阅读说明：本技术 一种二维起伏地表下的sh波曲线网格模拟方法 (SH wave curve grid simulation method under two-dimensional undulating surface ) 是由 刘炯 刘喜武 刘宇巍 霍志周 钱恪然 张金强 于 2019-03-01 设计创作，主要内容包括：本发明涉及一种二维起伏地表下的SH波曲线网格模拟方法,包括以下步骤：步骤1：建立不规则的物理空间和规则的计算空间之间的映射关系；步骤2：将所述规则的计算空间中的SH波进行波场模拟；步骤3：将所述波场模拟的结果通过所述映射关系映射到所述物理空间中,得到所述物理空间中SH波沿起伏地表的波场模拟结果。本发明方法稳定且不会产生传统直角网格“阶梯”离散产生的数值噪音,起伏地表的新模拟精度较高。在解决自由面模拟的同时,保证了模拟的高精度。本发明较好解决了起伏地表用传统直角网格模拟时认为数值噪音误差,不会出现模拟不稳定的现象,有效保证了波场数值模拟的精度。(The invention relates to a method for simulating SH wave curve grids under a two-dimensional undulating surface, which comprises the following steps: step 1: establishing a mapping relation between an irregular physical space and a regular calculation space; step 2: carrying out wave field simulation on the SH wave in the regular calculation space; and step 3: and mapping the result of the wave field simulation to the physical space through the mapping relation to obtain a wave field simulation result of the SH waves in the physical space along the undulating surface. The method is stable, numerical noise generated by the 'ladder' dispersion of the traditional right-angle grid can not be generated, and the new simulation precision of the undulating surface is higher. The method solves the problem of free surface simulation and ensures the high precision of simulation. The method well solves the problem that numerical noise errors are considered when the undulating surface is simulated by using the traditional right-angle grid, the phenomenon of unstable simulation cannot occur, and the precision of numerical simulation of the wave field is effectively ensured.)

1. A method for simulating SH wave curve grids under a two-dimensional undulating surface is characterized by comprising the following steps:

step 1: establishing a mapping relation between an irregular physical space and a regular calculation space;

step 2: carrying out wave field simulation on the SH wave in the regular calculation space;

and step 3: and mapping the result of the wave field simulation to the physical space through the mapping relation to obtain a wave field simulation result of the SH waves in the physical space along the undulating surface.

2. The method according to claim 1, wherein the mapping relationship is established by generating a curve grid in step 1.

3. The method of claim 2, wherein the curve mesh generation method comprises algebraic methods, conformal transformation methods, and differential equation methods.

4. The method of simulating an SH wave curve grid in a two-dimensional undulating subsurface according to claim 2, wherein the step of generating the curve grid includes:

(1) establishing a poisson equation set;

(2) obtaining a control equation set for generating a curve grid according to the Poisson equation set;

(3) and establishing a mapping relation between the physical space and the calculation space by solving numerical values of a control equation set for generating the curve grid.

5. The method for mesh simulation of an SH wave curve in a two-dimensional undulating surface according to claim 4, wherein the system of Poisson's equations in the step (1) is as follows:

wherein x and z are independent variables of physical coordinates (xoz) in an irregular physical space, and xi and eta are independent variables of curve grid coordinates (xi o eta) in a regular calculation space; p, Q is a coordinate control function for realizing control of grid distribution in physical coordinates (xoz);

and changing the independent variable (x, z) in the Poisson equation set to be (xi, eta) by utilizing the mapping relation between the physical coordinate and the curve grid coordinate xi, xi (x, z) and eta, eta (x, z), wherein xi, xi (x, z) and eta, eta (x, z) are functions expressed by xi and eta by the independent variables x and z in the equations.

6. The method for simulating an SH wave curve grid under a two-dimensional undulating surface according to claim 5, wherein the step (2) is specifically as follows: utilizing a composite function derivation rule to conduct derivation on the Poisson equation set to obtain a control equation set for generating a curve grid:

α·x_ξξ-2β·x_ξη+γ·x_ηη+J2(P·x_ξ+Q·x_η)＝0

α·z_ξξ-2β·z_ξη+γ·z_ηη+J²(P·z_ξ+Q·z_η)＝0

in the formula (I), the compound is shown in the specification,β＝x_ξx_η+z_ξz_η，

in the step (3), the control equation set of the curve grid may be numerically solved by a finite difference method, a finite element method, or a pseudo spectrum method, so as to obtain ξ (x, z), η ═ η (x, z).

