Mixed Spectral-Element Method for Overcoming the Low-Frequency Breakdown Problem in Subsurface EM Exploration

One fundamental difficulty in low-frequency subsurface electromagnetic exploration is the low-frequency breakdown phenomenon in numerical computation. It makes the discretized linear system very poorly conditioned and thus difficult to solve. This issue is present in both integral equation and partial differential equation solution methods, and thus has attracted many researchers who have proposed various methods to overcome this difficulty. In this paper, we propose a new mixed spectral element method (mixed SEM) to eliminate this low-frequency breakdown problem and apply this method to solve the subsurface electromagnetic exploration problem. Since Gauss' law is now explicitly enforced in the mixed SEM to make the system matrix well-conditioned even at extremely low frequency, we can solve the linear system from dc to high frequencies. With the proposed method, we study the surface-to-borehole electromagnetic system for hydrocarbon exploration. Numerical examples show that the mixed SEM is accurate and efficient, and has significant advantages over conventional methods.