Well-ground electromagnetic forward simulation method and device and storage medium
By combining octree meshes and higher-order shape functions, the problems of excessive mesh elements and high computational cost in well-to-ground electromagnetic methods are solved, enabling accurate simulation and high-precision electromagnetic response calculation of complex anomalies and wellbores.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- CHINA PETROLEUM & CHEMICAL CORP
- Filing Date
- 2025-01-09
- Publication Date
- 2026-07-10
AI Technical Summary
Traditional well-to-ground electromagnetic methods suffer from problems such as excessive mesh generation elements and high computational costs when simulating complex anomalies and wells. Furthermore, higher-order finite element methods have not received sufficient attention in electromagnetic forward modeling, making it difficult to improve the accuracy of numerical solutions.
By employing an octree mesh modeling algorithm and higher-order shape functions, combined with Gauss integration points and Gauss-Legendre-Lobatto integration points, the governing equations of the well-to-ground electromagnetic finite element method are derived. The iterative method with block diagonal preconditions is then used to solve the equations, achieving accurate modeling and high-precision calculation.
It significantly reduces the number of grid cells, lowers computational costs, and improves the accuracy of numerical solutions, enabling accurate simulation of the electromagnetic response of complex anomalies and wellbores.
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Figure CN122365971A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of geophysical exploration technology, specifically to a well-to-surface electromagnetic forward modeling method, apparatus, and storage medium. Background Technology
[0002] Geophysical exploration, or geophysical prospecting for short, refers to the study and observation of changes in various geophysical fields to detect geological conditions such as lithology and geological structure. Different rock layers that make up the Earth's crust often differ in density, elasticity, electrical conductivity, magnetism, radioactivity, and thermal conductivity. These differences cause corresponding local variations in the geophysical fields. By measuring the distribution and variation characteristics of these physical fields and analyzing them in conjunction with known geological data, the geological characteristics can be inferred. Research and practical experience in the industry have revealed that traditional surface exploration methods, due to the presence of complex influences such as air waves, struggle to accurately obtain relevant electromagnetic response signals from underground strata.
[0003] Well-to-surface electromagnetic (WSEM) is an effective technique for exploring deep-seated resources, offering advantages such as a wide exploration range and sensitivity to high-resistivity anomalies. However, several problems remain with 3D forward modeling methods: First, the widely used regular meshes have significant drawbacks when dealing with complex anomalies or undulating terrain. When anomalies in the strata have complex boundaries or exhibit very small scale variations in space, traditional hexahedral meshes struggle to simulate these situations, requiring global mesh refinement and resulting in an excessive number of generated cells. This method does not maximize forward modeling accuracy and increases the consumption of computer physical memory. For example, in practical applications of WSEM, wellbores are typically encased in metal casing or filled with conductive fluid to protect the wellbore and emission source from groundwater. The conductive medium in the wellbore generally has extremely high conductivity, requiring precise modeling of the wellbore to accurately simulate the electromagnetic response. However, the wellbore sizes used in actual well-to-surface exploration are very small, typically only tens of centimeters in diameter. There is a huge scale difference between the wellbore size and the entire computational domain, which is tens of kilometers in size, making it very difficult to accurately model the wellbore using tetrahedral meshes and traditional coordinated hexahedral meshes.
[0004] Secondly, according to finite element theory, using finer meshes or higher-order basis functions can improve the accuracy of numerical solutions. When the solution is smooth, using higher-order basis functions is more effective than using finer meshes. In most computational domains with uniform resistivity, such as air and boundary regions, the numerical solution is smooth. In such cases, using higher-order shape functions is expected to improve the accuracy of the numerical solution. However, in the field of electromagnetic forward modeling, higher-order finite element methods have not received sufficient attention. Summary of the Invention
[0005] To address the aforementioned technical problems in existing technologies, this invention provides a well-to-ground electromagnetic forward modeling method, apparatus, and storage medium. By developing an octree mesh modeling algorithm, complex anomaly models are accurately modeled. Higher-order shape functions are used in the forward modeling calculations to improve the accuracy of the numerical solution, ultimately generating an application program.