7. The method for simulating an SH wave curve grid under a two-dimensional undulating surface according to claim 6, wherein the step 2 further comprises the following steps:

step 2.1: establishing a control equation set for SH wave propagation in a physical coordinate system;

step 2.2: obtaining a control equation set of the SH wave transmitted under a curve grid coordinate system according to the control equation set transmitted under the physical coordinate system;

step 2.3: and carrying out wave field simulation on the control equation set propagated under the curve grid coordinate system by using a numerical method in the regular calculation space.

8. The method for simulating an SH wave curve grid under a two-dimensional undulating surface according to claim 7, wherein in step 2.1, the step of establishing the control equation system for SH wave propagation under a physical coordinate system is as follows: according to the principle of elastic mechanics, an SH wave propagation equation set of a solid medium in a two-dimensional x-z space is established:

wherein V represents the velocity of the solid medium in the y-direction perpendicular to the x-z plane; tau is_xy、τ_yzIs a stress component; f. of_yIs an external force in the y-direction; t is time; ρ represents the density of the solid medium; μ is the shear modulus of the solid medium.

9. The method according to claim 8, wherein in step 2.2, the control equations propagated in the physical coordinate system are differentiated according to a complex function derivation rule to obtain the SH wave equations in the curved grid coordinate system:

in the formula (f)_y(ξ) is the external force in space y direction in curvilinear coordinates (ξ o η);

10. the method of mesh simulation of an SH wave curve in a two dimensional undulating subsurface according to claim 9, wherein the numerical method in step 2.3 includes finite difference or pseudo-spectral methods.

Technical Field

The invention belongs to the technical field of oil and gas geophysical, and particularly relates to a method for simulating SH wave curve grids under a two-dimensional undulating surface.

Background

Depending on the direction of vibration of the wave particles and the direction of propagation of the wave, the wave can be divided into transverse waves (i.e., S-waves) and longitudinal waves (i.e., P-waves). The wave with the particle vibration direction same as the propagation direction is a longitudinal wave; waves whose particle vibration direction is perpendicular to the propagation direction are called shear waves. A wave in which particle vibration occurs in a plane perpendicular to a wave propagation plane is an SV wave, and a wave in which particle vibration occurs in a plane parallel to the wave propagation plane is an SH wave.

The undulating surface is one of the most important interference factors for mountain seismic exploration, and the effect of seismic exploration can be improved by identifying and eliminating the interference by adopting a numerical simulation method, so that a plurality of scholars at home and abroad develop the research on the seismic wave propagation rule under the undulating surface by utilizing wave equation numerical simulation.

The finite element method can well describe the shape of the free surface of the irregular earth surface, and the condition of the free surface can be automatically met in the finite element method simulation, so that the method is suitable for the problem of complicated free surface in principle. However, the low-order finite element method has strong dispersion, the common high-order finite element method generates pseudo waves, and the finite element method has high memory requirement and large calculation amount, so that the method is relatively rarely used in the oil and gas exploration field with large calculation scale.

Finite difference method and pseudo-spectrum method are two methods which are most frequently used in seismic simulation in the geophysical field at present. The traditional finite difference method and the pseudo-spectrum method are simulated on a right-angle grid. Many researchers first applied these two simulation methods directly to seismic wave simulation of undulating surfaces. They approximate and then complement a portion of the space on the free surface, making the whole a regular rectangular space, and then discretize the computational model space with a rectangular grid, which results in the relief being discretized into a "staircase" shape in the mathematical model; and finally, simulating the propagation of the seismic wave by adopting a traditional finite difference method or a pseudo-spectrum method in a regular model space. However, the use of a conventional rectangular grid to perform step discretization of the relief surface can introduce large errors into the shape of the model, which can introduce large errors into the simulation of the physical quantities, and the discretized "steps" can produce strong scatter values. In addition, the processing of the fluctuant free interface generally implies the boundary condition of the earth surface and the upper space, and the difference between the physical property of the upper space and the actual underground property is obvious, so that the unstable phenomenon easily occurs in the simulation process, and the simulation process fails. And the accuracy of the interface can only reach second order at most.

