This article presents a new reduced 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.

A B S T R A C TAn artificial neural network method is proposed as a computationally economic alternative to numerical simulation by the Biot theory for predicting borehole seismoelectric measurements given a set of formation properties. Borehole seismoelectric measurements are simulated using a finite element forward model, which solves the Biot equations together with an equation for the streaming potential. The results show that the neural network method successfully predicts the streaming potentials at each detector, even when the input pressures are contaminated with 10% Gaussian noise. A fast inversion methodology is subsequently developed in order to predict subsurface material properties such as porosity and permeability from streaming potential measurements. The predicted permeability and porosity results indicate that the method predictions are more accurate for the permeability predictions, with the inverted permeabilities being in excellent agreement with the actual permeabilities. This approach was finally verified by using data from a field experiment. The predicted permeability results seem to predict the basic trends in permeabilities from a packer test. As expected from synthetic results, the predicted porosity is less accurate. Investigations are also carried out to predict the zeta potential. The predicted zeta potentials are in agreement with values obtained through experimental self potential measurements.