[0006] To achieve the above objectives, the technical solution of the present invention is as follows:
[0007] On one hand, the present invention provides a well-to-ground electromagnetic forward modeling method, comprising:
[0008] S1. Modeling based on octree mesh;
[0009] S2. Based on the Gauss integration points and Gauss-Legendre-Lobatto (GLL) integration points set in the element, construct the vector Nédélec basis functions on the element;
[0010] S3. Starting from Maxwell's equations, derive the governing equations of the well-to-ground electromagnetic finite element method;
[0011] S4. Construct preconditions for the block diagonals and solve them using the iterative method;
[0012] S5. After exiting the iteration, calculate the electromagnetic response using the iterative model.
[0013] Furthermore, the specific method for modeling based on the octree mesh in step S1 is as follows:
[0014] S101. Create an initial 3D model using a coarse mesh.
[0015] S102. Traverse all cells. For each cell, determine whether it meets the refinement condition. The refinement condition is whether the cell is located in different medium regions. Then determine whether to refine the cell and mark it.
[0016] S103. Use an octree mesh to subdivide all the cells marked as refined in step S102;
[0017] S104. Repeat steps S102 and S103 until all cells no longer meet the refinement conditions in step S102.
[0018] Furthermore, in step S102, determining whether the refinement conditions are met specifically includes: selecting several points in the cell, determining whether the region's materialia_id attribute is the same for the regions where the several points are located, and if the materialia_id attribute of all selected points is not unique, and the level of the cell is less than the predefined maximum number of encryptions, then the cell is marked as refined.
[0019] Furthermore, several points were selected as the center point and eight vertices.
[0020] Furthermore, after step S1 and before step S2, the process also includes: demonstrating the octree mesh modeling process.
[0021] Furthermore, the governing equation in step S3 is:
[0022]
[0023] In the above formula, V is the vector Nédélec basis function constructed in step S2, E is the electric field to be determined, Ω represents the calculation region, i is the imaginary unit, ω represents the angular frequency, μ0 represents the free magnetic permeability, σ represents the dielectric conductivity, and J s This represents the field source term.
[0024] Furthermore, the integral of each element is calculated using the Gaussian numerical integration method, and the matrices of all elements are assembled to obtain the linear equations of the finite element method, which are:
[0025] Ke = s
[0026] In the above formula, K is a matrix, e represents the electric field interpolation coefficient, and s represents the vector on the right-hand side, which includes the contribution of the field source.
[0027] Furthermore, the preconditions for the diagonal partitioning are:
[0028] B=K r +K i
[0029] In the above formula, P represents a block diagonal matrix, and K r Let K represent the real part of matrix K. i Let K represent the imaginary part of matrix K.
[0030] Furthermore, in step S4, the exit condition for solving using the iterative method is: the number of iterations is greater than the maximum number of iterations, or the residual is less than the set residual threshold.
[0031] On the other hand, the present invention also provides a well-to-ground electromagnetic forward modeling device, including an operating system and a display module. The operating system is equipped with a processor and a memory. The memory stores a computer program. When the computer program is executed by the processor, it implements the above-mentioned well-to-ground electromagnetic forward modeling method.
[0032] On the other hand, the present invention also provides a computer-readable storage medium storing a computer program that, when executed by a processor, implements the above-described well-to-ground electromagnetic forward modeling method.
[0033] Compared with the prior art, the present invention has the following beneficial effects:
[0034] The well-to-surface electromagnetic forward modeling method provided by this invention employs an octree mesh, one of the non-uniform mesh types, to improve the modeling of complex models. This significantly reduces the number of elements in the mesh while maintaining model accuracy, thereby lowering computational costs. Higher-order shape functions are constructed and used in the forward modeling calculations to improve the accuracy of the numerical solution. Attached Figure Description
[0035] Figure 1 This is a flowchart of the well-to-ground electromagnetic forward modeling method of the present invention.
[0036] Figure 2 Flowchart for modeling an octree mesh.
