Reduced order borehole induction modelling

This article presents a newreduced order model based on proper orthogonal decomposition (POD) for solving the electromagnetic
equation for borehole modelling applications. The method aims to accurately and efficiently predict the electromagnetic
fields generated by an array induction tool – an instrument that transmits and receives electrical signals along different
positions within a borehole. The motivation for this approach is in the generation of an efficient ‘forward model’ (which
provides solutions to the electromagnetic equation) for the purpose of improving the efficiency of inversion calculations
(which typically require a large number of forward solutions) that are used to determine surrounding material properties.
This article develops a reduced order model for this purpose as it can be significantly more efficient to compute than standard
models, for example, those based on finite elements. It is shown here how the POD basis functions are generated from the
snapshot solutions of a high resolution model, and how the discretised equations can be generated efficiently. The novelty
is that this is the first time such a POD model reduction approach has been developed for this application, it is also unique
in its use of separate POD basis functions for the real and complex solution fields. A numerical example for predicting the
electromagnetic field is used to demonstrate the accuracy of the POD method for use as a forward model. It is shown that
the method retains accuracy whilst reducing the costs of the computation by several orders of magnitude in comparison to
an established method.