Aiming at the problems of the traditional rectangular coordinate discrete undulating surface, the foreign scholars Tessemer (1992) introduces the processing idea of Fornberg (1988) underground irregular surfaces, maps irregular free surfaces onto regular flat surfaces by using a coordinate mapping technology, and then calculates. Some scholars at home and abroad then perform numerical simulation of the relief subsurface converted wave (P-SV wave) in the same manner. However, these methods only perform simple coordinate mapping in the vertical direction, and the horizontal direction is kept unchanged, and when the surface relief is severe, the processing may cause the curve grid in the physical space to be too distorted, thereby resulting in a large error.

Disclosure of Invention

In order to solve the problems, the invention provides a method for simulating SH wave curve grids under a two-dimensional undulating surface, which is stable, does not generate numerical noise generated by the step dispersion of the traditional right-angle grids, and has higher new simulation precision of the undulating surface. The method solves the problem of free surface simulation and ensures the high precision of simulation. The method well solves the problem that numerical noise errors are considered when the undulating surface is simulated by using the traditional right-angle grid, the phenomenon of unstable simulation cannot occur, and the precision of numerical simulation of the wave field is effectively ensured.

The invention provides a method for simulating SH wave curve grids under a two-dimensional undulating surface, which comprises the following steps:

step 1: establishing a mapping relation between an irregular physical space and a regular calculation space;

step 2: carrying out wave field simulation on the SH wave in the regular calculation space;

and step 3: and mapping the result of the wave field simulation to the physical space through the mapping relation to obtain a wave field simulation result of the SH waves in the physical space along the undulating surface.

In one embodiment, in step 1, the mapping relationship is established by generating a curve grid.

In one embodiment, the generation method of the curve grid comprises an algebraic method, a conformal transformation method and a differential equation method.

In one embodiment, the step of generating the curved grid comprises:

(1) establishing a poisson equation set;

(2) obtaining a control equation set for generating a curve grid according to the Poisson equation set;

(3) and establishing a mapping relation between the physical space and the calculation space by solving numerical values of a control equation set for generating the curve grid.

In one embodiment, the poisson equation set in step (1) is:

wherein x and z are independent variables of physical coordinates (xoz) in an irregular physical space, and xi and eta are independent variables of curve grid coordinates (xi o eta) in a regular calculation space; p, Q is a coordinate control function for realizing control of grid distribution in physical coordinates (xoz);

and changing the independent variable (x, z) in the Poisson equation set to be (xi, eta) by utilizing the mapping relation between the physical coordinate and the curve grid coordinate xi, xi (x, z) and eta, eta (x, z), wherein xi, xi (x, z) and eta, eta (x, z) are functions expressed by xi and eta by the independent variables x and z in the equations.

In one embodiment, the step (2) is specifically: utilizing a composite function derivation rule to conduct derivation on the Poisson equation set to obtain a control equation set for generating a curve grid:

α·x_ξξ-2β·x_ξη+γ·x_ηη+J²(P·x_ξ+Q·x_η)＝0

α·z_ξξ-2β·z_ξη+γ·z_ηη+J²(P·z_ξ+Q·z_η)＝0

in the formula (I), the compound is shown in the specification,

β＝x_ξx_η+z_ξz_η，x_ξis the first derivative of x with respect to ξ_ηIs the first derivative of x with respect to η_ξξIs the second derivative of x with respect to ξ_ξηIs the second partial derivative of x with respect to ξ and η_ηηIs the second derivative of x with respect to η, z_ξIs the first derivative of z with respect to ξ_ηIs the first derivative of z with respect to η_ξξIs the second derivative of z with respect to ξ_ξηIs the second partial derivative of z with respect to ξ and η, z_ηηIs the second derivative of z with respect to η, the argument ξ is defined over a regular rectangular area, J is the Jacobian from the curve grid coordinates to the physical coordinates:

in the step (3), the control equation set of the curve grid may be numerically solved by a finite difference method, a finite element method, or a pseudo spectrum method, so as to obtain ξ (x, z), η ═ η (x, z).

In an embodiment, the step 2 further specifically includes the following steps:

step 2.1: establishing a control equation set for SH wave propagation in a physical coordinate system;

step 2.2: obtaining a control equation set of the SH wave transmitted under a curve grid coordinate system according to the control equation set transmitted under the physical coordinate system;

step 2.3: and carrying out wave field simulation on the control equation set propagated under the curve grid coordinate system by using a numerical method in the regular calculation space.

