A three-dimensional numerical simulation method of magnetic induction intensity in time domain electromagnetic method

Abstract

The present application relates to a kind of magnetic induction intensity three-dimensional numerical simulation method in time domain electromagnetic method, by deducing the magnetic potential control equation of static magnetic time, the relationship between residual magnetization and induced magnetization is established, the three-dimensional static magnetic field when the magnetic target body exists is calculated.Initial magnetic field, initial induced magnetization and induced electromotive force are loaded into double spin induced magnetization diffusion equation, and the properties of magnetic target body after transient electromagnetic excitation are characterized by magnetic susceptibility Cole-Cole model, after rational function approximation in time domain, three-dimensional numerical simulation of transient magnetic field in time domain containing source is realized by finite difference method and convolution recursive algorithm.The present application aims to overcome the current research method can only carry out three-dimensional numerical simulation to the induced electromotive force of time-varying magnetic field, by establishing the relationship between magnetic method and time domain electromagnetic method under magnetic environment, three-dimensional numerical simulation of magnetic induction intensity in time domain electromagnetic method is realized.

Description

Technical Field

[0001] This invention relates to a three-dimensional numerical simulation method for magnetic induction intensity in the time-domain electromagnetic method in the field of geophysical exploration. Background Technology

[0002] The Time-Domain Electromagnetic Method (AEM) has advantages such as speed, efficiency, and wide detection range, and has been widely used in various fields such as mineral resource exploration, environmental monitoring, and basic geological surveys (Lei, 2006; Zhang, 2005). In the process of detecting magnetic environments using the transient electromagnetic method, the on-time phase of the emitted current magnetizes underground magnetic particles; after the current is turned off, the magnetic particles lose their magnetization, and the receiving system observes a decaying signal with an approximately -1 power rate. This process is called the induction-magnetization effect. Furthermore, with the increasing popularity of superconducting time-domain electromagnetic methods, longer time windows and higher precision magnetic field time-domain electromagnetic data are attracting growing attention. Currently, three-dimensional forward modeling using time-domain electromagnetic methods is limited to simulating the induced electromotive force of the magnetic field. However, processing superconducting magnetic field data by taking partial derivatives not only results in the loss of some magnetic field information but also introduces errors due to time-point discrepancies. Therefore, achieving direct three-dimensional numerical simulation of the magnetic field in time-domain electromagnetic methods is crucial for understanding the characteristics of magnetic field data and performing magnetic field inversion using time-domain electromagnetic methods. Magnetic detection is also a widely used method, effectively identifying and detecting magnetic targets in static magnetic environments. Therefore, introducing forward modeling of magnetic methods can accurately determine the initial values ​​of the static magnetic field in magnetic environments, facilitating the numerical simulation of magnetic induction intensity using time-domain electromagnetic methods.

[0003] Chinese patent CN113887106A discloses a three-dimensional numerical simulation method for the induction-magnetization effect based on the Chikazumi model. Based on the frequency domain finite-difference method, it introduces the Chikazumi magnetic susceptibility model to re-establish the scientific mapping matrix between permeability and magnetic field response, achieving a three-dimensional solution for the induction-magnetization effect using the frequency domain finite-difference method. Furthermore, it utilizes a sine-cosine transform numerical filtering algorithm to transform the frequency domain results to the time domain. This method can achieve three-dimensional numerical simulation of the induction-magnetization effect; however, the model is relatively simple, and it can only calculate the response of the magnetic field-induced electromotive force under a step waveform.

[0004] Chinese patent CN113779853A discloses a fractional-order three-dimensional numerical simulation method for time-domain electromagnetic induction-magnetization effects. By introducing the Cole-Cole fractional-order magnetic susceptibility model into the passive Maxwell equations, and employing a rational function approximation algorithm and a convolution recursion method, a fractional-order three-dimensional numerical simulation of the time-domain electromagnetic induction-magnetization effect is achieved. This method can realize the three-dimensional numerical simulation of fractional-order induction-magnetization effects in the time domain, but it cannot perform source-containing calculations and can only calculate the electromotive force induced by the magnetic field.

[0005] Chinese patent CN115935746A discloses a three-dimensional numerical simulation method for time-domain induction-magnetization-polarization effects. By deriving the source-diffusion equation for induction-magnetization-polarization and introducing the Cole-Cole model for magnetic susceptibility and the Cole-Cole model for electrical conductivity, a three-dimensional source-containing numerical simulation of the time-domain electromagnetic induction-magnetization-polarization effect is achieved. This method can realize three-dimensional numerical simulation of the magnetic field-induced electromotive force of the fractional-order induction-magnetization-polarization effect under different waveforms in the time domain.

