3D controlled-source electromagnetic modeling in anisotropic medium using edge-based finite element method
Introduction
Controlled-source electromagnetic (CSEM) method has been widely used in geophysical exploration on land for decades (Ward and Hohmann, 1988, Zhdanov and Keller, 1994). The marine controlled-source electromagnetic (MCSEM) method was also applied for the off-shore hydrocarbon (HC) exploration (Srnka et al., 2006, Constable and Srnka, 2007, Um and Alumbaugh, 2007, Andréis and MacGregor, 2008, Zhdanov, 2010). The subsurface conductivity structure could be very complex due to bathymetry and a lateral variation of the conductivity of the sea-bottom sediments. In this case, a full 3D modeling of diffusive electromagnetic data is desirable to correctly interpret the field MCSEM data (Silva et al., 2012). The 3D electromagnetic modeling requires solving the diffusive Maxwell's equations in a discretized form. The most popular numerical techniques for EM forward modeling are integral equation (IE), finite difference (FD), and finite element (FE) methods.
Compared to the integral equation and finite difference methods, the finite element method is more suitable for modeling of EM response in a complex geoelectrical structure. In a 3D scenario, the subsurface can be discretized using either regular brick, hexahedral, or tetrahedron elements. The electric and magnetic fields within each element can be approximated by either linear or higher order polynomial functions. Since the support of the finite element basis function is small, the resulting stiffness matrix is very sparse, which makes it easy to store. In the paper, we also compared the sparsity pattern of the stiffness matrix created by our finite element method and that from the finite difference method. Although the finite element stiffness matrix is less sparse than the finite difference stiffness matrix for the same model, the bandwidth of finite element stiffness matrix is much narrower.
The node-based finite element method was applied in the past to model EM data by solving the coupled equations for the vector and scalar potentials and also by solving Maxwell's equations for electric and magnetic fields (e.g., Zhdanov, 2009). However, for accurate computations, the divergence free condition for the electric and magnetic fields in the source free regions needs to be addressed by an additional penalty term to alleviate possible spurious solutions (e.g., Jin, 2002).
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. The basis function of the Nédélec element is a vector function defined along the element edges and at the center of each edge. 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). In this paper, we present the formulation of Maxwell's equation for the electric field directly using edge-based finite element approach and the continuity of tangential electric field can be imposed directly. We can also formulate Maxwell's equation for magnetic field in the same way and the continuity of tangential magnetic field will be imposed directly in the formulation. In spite of the fact that the edge-based finite element method was widely used in electrical engineering for over 30 years, it started gaining the interest from the geophysical community recently only. Mitsuhata and Uchida (2004) implemented an edge-based finite element modeling algorithm for solving 3D magnetotelluric problem. Schwarzbach et al. (2011) applied linear and higher order edge element for the modeling of marine CSEM data using tetrahedron discretization to better simulate the seafloor bathymetry. Silva et al. (2012) proposed a finite element multifrontal method which is very efficient for 3D CSEM modeling in the frequency domain. One needs to note that all these formulations of the 3D CSEM problem assume the subsurface conductivity to be isotropic.
In the marine environment, the subsurface conductivity is usually characterized by strong anisotropy due to sedimentation. Generally the subsurface is more conductive in the horizontal direction than in the vertical direction for a horizontally stratified medium. The anisotropy of conductive sediment can affect the response of the electromagnetic field in a marine CSEM survey and this effect has already been well studied (Ramananjaona et al., 2011, Ellis et al., 2010, Brown et al., 2012, Newman et al., 2010). Obviously, to accurately interpret the marine CSEM data, the conductivity anisotropy needs to be considered in the forward modeling. There are already series of papers published on 2.5D and 3D modeling of marine CSEM data in the anisotropic medium using node-based finite element and finite difference methods (Kong et al., 2008, Weiss and Newman, 2002).
Meanwhile, another challenge arising in the interpretation of MCSEM data is strong distortion of the data by the effect of seafloor bathymetry (e.g., Sasaki, 2011). For accurate interpretation of the subsurface structure using the MCSEM method, the bathymetry effect should be accurately simulated. The finite element method is very well suited to solve this problem.