In one embodiment, in step 2.1, the step of establishing the system of control equations for the SH wave propagating in the physical coordinate system is: according to the principle of elastic mechanics, an SH wave propagation equation set of a solid medium in a two-dimensional x-z space is established:

wherein V represents the velocity of the solid medium in the y-direction perpendicular to the x-z plane; tau is_xy、τ_yzIs a stress component; f. of_yIs an external force in the y-direction; t is time; ρ represents the density of the solid medium; μ is the shear modulus of the solid medium.

In one embodiment, in step 2.2, the derivation is performed on the control equation system propagated in the physical coordinate system according to a complex function derivation rule, so as to obtain an SH wave equation system in the curved grid coordinate system:

in the formula (f)_y(ξ) is the external force in space y direction in curvilinear coordinates (ξ o η);

is a measurement coefficient transformed from a physical coordinate system to a curved grid coordinate system, wherein,

in one embodiment, the numerical method in step 2.3 comprises finite difference or pseudo-spectral methods.

Compared with the prior art, the invention has the advantages that: the invention introduces a curve mesh subdivision technology of an irregular mould surface and provides an SH wave simulation method under a general curve coordinate. The method comprises the steps of firstly smoothly mapping an actual undulating surface physical space to a regular rectangular calculation space through a new curve grid splitting method, establishing a corresponding relation between the actual physical space and the regular calculation space, then simulating SH waves in the regular calculation space, and finally mapping a simulation result of the calculation space back to the actual physical space to obtain a wave field transmission process of the SH waves along the irregular undulating surface. The method is stable, numerical noise generated by the 'ladder' dispersion of the traditional right-angle grid can not be generated, and the new simulation precision of the undulating surface is higher. The method solves the problem of free surface simulation and ensures the high precision of simulation. The method well solves the problem that numerical noise errors are considered when the undulating surface is simulated by using the traditional right-angle grid, the phenomenon of unstable simulation cannot occur, and the precision of numerical simulation of the wave field is effectively ensured.

Drawings

The invention will be described in more detail hereinafter on the basis of embodiments and with reference to the accompanying drawings. Wherein:

FIG. 1 is a schematic diagram of the irregular physical space curve grid discretization of the present invention;

FIG. 2 is a diagram of a two-dimensional inclined terrain model according to the present invention (embodiment one);

FIG. 3 is a graph of a curved grid dispersion of an inclined earth surface according to the present invention (embodiment one);

FIG. 4 is a wavefield snapshot at a time of 0.55 seconds at an inclined surface according to the present invention (example one);

FIGS. 5 to 7 are the comparison between the numerical simulation results and the theoretical results of 3 virtual detectors according to the present invention (example I);

FIG. 8 is a diagram of an irregular moat model according to the present invention (example II);

FIG. 9 is a curve grid dispersion diagram of the irregular moat model according to the present invention (example II);

FIG. 10 is a wavefield snapshot at time 0.2 second for the present invention cutting model (example two).

In the drawings like parts are provided with the same reference numerals. The figures are not drawn to scale.

Detailed Description

The invention will be further explained with reference to the drawings. Therefore, the realization process of how to apply the technical means to solve the technical problems and achieve the technical effect can be fully understood and implemented. It should be noted that the technical features mentioned in the embodiments can be combined in any way as long as no conflict exists. It is intended that the invention not be limited to the particular embodiments disclosed, but that the invention will include all embodiments falling within the scope of the appended claims.

The invention provides a method for simulating SH wave curve grids under a two-dimensional undulating surface, which comprises the following steps:

step 1: establishing a mapping relation between an irregular physical space and a regular calculation space;

step 2: carrying out wave field simulation on the SH wave in the regular calculation space;

and step 3: and mapping the result of the wave field simulation to the physical space through the mapping relation to obtain a wave field simulation result of the SH waves in the physical space along the undulating surface.

In step 1, the establishment of the mapping relationship is a generation process of a curve grid, and the mapping relationship is established by generating the curve grid. A curvilinear coordinate grid technique is applied to the wavefield simulation. That is, first, the calculation region of the irregular undulating surface (physical space) is discretized by the curved grid (as shown in fig. 1), and the irregular space is discretized by the curved grid (as shown in the left diagram of fig. 1), and at the same time, the mapping relationship between the physical coordinates (xoz) in the irregular physical space and the calculated coordinates (ξ η) of the curved grid in the regular calculation space (as shown in the right diagram of fig. 1) is established. Wave field simulation is carried out in a regular calculation space, and the final simulation result is returned to a physical space according to the mapping relation, so that the wave field simulation result of the SH waves along the undulating surface in the irregular physical space is obtained.