[0006] The three-dimensional numerical simulation method for the induced electromotive force of the time-domain magnetic field induction effect disclosed above has not yet been addressed in domestic and foreign patents for direct three-dimensional numerical simulation and source calculation of the magnetic field in the time-domain electromagnetic method. Therefore, this invention performs three-dimensional numerical simulation based on the diffusion equation of the double-curl induced magnetization intensity. On the basis of the existing three-dimensional numerical simulation of induced electromotive force, it enters the forward modeling of the magnetic method and derives a new form of the diffusion equation of the double-curl induced magnetization intensity. Based on the finite difference method and convolution recursive algorithm, it realizes the direct three-dimensional numerical simulation of the time-varying magnetic induction intensity in the time-domain electromagnetic method under magnetic environment. Summary of the Invention

[0007] The technical problem to be solved by this invention is to provide a three-dimensional numerical simulation method for magnetic induction intensity in the time-domain electromagnetic method. By deriving the magnetic potential control equation under static magnetization, the relationship between remanent magnetization and induced magnetization is established, and the three-dimensional static magnetic field in the presence of a magnetic target is calculated. The initial magnetic field, initial induced magnetization, and induced electromotive force are loaded into the double-curvature induced magnetization diffusion equation, and the properties of the magnetic target after transient electromagnetic excitation are characterized by the Cole-Cole model of magnetic susceptibility. After approximating the model with rational functions in the time domain, the three-dimensional numerical simulation of the transient magnetic field in the time-domain with a source is finally achieved through the finite difference method and convolutional recursive algorithm. This invention is implemented as follows: A three-dimensional numerical simulation method for magnetic induction intensity in the time-domain electromagnetic method.

[0008] Includes the following steps:

[0009] 1) Derive the equations containing remanent magnetization M using Maxwell's equations for static magnetic fields and the governing equations for magnetization M.r The scalar magnetic potential control equation, and through the magnetic scalar V m Solve for the magnetic flux density component B of the static magnetic field. x B y B z ;

[0010] 2) The Cole-Cole model of magnetic susceptibility is introduced to characterize the change process of the magnetized medium under transient electromagnetic excitation conditions. The Cole-Cole model of magnetic susceptibility is approximated in the time domain by the multiple zero-pole construction approximation method.

[0011] 3) Using Maxwell's equations with a source, combined with the induced magnetization M i The governing equations were derived by taking double curl on both sides of the equations, and the source magnetic field diffusion equation containing double curl magnetization was derived as the governing equations for the time-varying magnetic field solution stage.

[0012] 4) Establishing a magnetic induction electromotive force With induced magnetization M i The iterative relationship is used to load the calculated magnetic induced electromotive force, and the induced magnetization M at the next time step is calculated using a convolution recursive operation method. i Perform calculations;

[0013] 5) Regarding magnetic susceptibility χ and magnetic induction electromotive force The convolution operation is substituted into the double curl magnetization diffusion equation. The computational domain is divided using a non-uniform three-dimensional mesh. Based on the finite difference method, the differential iteration format of the governing equation is derived for the entire computational domain.

[0014] 6) The governing equations are rearranged into the form Ax = b, and the double conjugate gradient method is used to solve the governing equations, thereby realizing the discrete solution operation of the time-domain magnetic induction intensity B.

[0015] 7) A variable time step iteration method is adopted to speed up the calculation time. After the calculation is completed, the response of each component of magnetic induction intensity B is extracted, and the calculation results are plotted and analyzed.

[0016] In step 1), the Maxwell equations for the static magnetic field and the governing equations for the magnetization M are expressed as follows:

[0017]

[0018] In equation (1), E represents the electric field strength, B represents the magnetic induction intensity, H represents the magnetic field strength, and M represents the magnetization intensity. i M represents the induced magnetization intensity. r σ represents the remanent magnetization, μ0 represents the conductivity, and μ0 represents the free permeability.

[0019] Combining magnetic field strength H and induced magnetization M i Based on the relationship, by deriving equation (1), we can obtain the solution for the magnetic scalar potential V. m The governing equations:

[0020]

[0021] Where χ is the magnetic susceptibility of the magnetic medium, and further, the initial magnetic flux density component B of the static magnetic field can be obtained by solving the divergence of the magnetic potential. x B y B z ;

[0022] The Cole-Cole model for magnetic susceptibility in step 2) is:

[0023]

[0024] Where x0 is the zero-frequency magnetic susceptibility, α is the fractional order; and through the induced magnetization M i From the governing equations, the formula for the rate of change of magnetization, characterized by the magnetic induction electromotive force, can be derived as follows:

[0025]

[0026] And by approximating it in the time domain, it can be expressed as:

[0027]

[0028] Where r1, r2…r m p1, p2...p m These are the rational function approximation coefficients of the Cole-Cole model of magnetic susceptibility, where m is the order of approximation. This formula can be used to characterize the change in induced magnetization in the time domain.

