# 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 (3 min read)

### 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.

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##### Citations

45 citations

### Cites background or methods from "3D controlled-source electromagneti..."

...The electric field inside each element can be represented as: ∑ EE N= .e i i e i e =1 6 (7) The vector basis functions are continuous on element boundaries and the continuity conditions are imposed directly (Jin, 2002, 2014; Silva et al., 2012; Cai et al., 2014, 2015)....

[...]

...The iterative solvers were widely used for solving 3D EM forward modeling problem for less memory requirement (Axelsson, 1994; Badea et al., 2001; Cai et al., 2014; Freund and Nachtigal, 1991; Puzyrev et al., 2013; Saad, 2003)....

[...]

...For simplicity, we consider our coordinate axes coincide with the principal axes of the conductivity tensor (Cai et al., 2014)....

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...…online 22 November 2016 crossmark have observed that the edge-based FE starts to gain more interests from geophysical community (Mitsuhata and Uchida, 2004; Nam et al., 2007; Mukherjee and Everett, 2011; Schwarzbach et al., 2011; Silva et al., 2012; Kordy et al., 2016; Cai et al., 2014, 2015)....

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##### References

12,575 citations

### "3D controlled-source electromagneti..." refers methods in this paper

...However, this method requires large memory, because it needs all the previously generated Arnoldi vectors to be saved (Saad, 2003; Puzyrev et al., 2013)....

[...]

4,439 citations

### "3D controlled-source electromagneti..." refers methods in this paper

...BiCGStab (Van der Vorst, 1992) and QMR (Freund and Nachtigal, 1991) are both Lanczos-based approaches....

[...]

3,699 citations

### "3D controlled-source electromagneti..." refers background or methods in this paper

...…E N E E N E, , , (9) x e i xi e xi e y e i yi e yi e z e i zi e zi e 1 4 1 4 1 4 where the edge basis functions are defined by the following expressions (Jin, 2002): = + − + −N l l y l y z l z 1 2 2 , (10) x e y e z e c e y e c e z e 1 ⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟ = − + + −N l l y y l z l z 1 2 2 ,…...

[...]

...The integrals in (27) and (28) can be calculated analytically for the rectangular elements (Jin, 2002)....

[...]

...Following the work of Jin (2002) and Silva et al. (2012), we consider the homogeneous Dirichlet boundary conditions in the edge element formulation | =Ω∂e 0 (30) which holds approximately for the anomalous electric field at a distance from the domain with the anomalous conductivity....

[...]

...Similar to the conventional node-based finite element method, the modeling domain can be discretized using rectangular, tetrahedron, hexahedron or other complex elements (Jin, 2002)....

[...]

...The tangential continuity of either electric or magnetic field is imposed automatically on the element's interfaces while the normal components are still can be discontinuous (Jin, 2002)....

[...]

2,774 citations

### "3D controlled-source electromagneti..." refers methods in this paper

...The advantage of the edge-based finite element method, introduced by Nédélec (1980), is that the divergence free conditions are satisfied automatically by an appropriate selection of the basis functions....

[...]

2,182 citations

### "3D controlled-source electromagneti..." refers background or methods in this paper

...…E N E E N E, , , (9) x e i xi e xi e y e i yi e yi e z e i zi e zi e 1 4 1 4 1 4 where the edge basis functions are defined by the following expressions (Jin, 2002): = + − + −N l l y l y z l z 1 2 2 , (10) x e y e z e c e y e c e z e 1 ⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟ = − + + −N l l y y l z l z 1 2 2 ,…...

[...]

...The integrals in (27) and (28) can be calculated analytically for the rectangular elements (Jin, 2002)....

[...]

...The transformation can be described by the following formulas (Jin, 2002):...

[...]

...Following the work of Jin (2002) and Silva et al. (2012), we consider the homogeneous Dirichlet boundary conditions in the edge element formulation | =Ω∂e 0 (30) which holds approximately for the anomalous electric field at a distance from the domain with the anomalous conductivity....

[...]

...Similar to the conventional node-based finite element method, the modeling domain can be discretized using rectangular, tetrahedron, hexahedron or other complex elements (Jin, 2002)....

[...]

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##### Frequently Asked Questions (2)

###### 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.