3D Modeling Of Geo-Electromagnetic Fields Using Continuous And Discontinuous Finite Element Methods

As an important branch of geophysical prospecting techniques, electromagnetic (EM) sounding detects the underground electrical structure by measuring and analyz-ing the EM field propagating in the conductive earth media, in which reflection, re-fraction and attenuation of the EM field occur where heterogeneity exists. The EM sounding techniques, which were first applied for near surface exploration more than a century ago, have been widely used in various prospecting areas. With the dwindling of oil, gas and solid mineral reserves, geophysical prospecting techniques are facing new challenges,such as greater prospecting depth and more complicated geological cir-cumstance. New construction techniques and quality control in engineering also require more precise detection methods. These new trends require not only high-precision in-struments for EM sounding, but also accurate and efficient simulation algorithms which are able to handle complicated geological models. This thesis is focusing on the latter, the advanced algorithms for direct-current (DC) resistivity modeling and frequency-and time-domain EM simulations. Finite element (FE) methods on unstructured grids, including continuous and discontinuous FE methods, have been applied in the simu-lations. The main difference between the continuous and discontinuous FE is that the former requires a continuous finite-dimensional solution space while the latter allows discontinuity in the solution.There have been vast researches in the simulation and application for point elec-trode (PE) sources. The numerical simulation for long electrode (LE) source, however, is rare, as some theoretical obstacles are not yet overcome. The biggest problem is that the current density distribution on the surface of the electrode is extremely hard to determine when it is buried in heterogeneous media. In addition, the construction of a vimineous vertical electrode in a big mesh model is tedious, let along arbitrary LEs penetrating through heterogeneous blocks. When FE method is applied, the sharp re-sistivity contrast between the metallic electrode and the surrounding media will also lead to an ill-posed linear system and result in slow convergence and degraded accu-racy. In this thesis, the LE source is regarded as a perfectly conducting line source with finite length, which not only gets rid of the convergence problem resulting from the resistivity contrast but also leaves the construction of the electrode out of the mesh model. A continuous FE method with nodal basis is applied to the simulation for ar-bitrary LE sources on unstructured grids, combined with an approximated equation for determining the current density along the electrode which allows the electrode model to penetrate through heterogeneity in the simulations. Both total- and secondary field methods are applied in the simulation and the numerical solution is verified using the analytic solutions. The apparent resistivity formulas of LE source arrays are proposed for the first time. The electrical fields excited by PE and LE sources are compared and the simulation of ground and borehole models show that the arrays with LE sources are more sensitive to the resistivity anomalies than those with PE sources. The mixed bipole source, which is a compromise in the case that the LEs are insufficient in work sites, is also tested and the simulation result indicates an enhanced detection ability of the mixed bipole array. A time-lapsed LE source simulation in a hydraulic fracturing model is also performed to investigate the capability of the LE array for oil field application and the result shows that the LE array can be used for monitoring the hydrofracturing process.The main difficulties in the frequency-domain EM simulation with FE method are the normal discontinuity and divergence-free condition of the electrical field. The the-ory of the nodal FE method is solid and it is widely used in numerical simulations, but it is not suitable for simulating EM field directly in heterogeneous media. Instead, the equations of the continuous vector and scalar potentials need to be solved first and then the solutions can be converted to EM field components. While solving the potentials, the divergence condition still needs to be satisfied in order to eliminate the spurious solution. In a word, it is cumbersome to solve the EM field problem in heterogeneous media using nodal FE method. To overcome these difficulties, the vector FE method is chosen for the 3D simulation of frequency-domain controlled-source electromagnetics (CSEM). The degree of freedoms is defined on the edges of the elements in vector FE method, which only requires the continuity of the tangential components. Additionally, the edge vector basis satisfies the divergence-free condition naturally. Consequently, it is a better choice to use vector FE method for EM simulation. The curl-curl equation of the electrical field is used as the governing equation and the general variation principle is applied to obtain the equivalent variational problem. The vector FE method is then applied to the discretization of the variational problem and the linear system is solved to obtain the frequency-domain EM solution. Sommerfeld radiation boundary condition, which offers better absorbing efficiency, is imposed for the domain truncation. The stability and accuracy of the vector FE algorithm are verified by comparing with a ID analytic solution and a 2D FE solution. The influence of the submarine topography to the CSEM exploration is investigated with numerical simulations and the results show that submarine topography will give rise to significant anomalous signal and may cover the signal from the oil and gas reservoir. With the use of unstructured tetrahedral grids, the vector FE algorithm is able to handle extremely complicated submarine models.As it is easy to implement high order interpolation and has the capability to handle discontinuous solutions, together with high parallel performance, discontinuous finite method (also known as discontinuous Galerkin method) has become one of the most at-tractive numerical methods during the last two decades. However, this advanced numer-ical method is not yet employed to the simulation of low frequency geo-electromagnetic field simulation. We have accomplished the discontinuous FE algorithm for time do-main EM field simulation in high and low frequency regimes. For the geophysical purpose, the algorithm can be applied to the simulations of ground penetrating radar (GPR) and transient electromagnetics (TEM). The TEM fields excited by electric and magnetic dipoles and arbitrary large loops loaded with various current pulses can be simulated. The uniaxial perfectly matched layer (UPML) is employed to truncate the calculation domain, which leads to a frequency dependency in the governing equation. In order to decouple the dependency,12 auxiliary variables are introduced and the cor-responding Maxwell’s equation in UPML is derived. The UPML is able to absorb the out-going wave and reduce the reflection from the artificial boundary efficiently. As a result, the calculation domain can be shrank greatly. A class of hierarchical orthogonal bases is chosen for spatial discretization while the Runge-Kutta method is applied for time integration. The computational complexity is greatly diminished with the use of the orthogonal bases, as the elemental mass matrix become a diagonal and the stiffness matrix maintains well-conditioned even for high order basis functions. The simulation result shows that the accuracy of the discontinuous FE solution is close to order k+1 when k-th order basis functions are applied. The discontinuous FE algorithm is imple-mented in parallel with OpenMP library and the parallel test shows superior parallel performance of the algorithm.