[0037] Figure 3 Example of modeling an octree mesh.
[0038] Figure 4 To use an octree mesh to discretize the vertical wellbore model.
[0039] Figure 5 The response of borehole models with different resistivity. Detailed Implementation
[0040] The technical solution of the present invention will be clearly described below with reference to the accompanying drawings. Obviously, the described embodiments are not all embodiments of the present invention. All other embodiments obtained by those skilled in the art without creative effort are within the protection scope of the present invention.
[0041] It should be noted that, unless otherwise specifically stated, the relative arrangement and numerical expressions of the components and steps described in these embodiments should not be construed as limiting the scope of the invention.
[0042] The following description of exemplary embodiments is merely illustrative and is not intended to limit the invention or its application or use in any way. Techniques, methods, and apparatus known to those skilled in the art may not be discussed in detail herein, but where applicable, such techniques, methods, and apparatus should be considered part of this specification.
[0043] Example 1
[0044] This embodiment provides a well-to-ground electromagnetic forward modeling method, such as... Figure 1 As shown, it includes:
[0045] S1. Modeling based on octree meshes: By dividing hexahedral elements into eight smaller elements to locally refine the mesh, the accuracy of forward modeling can be maintained while using fewer cells, thus reducing computational costs. The octree mesh-based precise modeling method enables the modeling of small-scale vertical wellbore models and various complex anomaly models. The octree mesh modeling process is as follows: Figure 2 As shown, the specific method is as follows:
[0046] S101. Create an initial 3D model using a coarse mesh.
[0047] First, create an initial 3D model using a coarse mesh. The length, width, and height of the initial model should be as balanced as possible, because excessive stretching in one direction will increase the order of the coefficient matrix of the solution equations and may cause the linear solver to become scattered.
[0048] S102. Traverse all units. For each unit, determine whether it meets the refinement condition. The refinement condition is whether the unit is located in different medium regions. Then determine whether to refine and mark the unit. Specifically, select several points in the unit and determine whether the materialia_id attribute of the regions where the several points are located is the same. If the materialia_id attribute of all selected points is not unique, and the level of the unit is less than the predefined maximum number of encryption times, then mark the unit as refined.
[0049] Among them, several points were selected as the center point and eight vertices.
[0050] S103. Use an octree mesh to refine all elements marked as refined in step S102. Note that to ensure the level difference between adjacent elements does not exceed 1, some unmarked elements will also be refined. This condition is also known as the 2:1 rule.
[0051] S104. Repeat steps S102 and S103 until all cells no longer meet the refinement conditions in step S102.
[0052] In a preferred embodiment, after step S1 and before step S2, the process further includes demonstrating the octree mesh modeling workflow. For example, using... Figure 3 The complex oil and gas model shown is an example demonstrating the octree mesh modeling process. The model structure is as follows: Figure 3As shown in Figure (a), an oil and gas reservoir with a resistivity of 50 exists at a depth of approximately 1.5 km. The reservoir is surrounded by water-filled formation fractures with a resistivity of 5. Above the oil and gas reservoir is a geological fault zone extending laterally for approximately 10 km with a resistivity of 500. The aforementioned anomaly extends 2 km in the directional direction. The initial model consists of 202016 elements, with a core element size of 1000 m. The element scaling factor is set to 1.3, and five elements are added outwards from the core, resulting in a final computational domain of approximately 33 km, 33 km, and 10 km. During the octree mesh modeling process, the method in modeling step S102 is used to determine whether each element needs refinement, and then the elements marked for refinement are divided into eight sub-elements. The mesh after the first element traversal and refinement is shown below. Figure 3 As shown in Figure (b), it consists of 6350 units. Then, the above refinement process is repeated 6 times, as follows: Figure 3 Figures (c)-(f) show the final mesh, which contains 372,808 elements, with the smallest element being approximately 15m in size. By comparing the final mesh with the initial example model, it is demonstrated that the complex boundaries and shape features of the anomaly can be fully constructed using an octree mesh. Therefore, using elements of approximately 15m in size is sufficient to accurately simulate the complex anomaly in the model.
