3D controlled-source electromagnetic modeling in anisotropic medium using edge-based finite element method

TL;DR: The method uses the edge-based vector basis functions, which automatically enforce the divergence free conditions for electric and magnetic fields, which is effective in modeling the seafloor bathymetry using hexahedral mesh.

Abstract: This paper presents a linear edge-based finite element method for numerical modeling of 3D controlled-source electromagnetic data in an anisotropic conductive medium. We use a nonuniform rectangular mesh in order to capture the rapid change of diffusive electromagnetic field within the regions of anomalous conductivity and close to the location of the source. In order to avoid the source singularity, we solve Maxwell's equation with respect to anomalous electric field. The nonuniform rectangular mesh can be transformed to hexahedral mesh in order to simulate the bathymetry effect. The sparse system of finite element equations is solved using a quasi-minimum residual method with a Jacobian preconditioner. We have applied the developed algorithm to compute a typical MCSEM response over a 3D model of a hydrocarbon reservoir located in both isotropic and anisotropic mediums. The modeling results are in a good agreement with the solutions obtained by the integral equation method. HighlightsThis paper develops a novel formulation of the edge-based finite element method for 3D modeling of marine CSEM data in anisotropic conductive medium.The method uses the edge-based vector basis functions, which automatically enforce the divergence free conditions for electric and magnetic fields.The developed method is effective in modeling the seafloor bathymetry using hexahedral mesh.

Summary

1. Introduction

• For accurate interpretation of the subsurface structure using the MCSEM method, the bathymetry effect should be accurately simulated.
• The edge-based finite element method uses vector basis functions defined on the edges of the corresponding elements.
• More advanced preconditioners based on the approximated inverse of the stiffness matrix can be used to speed up the solvers.

4. Comparison with semi-analytical solution for a horizontally layered geoelectrical model

• The frequency of the harmonic electric source is 0.5 Hz.
• The grid is refined nearby the source, target layer and the surface of observation (see Fig. 3).
• Fig. 4 shows a comparison of the anomalous electric field between the finite element solution and the 1D semi-analytical solution.
• One can see that the finite element results are in a good agreement with the semi-analytical solution.
• For this model with the specified source configuration, the y component of secondary electric field, the x and z components of secondary magnetic field are equal to 0 at y¼0.

5. Model of an off-shore hydrocarbon reservoir

• Hz, which is a typical frequency for marine CSEM.
• The EM receivers are located on the seafloor.
• The sparsity pattern for the finite element stiffness matrix is shown in panel (a) of Fig.
• From this figure, one can see that the matrix is very sparse, although the problem size is huge.
• In the second model, the authors consider an anisotropic background conductivity and isotropic anomalous conductivity for the reservoir.

5.1. Model 1: isotropic background and isotropic reservoir

• The numerical result obtained by the edge-based finite element method was compared with the integral equation solution.
• One can clearly see an anomaly around x¼2 km where the offset is about 5 km.
• Due to the page limits, the authors will only show the numerical modeling result for the frequency of 1 Hz in the following sections.
• Due to the page limits, the authors only show a comparison of the convergences of QMR, GMRES and BiCGSTAB solvers for Model 1.
• It took about 20 min to solve this model using the finite element method and about 3 min for the integral equation method on a PC with 2.6 GHz CPU.

5.2. Model 2: anisotropic background with isotropic reservoir

• In a marine environment, the conductivity of sediments shows a strong transverse anisotropy due to the process of sedimentation with the longitudinal conductivity being larger than the transverse conductivity (Ellis et al., 2010; Ramananjaona et al., 2011).
• The macroanisotropy is mainly caused by thin layering when bulk resistivity is measured so that the electric current prefers to travel parallel to the bedding planes (Ellis et al., 2010).
• This transverse anisotropy could have a strong effect on the primary field and the anomalous field could also be distorted significantly.
• The conductivity of seawater and air stays unchanged compared to the previous model.
• As in the previous model, the authors compare the finite element result with the integral equation solution.

5.3. Model 3: isotropic background with anisotropic reservoir

• In practical applications of the MCSEM method, not only the conductivity of the sea-bottom sediments, but also the reservoir conductivity can be anisotropic.
• The reservoir anisotropy is usually weak in comparison to the background conductivity anisotropy (Brown et al., 2012).
• The authors can see that the result obtained by the integral equation method is practically the same as the finite element solution for the anisotropic reservoir model.
• Ex components of the background and total electric field with the frequency of 1 Hz.
• The total memory requirement for solving this problem using finite element and integral equation methods were practically the same as for Model 1.

5.4. Model 4: anisotropic background with anisotropic reservoir

• Finally, the authors study Model 4 having transverse anisotropy of both the background and reservoir conductivities.
• Hz, with the anomaly observed around a point x¼3 km, which corresponds to the 6 km offset.
• The total memory requirement of this problem for finite element and integral equation methods are the same as Model 1.
• The effect of the anisotropy of background conductivity is manifested by shifting the anomaly to larger offset, while the increase of the anisotropy coefficient of the reservoir increases the amplitude of the anomaly without shifting the anomaly significantly.
• Thus, their numerical modeling results confirm ones again that quantitative interpretation of the MCSEM data requires taking into account the effect of anisotropy on the observed data.

