Asymptotic models for the electric potential across a highly conductive casing

Abstract

We analyze a configuration that involves a steel-cased borehole, where the casing that covers the borehole is considered as a highly conductive thin layer. We develop an asymptotic method for deriving reduced problems capable of efficiently dealing with the numerical difficulties caused by the casing when applying traditional numerical methods. We derive several reduced models by employing two different approaches, each of them leading to different classes of models. We prove stability and convergence results for these models. The theoretical orders of convergence are supported by numerical results obtained with the finite element method.

Semi-analytical solutions for the problem of the electric potential set in a borehole with a highly conductive casing

“…Fourth-order Gap-ITCs model. In [5] it is observed that the choice of Impedance Transmission Conditions (ITCs) can lead to different asymptotics models. One of the proposed methods consists of computing the solution in the reduced domain Ω ε := R 3 \ Ω ε lay , thus the name Gap-ITCs.…”

“…In that case, the high conductivity of the casing, along with its thinness, greatly complicates numerical simulations. A natural way to circumvent this drawback is to consider asymptotic models avoiding the problems created by the casing [5].…”

“…In this paper, we consider the problem of the electric potential set in a borehole shaped domain surrounded by a metallic casing. This problem involves a realistic configuration, where the conductivity in the casing takes much higher values than those in the layered formation [14,5]. In this configuration, the casing can be seen as a thin layer of uniform thickness ε and its conductivity is proportional to the third negative power of ε.…”

“…As stated above, it is interesting to avoid this casing due to the numerical difficulties it induces. In [5], asymptotic techniques are employed to derive asymptotic models composed of impedance transmission conditions, which are specially designed to replace the casing. Stability and convergence results with respect to the parameter ε have been proved for these asymptotic models in [5].…”

“…In the present work, we will first concentrate on deriving semi-analytical solutions for different models: one reference model that considers the presence of the casing; one rigorous asymptotic model derived in [5] that avoids computations in the casing domain; and one formal asymptotic model from [7] that reduces the presence of the casing to an interface. The standard method we follow consists in employing cylindrical coordinates and assuming material homogeneity in the vertical and angular variables.…”