Parallel hp-Finite Element Simulations of 3D Resistivity Logging Instruments

We simulate electromagnetic (EM) measurements obtained by using one transmitter and one receiver antenna in a borehole environment. The measurements are used to assess electrical properties of rock formations. We have started from a problem where logging instruments as well as rock formation properties are assumed to exhibit axial symmetry around the axis of a vertical borehole. The initial computations are performed with a 2D self-adaptive goal-oriented hp-Finite Element Method (FEM) that delivers exponential convergence rates in terms of the quantity of interest against the CPU time. The 2D mesh obtained from the code delivers very accurate solution at the receiver antenna with minimal number of degrees of freedom in the mesh. The 3D hp finite element mesh is obtained by revolution of obtained 2D mesh. The layers of formation in the 3D mesh are then deviated by 30, 45 and 60 degrees. The 3D parallel hp finite element code is utilized to solve fully 3D problem in deviated wells. The 3D code has been verified by comparison of numerical results with the known exact solution of the problem of antenna radiating into a homogenous space, and by comparison of results of 2D and 3D code for the problem with axially symmetric layers in formation.