[0029] The governing equations in step 3) can be derived using Maxwell's equations with a source and the governing equations for induced magnetization:

[0030]

[0031] Where Δt represents the time step, B n+1 Let B be the magnetic field strength at time n+1. n Let H be the magnetic flux density at time n. n+1 E n+1 J sn+1 M in+1 Let n be the magnetic field strength, electric field strength, current strength, and induced magnetization at time n+1, respectively; thus, the diffusion equation based on the induced magnetization with double curl can be derived as follows:

[0032]

[0033] Furthermore, we can establish the magnetic induced electromotive force in step 4) using a convolutional recursive method. The induced magnetization intensity M at the previous moment in The iterative relationship between them

[0034]

[0035] Among them, G in K is the magnetic field convolution term, which can be written as:

[0036]

[0037]

[0038] ψ1、ψ2…ψ m The time-domain expressions representing the magnetic susceptibility and the magnetomotive force are respectively represented by the components of the expression. The convolution can be written in the form related to the convolution term of the previous time step as follows:

[0039]

[0040] And the induced magnetization intensity M i The initial value can be characterized as the zero-frequency induced magnetization, that is:

[0041]

[0042] Based on the finite difference method, the iterative format of the governing equation for magnetic induction B in step 5) for each component can be obtained as follows:

[0043]

[0044]

[0045] After solving the equations using the double conjugate gradient method, the components B of the time-varying magnetic induction intensity B in the time-domain electromagnetic method can be obtained. x B y B z Accurate response.

[0046] Compared with the prior art, the beneficial effects of this invention are as follows: it can overcome the current research methods that can only perform three-dimensional numerical simulation of the induced electromotive force of time-varying magnetic fields. By establishing the connection between the magnetic method and the time-domain electromagnetic method under magnetic conditions, it realizes the three-dimensional numerical simulation of magnetic induction intensity in the time-domain electromagnetic method. Attached Figure Description

[0047] Figure 1This is a schematic diagram of a three-dimensional numerical simulation method for magnetic induction intensity in the time-domain electromagnetic method;

[0048] Figure 2 This is a comparison of the numerical simulation results of the three-dimensional static magnetic field with the analytical solution response and error in the presence of a magnetic target;

[0049] Figure 3 This is a comparison of the response and error of three-dimensional magnetic field numerical simulation and analytical solution of magnetic field under a uniform half-space model.

[0050] Figure 4 This is a comparison of the response and error of the three-dimensional magnetic field numerical simulation after differentiation with the magnetization semi-analytical solution of the induced electromotive force under the half-space magnetization model.

[0051] Figure 5 This is a horizontal slice diagram of a three-dimensional magnetic field numerical simulation under a half-space magnetization model; Detailed Implementation

[0052] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0053] Example

[0054] See Figure 1 A three-dimensional numerical simulation method for magnetic induction intensity in the time-domain electromagnetic method includes:

[0055] 1) Set grid parameters and apply them to the calculation of the initial static magnetic field and the time-varying magnetic field. Combine the finite difference method to construct a grid difference scheme with a grid number of 29×29×29, where there are 29 grids in the x and y directions and 29 grids in the z direction. The scaling factor of adjacent grids is 2, and the minimum and maximum grid step sizes are 10m and 2560m, respectively.

[0056] 2) Define the position, shape, magnetic susceptibility, magnetic inclination, remanent magnetization, and other properties of the magnetic target, and ensure that they are consistent with those used in the calculation of the time-varying magnetic field.

[0057] 3) The static magnetic initial field under the condition of remanence is iterated in the whole space by the scalar magnetic potential control equation (Equation 2), and the induced magnetization intensity and induced electromotive force of the time-varying magnetic field at the initial moment are calculated.

[0058] 4) Load the relevant parameters into the time-varying magnetic field calculation program, and set a square loop source with a side length of 100m as the transmitting coil at the center of the surface layer. Set the transmitting current to 1A and the turn-off time to 10. -6 s.

[0059] 5) Set parameters such as conductivity and zero-frequency magnetization over the entire computational domain. In the example, the background conductivity is set to 0.01 Siemens / meter. The half-space magnetization model and the half-space polarization model are set for verification. The zero-frequency magnetization χ0 of the half-space magnetization model is set to 0.01, the time constant τ = 1, and the fractional order α = 0.5. The permeability on the grid plane can be replaced by the surface average of the permeability of two adjacent cubes.

[0060] 6) Using a multiple zero-pole approximation algorithm, the Cole-Cole model of magnetic susceptibility with corresponding parameters is approximated in the time domain. The double curl induction magnetization diffusion equation (Equation 7) is selected as the time-varying control equation, and the magnetization parameters are substituted into it for iterative calculation.

