Computational Geosciences manuscript No. (will be inserted by the editor) Fast 2.5D Finite Element Simulations of Borehole Resistivity Measurements Ángel Rodríguez-Rozas · David Pardo · Carlos Torres-Verdín Received: May 1, 2018/ Accepted: date Abstract We develop a rapid 2.5-dimensional (2.5D) finite element method for simulation of borehole resis- tivity measurements in transversely isotropic (TI) me- dia. The method combines arbitrary high-order H 1 - and H(curl)-conforming spatial discretizations. It solves problems where material properties remain constant along one spatial direction, over which we consider a Fourier series expansion and each Fourier mode is solved independently. We propose a novel a priori method to construct quasi-optimal discretizations in physical and Fourier space. This construction is based on examining the an- alytical (fundamental) solution of the 2.5D formulation over multiple homogeneous spaces and assuming that some of its properties still hold for the 2.5D problem over a spatially heterogeneous formation. In addition, a simple parallelization scheme over multiple measure- ment positions provides efficient scalability. Our method yields accurate borehole logging simu- lations for realistic synthetic examples, delivering sim- ulations of borehole resistivity measurements at a rate faster than 0.05 seconds per measurement location along the well trajectory on a 96-core computer. Ángel Rodríguez-Rozas* BCAM - Basque Center for Applied Mathematics, Bilbao, Spain E-mail: angel.rodriguez.rozas@gmail.com; *Corresponding author ORCiD: 0000-0002-9703-1043 David Pardo University of the Basque Country (UPV/EHU), Leioa, Spain BCAM - Basque Center for Applied Mathematics, Bilbao, Spain IKERBASQUE (Basque Foundation for Sciences), Bilbao, Spain Carlos Torres-Verdín The University of Texas at Austin, Austin, Texas, USA Keywords Borehole resistivity measurements · fast 2.5D simulations · finite element method · logging while drilling Mathematics Subject Classification (2000) 65Y05 · 65Y99 · 65T99 · 86A04 1 Introduction Borehole geophysical measurements, also known as well logs, often comprise hundreds or even thousands of mea- surements acquired at different locations along the well trajectory [14, 5, 16, 19]. Thus, their computer simula- tion involves the solution of multiple three-dimensional (3D) problems, which is computationally expensive. This prevents their real-time inversion [32,4,12], needed for well geosteering applications [17,5]. In order to reduce the computational cost, it is pos- sible to decrease the problem dimensionality by means of a Fourier or a Hankel series expansion along one or two spatial dimensions, leading to the so-called 2.5D [33,30,22,25,31,28,24,1] and 1.5D [4,27] formulations, respectively. 1.5D problems can be rapidly solved semi- analytically [18]; however, they are inaccurate when dealing with geological faults and/or other high-dimen- sional spatial features that may appear in rock forma- tions [4]. In this work, we focus on 2.5D formulations, which seem to provide an adequate balance between model accuracy and speed of simulations for a large number of borehole geophysical conditions. Efficient 2.5D simulations for real-time well geostee- ring inversion require spatial discretizations (grids) that are often challenging to design. One possibility is the use of a posteriori goal-oriented adaptive methods [10, 6,21]. Unfortunately, these adaptive processes require a