[0053] S2. Based on the Gauss integration points and Gauss-Legendre-Lobatto (GLL) integration points set in the element, construct the vector Néédéélec basis functions on the element;
[0054] S3. Starting from Maxwell's equations, derive the governing equations of the well-to-ground electromagnetic finite element method; the governing equations are:
[0055]
[0056] In the above formula, V is the vector Nédélec basis function constructed in step S2, E is the electric field to be determined, Ω represents the calculation region, i is the imaginary unit, ω represents the angular frequency, μ0 represents the free magnetic permeability, σ represents the dielectric conductivity, and J s This represents the field source term.
[0057] The integral of each element is calculated using the Gaussian numerical integration method. After assembling the matrices of all elements, the linear equations of the finite element method are obtained as follows:
[0058] Ke = s
[0059] In the above formula, K is a matrix, e represents the electric field interpolation coefficient, and s represents the vector on the right-hand side, which includes the contribution of the field source.
[0060] S4. Construct preconditions for the block diagonals and solve them using the iterative method;
[0061] After obtaining the linear equations of the well-to-surface electromagnetic finite element method, block diagonal preconditions are constructed, and an iterative solution method is used to solve them. This addresses the problem of excessive computational memory consumption in solving complex complex coefficient sparse equation systems using direct method solvers. The block diagonal preconditions are as follows:
[0062] B=K r +K i
[0063] In the above formula, P represents a block diagonal matrix, and K r Let K represent the real part of matrix K. i Let K represent the imaginary part of matrix K.
[0064] The exit conditions for solving using the iterative method are: the number of iterations is greater than the maximum number of iterations, or the residual is less than the set residual threshold.
[0065] S5. After exiting the iteration, calculate the electromagnetic response using the iterative model.
[0066] An example of forward modeling using the well-to-ground electromagnetic forward modeling method provided by this invention is shown, with a vertical wellbore model background of a uniform half-space and an air resistivity of 10⁻⁶. 8 The formation resistivity is 100 Ω·m. The vertical well is located at (0m, 0m, 0m), with a diameter of 0.25m and a depth of 300m. Using an octree mesh discretization model, the element size inside the well is 0.0625m, resulting in 183,452 elements. Figure 4 The xz-plane mesh and wellhead plan view of the wellbore model are shown respectively. Figure 4 Figure (a) is a grid diagram of the xz plane. Figure 4 Figure (b) shows the wellhead plan. Twenty-six receivers were placed along the x-axis on the Earth's surface, ranging from x = -500 m to x = 500 m, to observe the electric field E. r The components are spaced 40m apart from the receivers.
[0067] Since the wellbore and casing are considered as a single unit, the resistivity of the wellbore is set to 10. -2 Ω·m, 10 -4 Ω·m and 10 -6 The electromagnetic response was calculated in Ω·m to investigate the effect of the borehole having a conductive medium with different resistivity on the response. Figure 5 The graph shows the responses of three borehole models at transmission frequencies of 100Hz and 1000Hz. For comparison, the response of the half-space model is also included. The horizontal axis represents the receiver position, and the vertical axis represents E... rThe amplitude and phase responses of the wellbore are shown. Clearly, the presence of a conductive medium within the wellbore has a significant impact on the response. When the conductive medium filling the wellbore has a high conductivity, E... r The amplitude decreased significantly by several orders of magnitude. This phenomenon is caused by the rapid decay of the electromagnetic field due to the conductive medium within the well, resulting in a reduction in the amplitude of the surface response. These findings underscore the necessity of considering the presence of the wellbore when interpreting well-to-ground electromagnetic field data.
[0068] Example 2
[0069] This embodiment provides a well-to-surface electromagnetic forward modeling device, including an operating system and a display module. The operating system is equipped with a processor and a memory. The memory stores a computer program. When the computer program is executed by the processor, it implements the well-to-surface electromagnetic forward modeling method provided in Embodiment 1.