[0061] 7) By using the control equation (14), the induced electromotive force at the next moment and the induced magnetization intensity at the previous moment are used to perform iterative calculations on the induced magnetization intensity at the next moment.

[0062] 8) Substitute the magnetic field value and the induced magnetization value at the next moment into equation (13), and repeat step 6) to perform variable time step iteration calculation.

[0063] 9) After the calculation is completed, based on the magnetic field value at the selected time, display the results of the magnetic induction intensity B of the response, perform data analysis and image drawing.

[0064] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A three-dimensional numerical simulation method for magnetic induction intensity in the time-domain electromagnetic method, characterized in that, The steps include the following: 1) Derive the equations containing remanent magnetization M using Maxwell's equations for static magnetic fields and the governing equations for magnetization M. r The scalar magnetic potential control equation, and through the magnetic scalar V m Solve for the magnetic flux density component B of the static magnetic field. x B y B z ; 2) The Cole-Cole model of magnetic susceptibility is introduced to characterize the change process of the magnetized medium under transient electromagnetic excitation conditions. The Cole-Cole model of magnetic susceptibility is approximated in the time domain by the multiple zero-pole construction approximation method. 3) Using Maxwell's equations with a source, combined with the induced magnetization M i The governing equations were derived by taking double curl on both sides of the equations, and the source magnetic field diffusion equation containing double curl magnetization was derived as the governing equations for the time-varying magnetic field solution stage. 4) Establishing a magnetic induction electromotive force With induced magnetization M i The iterative relationship is used to load the calculated magnetic induced electromotive force, and the induced magnetization M at the next time step is calculated using a convolution recursive operation method. i Perform calculations; 5) Regarding magnetic susceptibility χ and magnetic induction electromotive force The convolution operation is substituted into the double curl magnetization diffusion equation, the computational domain is divided by a non-uniform three-dimensional mesh, and the differential iteration format of the governing equation is derived in the entire computational domain based on the finite difference method. 6) The governing equations are rearranged into the form Ax = b, and the double conjugate gradient method is used to solve the governing equations, thereby realizing the discrete solution operation of the time-domain magnetic induction intensity B. 7) A variable time step iteration method is adopted to speed up the calculation time. After the calculation is completed, the response of each component of magnetic induction intensity B is extracted, and the calculation results are plotted and analyzed. In step 1), the Maxwell equations for the static magnetic field and the governing equations for the magnetization M are expressed as follows: In equation (1), E represents the electric field strength, B represents the magnetic induction intensity, H represents the magnetic field strength, and M represents the magnetization intensity. i M represents the induced magnetization intensity. r σ represents the remanent magnetization, μ0 represents the conductivity, and μ0 represents the free permeability. Combining magnetic field strength H and induced magnetization M i Based on the relationship, by deriving equation (1), we can obtain the solution for the magnetic scalar potential V. m The governing equations: Where x is the magnetic susceptibility of the magnetic medium, and further, the initial magnetic flux density component B of the static magnetic field can be obtained by solving the divergence of the magnetic potential. x B y B z ; The Cole-Cole model for magnetic susceptibility in step 2) is: Where χ0 is the zero-frequency magnetic susceptibility, and α is the fractional order; and through the induced magnetization M i From the governing equations, the formula for the rate of change of magnetization, characterized by the magnetic induction electromotive force, can be derived as follows: And by approximating it in the time domain, it can be expressed as: Where r1, r2…r m p1, p2...p m These are the rational function approximation coefficients of the Cole-Cole model of magnetic susceptibility, where m is the order of approximation. This formula can be used to characterize the change in induced magnetization in the time domain. The governing equations in step 3) can be derived using Maxwell's equations with a source and the governing equations for induced magnetization: Where Δt represents the time step, B n+1 Let B be the magnetic field strength at time n+1. n Let H be the magnetic flux density at time n. n+1 E n+1 J sn+1 M in+1 Let n be the magnetic field strength, electric field strength, current strength, and induced magnetization at time n+1, respectively; thus, the diffusion equation based on the induced magnetization with double curl can be derived as follows: Furthermore, we can establish the magnetic induced electromotive force in step 4) using a convolutional recursive method. The induced magnetization intensity M at the previous moment in The iterative relationship between them Among them, G in K is the magnetic field convolution term, which can be written as: ψ1、ψ2…ψ m The time-domain expressions representing the magnetic susceptibility and the magnetomotive force are respectively represented by the components of the expression. The convolution can be written in the form related to the convolution term of the previous time step as follows: And the induced magnetization intensity M i The initial value can be characterized as the zero-frequency induced magnetization, that is: Based on the finite difference method, the iterative format of the governing equation for magnetic induction B in step 5) for each component can be obtained as follows: After solving the equations using the double conjugate gradient method, the components B of the time-varying magnetic induction intensity B in the time-domain electromagnetic method can be obtained. x B y B z Accurate response.

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