The generation method of the curve grid comprises an algebraic method, a conformal transformation method and a differential equation method.

In one embodiment, a method for generating a curvilinear mesh by solving a poisson equation system (belonging to differential equation method), specifically, the step of generating the curvilinear mesh comprises:

(1) establishing a poisson equation set;

(2) obtaining a control equation set for generating a curve grid according to the Poisson equation set;

(3) and establishing a mapping relation between the physical space and the calculation space by solving numerical values of a control equation set for generating the curve grid.

In one embodiment, the poisson equation set in step (1) is:

wherein x and z are independent variables of physical coordinates (xoz) in an irregular physical space, and xi and eta are independent variables of curve grid coordinates (xi o eta) in a regular calculation space; p, Q is a coordinate control function for realizing control of grid distribution in physical coordinates (xoz); by selecting P, Q coordinate control functions, different curve coordinate distributions are realized, P, Q can be selected according to requirements.

In one embodiment, the step (2) is specifically: utilizing a composite function derivation rule to conduct derivation on the Poisson equation set to obtain a control equation set for generating a curve grid:

α·x_ξξ-2β·x_ξη+γ·x_ηη+J²(P·x_ξ+Q·x_η)＝0

α·z_ξξ-2β·z_ξη+γ·z_ηη+J²(P·z_ξ+Q·z_η)＝0

in the formula (I), the compound is shown in the specification,

β＝x_ξx_η+z_ξz_η，x_ξis the first derivative of x with respect to ξ_ηIs the first derivative of x with respect to η_ξξIs the second derivative of x with respect to ξ_ξηIs the second partial derivative of x with respect to ξ and η_ηηIs the second derivative of x with respect to η, z_ξIs the first derivative of z with respect to ξ_ηIs the first derivative of z with respect to η_ξξIs the second derivative of z with respect to ξ_ξηIs the second partial derivative of z with respect to ξ and η, z_ηηIs the second derivative of z with respect to η, and the argument ξ is defined over a regular rectangular area, wherein the regular rectangular area is referred to asThe range of the argument ξ is designed over a rectangular area, is regular, and its boundaries are not irregular.

In the step (3), the control equation set of the curve grid may be numerically solved by various methods (including a finite difference method, a finite element method, a pseudo-spectrum method, and the like), and ξ (x, z) and η (x, z), that is, a mapping relationship between a physical coordinate point and a coordinate point of the curve grid, are obtained. And changing the independent variable (x, z) in the SH wave equation set into (xi, eta) by using the mapping relation xi (x, z) between the physical coordinate and the curve grid coordinate and eta (x, z). And then solving a coordinate control equation in the curve grid coordinate, and obtaining the wave field quantity to be solved in the physical space by utilizing the mapping relation between the physical coordinate points and the curve grid coordinate points obtained by the Poisson equation set.

In an embodiment, the step 2 further specifically includes the following steps:

step 2.1: establishing a control equation set for SH wave propagation in a physical coordinate system;

step 2.2: obtaining a control equation set of the SH wave transmitted under a curve grid coordinate system according to the control equation set transmitted under the physical coordinate system;

step 2.3: and carrying out wave field simulation on the control equation set propagated under the curve grid coordinate system by using a numerical method in the regular calculation space. The numerical method in step 2.3 includes finite difference or pseudo-spectral methods.

In one embodiment, in step 2.1, the step of establishing the system of control equations for the SH wave propagating in the physical coordinate system is: according to the principle of elastic mechanics, an SH wave propagation equation set of a solid medium in a two-dimensional x-z space is established:

wherein V represents the velocity of the solid medium in the y-direction perpendicular to the x-z plane; tau is_xy、τ_yzIs a stress component; f. of_yIs an external force in the y-direction; t is time; ρ represents the density of the solid medium; μ is the shear modulus of the solid medium.

In one embodiment, in step 2.2, the derivation is performed on the control equation system propagated in the physical coordinate system according to a complex function derivation rule, so as to obtain an SH wave equation system in the curved grid coordinate system:

in the formula (f)_y(ξ) is the external force in space y direction in curvilinear coordinates (ξ o η);

the measuring coefficients are transformed from a physical coordinate system to a curve grid coordinate system, and can be obtained by numerical value calculation when the curve grid is established according to the following formula, wherein,