[0070] Example 3
[0071] This embodiment provides a computer-readable storage medium storing a computer program. When the computer program is executed by a processor, it implements the well-to-ground electromagnetic forward modeling method provided in Embodiment 1.
[0072] The above specific embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit it. Although the present invention has been described in detail with reference to examples, those skilled in the art should understand that modifications or equivalent substitutions can be made to the technical solutions of the present invention without departing from the scope of the technical solutions of the present invention, and all such modifications or substitutions should be covered within the scope of the claims of the present invention.
Claims
1. A well-to-ground electromagnetic forward modeling method, characterized in that, include: S1. Modeling based on octree mesh; S2. Based on the Gauss integration points and Gauss-Legendre-Lobatto (GLL) integration points set in the element, construct the vector Nédélec basis functions on the element; S3. Starting from Maxwell's equations, derive the governing equations of the well-to-ground electromagnetic finite element method; S4. Construct preconditions for the block diagonals and solve them using the iterative method; S5. After exiting the iteration, calculate the electromagnetic response using the iterative model.
2. The well-to-ground electromagnetic forward modeling method according to claim 1, characterized in that, The specific method for modeling based on the octree mesh in step S1 is as follows: S101. Create an initial 3D model using a coarse mesh. S102. Traverse all cells. For each cell, determine whether it meets the refinement condition. The refinement condition is whether the cell is located in different medium regions. Then determine whether to refine the cell and mark it. S103. Use an octree mesh to subdivide all the cells marked as refined in step S102; S104. Repeat steps S102 and S103 until all cells no longer meet the refinement conditions in step S102.
3. The well-to-ground electromagnetic forward modeling method according to claim 2, characterized in that, Step S102 determines whether the refinement conditions are met, specifically including: selecting several points in the cell, determining whether the region's materialia_id attribute is the same for the regions where the several points are located, and if the materialia_id attribute of all selected points is not unique, and the level of the cell is less than the predefined maximum number of encryption times, then the cell is marked as refined.
4. The well-to-ground electromagnetic forward modeling method according to claim 3, characterized in that, Several points were selected as the center point and eight vertices.
5. The well-to-ground electromagnetic forward modeling method according to claim 1, characterized in that, After step S1 and before step S2, the process also includes: demonstrating the octree mesh modeling process.
6. The well-to-ground electromagnetic forward modeling method according to claim 1, characterized in that, The governing equation in step S3 is: In the above formula, V is the vector Nédélec basis function constructed in step S2, E is the electric field to be determined, Ω represents the calculation region, i is the imaginary unit, ω represents the angular frequency, μ0 represents the free magnetic permeability, σ represents the dielectric conductivity, and J s This represents the field source term.
7. The well-to-ground electromagnetic forward modeling method according to claim 6, characterized in that, The integral of each element is calculated using the Gaussian numerical integration method. After assembling the matrices of all elements, the linear equations of the finite element method are obtained as follows: K×e=s In the above formula, K is a matrix, e represents the electric field interpolation coefficient, and s represents the vector on the right-hand side, which includes the contribution of the field source.
8. The well-to-ground electromagnetic forward modeling method according to claim 1, characterized in that, The preconditions for the diagonal partitioning are: In the above formula, P represents a block diagonal matrix, and K r Let K represent the real part of matrix K. i Let K represent the imaginary part of matrix K.
9. The well-to-ground electromagnetic forward modeling method according to claim 1, characterized in that, In step S4, the exit condition for solving using the iterative method is: the number of iterations is greater than the maximum number of iterations, or the residual is less than the set residual threshold.
10. A well-to-surface electromagnetic forward modeling device, comprising an operating system and a display module, wherein the operating system is internally equipped with a processor and a memory, characterized in that, The memory stores a computer program, which, when executed by a processor, implements the well-to-ground electromagnetic forward modeling method according to any one of claims 1-9.
11. A computer-readable storage medium storing a computer program thereon, characterized in that, When the computer program is executed by the processor, it implements the well-to-ground electromagnetic forward modeling method according to any one of claims 1-9.