In this paper, we formulate the 3D marine CSEM problem using the linear edge-element method in the anisotropic medium. In a general case, we assume that the model has a triaxial conductivity anisotropy. In order to compare the EM response with integral equation solution (Zhdanov et al., 2006), we also consider transverse anisotropy in our model study. To avoid the source singularity, we solve Maxwell's equations with respect to anomalous electric field. The background EM field for the layered background model is computed using Hankel transforms (Anderson, 1989, Guptasarma and Singh, 1997). For simplicity, we use a rectangular element for the flat seafloor model. In order to simulate the bathymetry effect, the rectangular element is transformed into hexahedral one by shifting the vertical coordinate. The sparse finite element system of equations is solved using a quasi-minimum residual method (QMR) with a Jacobian preconditioner.
To validate our code, we first test it for a 1D model with an analytical solution. For a full 3D anisotropic problem, we compare the numerical results from our method and integral equation solutions.
Section snippets
Formulation of the EM field equations with respect to anomalous field in anisotropic medium
The low frequency electromagnetic field, considered in geophysical application, satisfies the following Maxwell's equations (Zhdanov, 2009):where we adopt the harmonic time dependence , ω is the angular frequency, is the free space magnetic permeability, is the distribution of source current, and the term is the induced current in the conductive earth, is the conductivity tensor which is defined as follows:In (3), are principle
Edge-based finite element analysis
The edge-based finite element method uses vector basis functions defined on the edges of the corresponding elements. 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).
For simplicity, we will discuss the rectangular grid first. Fig. 1 is an illustration of the bricks grid that we used with node and edge indexing. Following the work by Jin (2002), we denote the edge
Comparison with semi-analytical solution for a horizontally layered geoelectrical model
In this section, we validate our algorithm by considering an isotropic horizontally layered geoelectrical model as shown in Fig. 3. The background is a seawater-sediment model with air–earth interface at z=0 and the depth of seawater is 1000 m. The conductivities of air, seawater and sediments are , and , respectively. An infinite horizontal resistive layer with a conductivity of is embedded in the sediments from the depth of 1400 m to 1500 m. The electromagnetic
Model of an off-shore hydrocarbon reservoir
In this section, we consider a 3D model of a hydrocarbon (HC) reservoir in a marine environment. We consider a three-layered background model where the first layer is air with a conductivity of , the second layer is seawater with a conductivity of and the bottom layer is sediment with a horizontal conductivity of . The horizontal conductivity of the reservoir is set to be . The seawater depth is 1000 m and it is separated from the sediment by a horizontal flat plane.
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. The non-uniform rectangular elements can be applied to simulate very simple bathymetry by using a staircase approximation in a similar way as in the framework of finite difference or integral equation approaches. In order to simulate more complex seafloor topography, one needs to use more flexible hexahedral or tetrahedral elements. There are exist well-developed software
Conclusions
We have developed an edge-based finite element algorithm to solve the diffusive electromagnetic problem in the 3D anisotropic medium. This method can be specifically useful for modeling geophysical electromagnetic data observed by marine controlled-source electromagnetic (MCSEM) surveys in the areas with anisotropic conductivity of the sea-bottom geological formations and a complex bathymetry. We have considered a typical MCSEM survey with an electric dipole source. In order to avoid the source
Acknowledgement
The authors acknowledge the support of the University of Utah's Consortium for Electromagnetic Modeling and Inversion (CEMI) and TechnoImaging. Thanks to Dr. Gribenko for his suggestions and fruitful discussions. Thanks to our CEMI student Daeung Yoon for creating finite difference sparsity matrix presented in this paper. The project was partially supported by the National Natural Science Foundation of China with Grant codes of 40974077 and 41164004. The authors are thankful to both reviewers
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2022, Journal of Applied GeophysicsCitation Excerpt :A fast and high-precision 3D forward modeling algorithm is the important engine for 3D inversion. Typically, the mainstream 3D geoelectromagnetic numerical simulation methods include integral equation method (Xiong, 1992; Kuvshinov et al., 2002; Zhdanov et al., 2006), finite volume method (Jahandari and Farquharson, 2015; Jahandari et al., 2017; Lu and Farquharson, 2020), finite difference method (Haber et al., 2000; Weiss and Newman, 2003), and finite element method (Tang et al., 2010; Li et al., 2012; Ren et al., 2013, 2014; Cai et al., 2014; Grayver, 2015; Yin et al., 2016; Liu et al., 2018). Spectral element method is a high-precision numerical simulation method developed from the combination of finite element method and spectral method.