avoiding the grid division problem in traditional methods and significantly simplifying the calculation process. At the same time, the global characteristics of the Trefftz basis function are utilized to reduce the amount of calculation and significantly improve the calculation efficiency. The method is suitable for large-scale scientific computing and engineering simulation. The control equation is accurately satisfied by the complete basis function, and high-precision solutions can be obtained in both local areas and global ranges to meet the requirements of high-precision simulation. No grid division is required when solving the diffusion coefficient. The boundary conditions are directly applied through the point matching method, which simplifies the boundary condition processing and improves the accuracy and efficiency of the boundary condition processing. The method is suitable for complex geometric boundary problems and expands the application scope of numerical simulation.

[0131] It should be noted that for the aforementioned embodiments, for simplicity of description, they are all expressed as a series of action combinations. However, those skilled in the art should be aware that the present invention is not limited by the order of the actions described, because according to the present invention, certain steps may be performed in other orders or simultaneously. Secondly, those skilled in the art should also be aware that the embodiments described in this specification are all preferred embodiments, and the actions and modules involved are not necessarily required by the present invention.

[0132] The above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit the scope of protection of the invention. Obviously, the embodiments described are only some embodiments of the present invention, rather than all embodiments. Based on these embodiments, all other embodiments obtained by ordinary technicians in this field without making creative work are within the scope of protection of the present invention. Although the present invention has been described in detail with reference to the above embodiments, ordinary technicians in this field can still combine, add, delete or make other adjustments to the features in the various embodiments of the present invention according to the circumstances without conflict, without making creative work, so as to obtain different other technical solutions that do not deviate from the concept of the present invention in essence, and these technical solutions also fall within the scope of protection of the present invention.

Claims

1. A method for calculating the hydraulic diffusion coefficient of an aquifer based on a semi-analytical collocation method, characterized in that: The method comprises: S10, constructing a mathematical model of a two-dimensional seepage problem in a polar coordinate system, wherein the mathematical model of the two-dimensional seepage problem includes a groundwater seepage control equation and boundary conditions; S20, solve the groundwater seepage control equation by separation of variables method in polar coordinate system to obtain complete basis functions; S30, the field solution of the two-dimensional seepage problem is approximated by the linear superposition of the complete basis functions, points are arranged on the boundary of the space-time domain, a space-time point allocation constraint system is established, and a linear algebraic equation system with unknown coefficients is formed; S40, establishing an objective function with the hydraulic diffusion coefficient as an unknown parameter, and combining the boundary conditions to form a mathematical expression for the nonlinear optimization problem; S50, using the virtual time integration method FTIM to deal with the nonlinear optimization problem, iteratively updating the hydraulic diffusion coefficient, and terminating the iteration when the preset convergence criterion is met.

2. The method for calculating the hydraulic diffusion coefficient of an aquifer based on the semi-analytical collocation method according to claim 1, wherein: When constructing a mathematical model of a two-dimensional seepage problem in a polar coordinate system, the groundwater seepage control equation is expressed as: Where: D is the hydraulic diffusion coefficient, h is the hydraulic head; Among them, in the space-time coordinate system, the initial value and boundary value are both boundary conditions. The initial value is expressed as follows: h=h0(r,θ,t=0)(2) Where: h0 is the initial water head in the target study area; The boundary value expression is as follows: h=B D (r,θ,t)(3) Where: B D (r,θ,t) is the water head boundary condition, B N (r,θ,t) is the flow boundary condition.

3. The method for calculating the hydraulic diffusion coefficient of an aquifer based on the semi-analytical collocation method according to claim 2, wherein: The groundwater seepage control equation is solved by the separation of variables method to obtain the complete basis function. The linear combination of the complete basis function is used to obtain the general solution of the groundwater seepage control equation, which is expressed as follows: Where: m, s are the order, A1, A 2q ,A 3q ,A 4p ,A 5pq ,A 6pq ,B1,B 2p ,B 3q ,B 4q ,B 5pq ,B 6pq , C 1p ,C 2p ,C 3pq ,C 4pq ,C 5pq ,C 6pq The unknown coefficients of the control equation are arbitrary constants, I0, K0 are 0-order second-kind Bessel functions, I q ,K q is the qth order second kind Bessel function, J0, Y0 are the 0th order first kind Bessel functions, J q ,Y q is a Bessel function of the first kind of order q. In a simply connected region, the head value is approximated by a positive basis function: The flow value is approximated by linear superposition of the positive basis function normal: Where: n x and n y It is the Dharma phase quantity; In the space-time coordinate system, the initial value and boundary value are discretized into n1 points and n2 points to satisfy the initial conditions and boundary conditions of the study area as constraints, and the expression of the seepage problem in the target study area is obtained.

4. The method for calculating the hydraulic diffusion coefficient of an aquifer based on the semi-analytical collocation method according to claim 3, wherein: Arrange points under boundary conditions, discretize in the space-time domain, establish a space-time point constraint system, and form a linear algebraic equation system with unknown coefficients, including: Arrange points on the space-time boundary and bring the coordinates of the initial conditions and boundary conditions configuration points into the linear equation system composed of equations (6) and (7); Where: T 2(υ) ,T 3(υ) ,...,T 5(υ) is the T-type basis function corresponding to the flow boundary condition; Formula (8) is expressed in the form of the following matrix: Aα=B (9) Where A is a (n1+n2)×(2s+m+2ms+1) matrix composed of T-type basis, B is a (n1+n2)×1 matrix composed of boundary values ​​and initial values, and the undetermined coefficient α is a (2s+m+2ms+1)×1 matrix; Divide the system matrix constructed by the T-type basis function by the left of equation (9) to obtain the coefficient matrix, which can be expressed as α = A -1 B, after finding the coefficient, substitute it into formula (6) to find the value of any internal point.

5. The method for calculating the hydraulic diffusion coefficient of an aquifer based on the semi-analytical collocation method according to claim 4, wherein: Establish an objective function with the hydraulic diffusion coefficient as the unknown parameter and combine it with the boundary conditions to form a mathematical expression for the nonlinear optimization problem, including: In the area where hydraulic diffusion is unknown, a set of linear algebraic equations is formed by the boundary conditions and initial conditions of the target study area. Given an initial guess value D0, it is substituted into the linear algebraic equation (8) composed of the boundary conditions. A set of coefficient matrices can be obtained through the linear algebraic equation (8). Since the initial condition time term t = 0, t = 0 is substituted into equation (6) to obtain the linear algebraic equation of the initial condition. The obtained coefficient matrix is ​​substituted into the linear algebraic equation composed of the initial condition, which is expressed as follows: Where: R0 is the characteristic length, h(r i ,θ i , t=0) is the initial condition, is the D value of the κth iteration operation.

6. The method for calculating the hydraulic diffusion coefficient of an aquifer based on the semi-analytical collocation method according to claim 5, wherein: The virtual time integration method FTIM is used to deal with nonlinear optimization problems, iteratively update the hydraulic diffusion coefficient, and terminate the iteration when the preset convergence standard is met. include: Among them, the FTIM iterative calculation format is used to process the nonlinear algebraic equations generated by the spatial discretization of the Trefftz space-time collocation method. The iterative function used is: Given an initial guess value D0, a new hydraulic diffusion coefficient is obtained by formula (12): Will Substituting into formula (11), if The iteration ends when the following conditions are met: Where: ε represents the preset convergence tolerance.

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