Fast Frequency and Time Domain Integral Equation Modelling for Marine CSEM Applications

In this study we developed algorithms for fast frequency and time domain integral equation modelling for marine controlled-source electromagnetic (CSEM) applications. Solutions of integral equations for CSEM applications in a three-layered earth with an assumed reservoirs is examined using the conjugate gradient fast Fourier transformation (CG-FFT) method, which is used as a reference. For 3D configurations fast computational methods are relevant for both forward and inverse modelling studies. The Born approximation, extended Born approximation, and iterative extended Born approximation are implemented and compared with the full solution of the CG-FFT, even with a reservoir consisting of two separated compartments. We also showed that the approximate results are accurate at the receiver level, which is usually the sea bottom, and inside the reservoir. This iterative method is suitable as a modelling algorithm for solving the inverse scattering problem as well. These methods are based on an electric field domain integral equation formulation. With the help of this method sensitivity analysis using 3D modelling is possible in a timely manner, which is vital for CSEM applications. Our modelling studies investigate to which extent the iterative extended Born approximation method is fast and accurate for forward modelling and could be used for inverse modelling. Sensitivity analysis as a function of the source position and different reservoir sizes validates the accuracy of the iterative extended Born approximation. We also looked into the question in what model configurations two-and-a-half dimensional modelling is a good modelling choice for three-dimensional reservoir response to the diffusive EM field. It is investigated how the accuracy of 2.5D modelling compares with 3D modelling depending on the configurations. Because 2.5D modelling is much faster than 3D modelling, proper use of 2.5D CSEM modelling provides sensitivity analysis of subsurface parameters for many different scenarios in a limited amount of time. Depending on size and depth of the target, hydrocarbon thickness, and extension in cross-line dimension, the required frequency content of the emitted diffusive field and the number of offsets must be determined to assess if the target can be detected. At this stage a decision can be made on whether or not the target reservoir is detectable. From here an optimum survey can be designed based on the forward modelling results. Similarly, in this way the detectability from time-lapse monitoring data of changes in the reservoir due to production can be assessed. Finally we investigated how frequency-domain data can be efficiently transformed to the time-domain. CSEM methods are generally divided into frequency-domain electromagnetic (FDEM) and time-domain (transient) electromagnetic (TDEM) methods, depending on the waveform of the transmitted electrical current. We compared a quasi-analytical method to transform frequency-domain CSEM data back to the time-domain with a numerical transformation. The quasi-analytical method exploits the fact that the kernel of the integral equation has a known behavior as a function of frequency and that the solution to the integral equation can be written as a sum of repeated applications of the kernel to the incident field. A set of expansion functions is found, which have analytically known time domain counterparts, which need only a limited number of frequency values for the transformation back to time. We compared this quasi-analytical method, coined the Diffusive Expansion Method in CSEM applications with two other numerical methods, Gaver-Stehfest method and an optimized form of the fast Fourier transformation method where the data is required at an minimum number of discrete frequency values such that the data at intermediate frequency values can be accurately obtained by interpolation. We found that the Diffusion Expansion Method is a good candidate for CSEM frequency-to-time conversion of data for any kind of subsurface model and survey configuration.