The Inverse Problem of Electrical Capacitance Tomography and its Application to Gas-Oil 2-Phase Flow Imaging

Electrical-capacitance tomography (ECT) is a novel technique for the non-invasive internal visualization of industrial processes such as mixing, separation and multiphase flow, involving electrically nonconducting mixtures. One of the most promising potential applications of ECT is the determination of the amount of gas, oil and water produced by an oil well. The principle of ECT is to place a ring of between 8 and 16 adjacent electrodes (known as the sensor) around a particular cross-section of the process vessel or pipe (which must have insulating walls in that area), and to measure all the inter-electrode capacitances. The measurements are then used to generate an image of the electrical permittivity variations (reflecting the mixture composition) inside the sensor through a suitable inversion algorithm. In this work, a new algorithm based on a set of integral equations, which are equivalent to the differential model formulation, is presented. Using this integral relationships, the problem of identifying the unknown permittivity reduces to solving a quadratic optimization problem with quadratic constraints, for the coefficients of the expansion of the permittivity with respect to an orthonormal base conveniently chosen in a Sobolev space of functions defined in the sensor crosssection. Key-Words: Electrical Tomography, Capacitance, Inversion Algorithm, Imaging, Two-Phase Flow