Three-dimensional controlled-source electromagnetic modelling with a well casing as a grounded source: a hybrid method of moments and finite element scheme

Steel well casings in or near a hydrocarbon reservoir can be used as source electrodes in time-lapse monitoring using grounded line electromagnetic methods. A requisite component of carrying out such monitoring is the capability to numerically model the electromagnetic response of a set of source electrodes of finite length. We present a modelling algorithm using the finite-element method for calculating the electromagnetic response of a three-dimensional conductivity model excited using a vertical steel-cased borehole as a source. The method is based on a combination of the method of moments and the Coulomb-gauged primary-secondary potential formulation. Using the method of moments, we obtain the primary field in a half-space due to an energized vertical steel casing by dividing the casing into a set of segments, each assumed to carry a piecewise constant alternating current density. The primary field is then substituted into the primary-secondary potential finite-element formulation of the three-dimensional problem to obtain the secondary field. To validate the algorithm, we compare our numerical results with: (i) the analytical solution for an infinite length casing in a whole space, excited by a line source, and (ii) a three-layered Earth model without a casing. The agreement between the numerical and analytical solutions demonstrates the effectiveness of our algorithm. As an illustration, we also present the time-lapse electromagnetic response of a synthetic model representing a gas reservoir undergoing water flooding.