An iterative finite element time-domain method for simulating three-dimensional electromagnetic diffusion in earth

Abstract

Geophysical Journal International An iterative finite element time-domain method for simulating three-dimensional electromagnetic diffusion in earth Evan Schankee Um, 1 Jerry M. Harris 2 and David L. Alumbaugh 3 1 Earth Sciences Division, Geophysics Department, Lawrence Berkeley National Laboratory, Berkeley, CA, USA. E-mail: evanum@gmail.com USA 3 Chevron Energy Technology, 6001 Bollinger Canyon Road, San Ramon, CA 94583, USA 2 Geophysics Department, Stanford University, Stanford, CA, SUMMARY An iterative finite element time-domain (FETD) method has been developed for simulating transient electromagnetic fields in 3-D diffusive earth media and has been verified through comparisons with analytic and finite-difference time-domain solutions. The adaptive time step doubling (ATSD) method plays an important role in reducing solution run-time by allowing large time steps in late time when high-frequency electric fields are increasingly attenuated in the Earth. We demonstrate that for the ATSD method to work effectively, the conductivity of the air and a drop tolerance of a preconditioner should be carefully selected. The conductivity of the air should not be too large for accurate simulations but also not too small for avoid- ing ill conditioning that results in error amplification in the ATSD method. A proper drop tolerance keeps the eigenvalues of the preconditioned FETD matrices clustered when a time step size is successively doubled during the ATSD processes, resulting in the convergence of iterative solutions with the reasonable number of iterations. A rule of thumb for determining the conductivity of the air and the drop tolerance has been presented. We also present the simultaneous multiple-sources modelling (SM 2 ) approach. The SM 2 approach simultaneously advances the electric fields excited by multiple individual sources in a single time stepping loop. This approach allows multiple sources to share the same preconditioner in the time stepping loop and improves the simulation efficiency per a survey line. Key words: Numerical solutions; Electromagnetic theory; Marine electromagnetics. I N T R O D U C T I O N Transient electromagnetic (TEM) methods have been widely used in a wide range of geophysical research such as deep crustal stud- ies (e.g. Ho¨ rdt et al. 1992; Ho¨ rdt et al. 2002), energy explorations (e.g. Strack 1992), environmental investigations (e.g. Auken et al. ´ rnason et al. 2010). In- 2006) and geothermal explorations (e.g. A terpretations of TEM data in complex geological environments in- creasingly resort to multidimensional inverse modelling. Because forward modelling is a major numerical bottleneck for the inverse modelling, it is important to develop efficient and accurate forward modelling algorithms. Introduced by Yee (1966) in electrical engineering, finite- difference time-domain (FDTD) methods have become one of the standard tools used to simulate TEM fields in geophysics. Their popularity is mainly based on the fact that FDTD methods are relatively easy to implement, easy to use and can provide rea- sonably accurate solutions over a wide range of geophysics prob- lems. For example, Goldman & Stoyer (1983) present 2-D finite- difference modelling of TEM fields in simplified axially symmetric earth media. Wang & Hohmann (1993) develop a 3-D FDTD al- gorithm that advances TEM fields in time using an explicit time stepping approach. Commer & Newman (2004) introduce the par- allel version of Wang & Hohmann (1993) along with new algo- rithmic features such as parallel upward continuation procedures and parallel time stepping. The parallel version has been em- bedded to a 3-D TEM inversion algorithm (Newman & Commer Despite their popularity, the 3-D FDTD methods also have well known disadvantages. For example, their explicit time stepping ap- proach requires small time step sizes to satisfy stability conditions especially when there are large conductivity contrasts. To overcome this issue, Haber et al. (2004) use an unconditionally stable implicit time discretization scheme in their finite volume approach (we also employ an implicit time discretization scheme in this paper). Be- sides, in FDTD modelling, complex geological structures such as seafloor bathymetry, reservoirs and salt domes need to be approx- imated by small stair steps when the structures do not confirm to rectangular cells. Therefore, this approximation approach can quickly increase FDTD problem sizes; FDTD problems resulting from realistic 3-D earth models are usually solved in massively parallel computing environments (Commer & Newman 2004; Newman & Commer 2005; Commer et al. 2008). Furthermore, such stair steps can introduce discretization errors into numerical modelling results especially when sources and receivers are placed on or very close to the complex surface described by the stair steps.