6. Modeling the effect of the bathymetry on the EM data

• One of the advantages of the edge-based finite element method is its ability to model the bathymetry effect on the EM data.
• Fig. 23 shows the hexahedral grid for the bathymetry model without a reservoir in the X–Z section at y¼0.
• Fig. 26 shows a comparison of the anomalous field at y¼0 along the bathymetry, with the frequency of 1 Hz, computed using both the edge-based finite element and integral equation methods.
• One can see that the results produced by these two methods show a good agreement.
• Some minor difference may be related to the staircase approximation used in the integral equation method.

7. Conclusions

• The authors have developed an edge-based finite element algorithm to solve the diffusive electromagnetic problem in the 3D anisotropic medium.
• The authors use the edge-based vector basis functions, which automatically enforce the divergence free conditions for electric and magnetic fields.
• The sparse finite element system is solved using the quasiminimum residual method with a Jacobian preconditioner.
• The results of numerical study confirm the accuracy and the efficiency of a new code.

Figures

Fig. 5. A comparison between the finite element result and the 1D semi-analytical solution for the secondary magnetic field with a frequency of 0.5 Hz on the seafloor. The upper panel shows a comparison for the x component of secondary magnetic field at = −y 50 m; the middle panel shows a comparison for the y component of secondary magnetic field at y¼0; the lower panel shows a comparison for the z component of secondary magnetic field at = −y 50 m.

Fig. 8. Plane view of the grid used for the model of the off-shore HC reservoir. The blue area indicates the reservoir and the red star represents the location of a dipole source. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

Fig. 6. 3D view of a model of the off-shore HC reservoir. The resistivity is shown in logarithmic scale.

Fig. 7. Vertical section at y¼0 of the grid used for the model of the off-shore HC reservoir. The blue area indicates the reservoir and the red star represents the location of a dipole source. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

Fig. 28. Bathymetry model with the HC reservoir. A comparison of the x component of the anomalous electric field computed using the finite element (red circles) and integral equation (blue line) methods at y¼0 along the sea floor for a frequency of 1 Hz. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

Fig. 27. X–Z section of the hexahedral grid for bathymetry model with a reservoir. The red star indicates the location of the dipole source oriented in the x direction. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

Fig. 26. Bathymetry model without HC reservoir. A comparison of the x component of the anomalous electric field computed using the finite element (red circles) and integral equation (blue line) methods at y¼0 along the sea floor for a frequency of 1 Hz. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

Fig. 24. A staircase approximation of the bathymetry model use in the framework of the integral equation method.

Fig. 25. Bathymetry model without HC reservoir. A comparison of the x component of the anomalous electric field computed using the finite element (red circles) and integral equation (blue line) methods at y¼0 and z¼500 m for a frequency of 1 Hz. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

Fig. 11. Model 1. A comparison of the x component of the background and total electric fields at y¼400 m for a frequency of 1 Hz. The solid blue line shows the background field while the dashed red line represents the total field. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

Fig. 10. Model 1. A comparison of the x component of the anomalous electric field computed using the finite element (red circles) and integral equation (blue line) methods at y¼0 for a frequency of 1 Hz. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

Fig. 9. Sparsity patterns of the FEM and FDM stiffness matrices for the 3D reservoir model. Panel (a) shows the sparsity pattern of edge-based finite element stiffness matrix (with 41,412,819 nozero entries), while panel (b) presents the sparsity of finite difference stiffness matrix (with 17,568,102 nozero entries) for the same model.

Fig. 23. X–Z section of the hexahedral grid for bathymetry model without a reservoir. The red star indicates the location of the dipole source oriented in the x direction. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

Fig. 21. A comparison of normalized anomalous field at y¼0 for different 3D models for a frequency of 1 Hz.

Fig. 22. A 3D view of a trapezoidal-type hill model of the seafloor bathymetry.

Fig. 20. Model 4. A comparison of the x component of the background and total electric fields at y¼400 m for a frequency of 1 Hz. The solid blue line shows the background field while the dashed red line represents the total field. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

Fig. 1. An illustration of rectangular brick element. The number with circle indicates the node index and the number without circle is the index of edges.

Fig. 14. Convergence plot of QMR solver for 3D reservoir Model 1 for the data with the frequency of 1 Hz compared to the convergence plots of the GMRES and BiCGSTAB solvers.

Fig. 12. Model 1. A comparison of the x component of the anomalous electric field computed using the finite element (red circles) and integral equation (blue line) methods at y¼0 for a frequency of 0.1 Hz. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

Fig. 13. Model 1. A comparison of the x component of the anomalous electric field computed using the finite element (red circles) and integral equation (blue line) methods at y¼0 for a frequency of 0.5 Hz. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

Fig. 15. Model 2. A comparison of the x component of anomalous electric field computed using the finite element (red circles) and integral equation (blue line) methods at y¼0 for a frequency of 1 Hz. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

Frequently asked questions

Q1. What have the authors contributed in "3d controlled-source electromagnetic modeling in anisotropic medium using edge-based finite element method" ?

This paper presents a linear edge-based finite element method for numerical modeling of 3D controlledsource electromagnetic data in an anisotropic conductive medium. The authors use a nonuniform rectangular mesh in order to capture the rapid change of diffusive electromagnetic field within the regions of anomalous conductivity and close to the location of the source. 

Q2. What are the future works in "3d controlled-source electromagnetic modeling in anisotropic medium using edge-based finite element method" ?

Future work will be aimed at the implementation of the high order finite elements and at the use of the unstructured tetrahedral and hexahedron meshes to include seafloor bathymetry and complex geoelectrical structures in the modeling of the MCSEM data.