In some geological formations, borehole resistivity measurements can be simulated using a sequence of 1D models. By considering a 1D layered media, we can reduce the dimensionality of the problem from 3D to 1.5D via a Hankel transform. The resulting formulation is often solved via a semi-analytic method, mainly due to its high performance. However, semi-analytic methods have important limitations such as, for example, their inability to model piecewise linear variations on the resistivity. Herein, we develop a multi-scale finite element method (FEM) to solve the secondary field formulation. This numerical scheme overcomes the limitations of semi-analytic methods while still delivering high performance. We illustrate the performance of the method with numerical synthetic examples based on two symmetric logging-while-drilling (LWD) induction devices operating at 2 MHz and 500 KHz, respectively.

Biological diversity is essential for the maintenance of the ecosystem functions that support life on the planet. Inherent to this diversity is the seemingly endless way in which the biological entities of a natural system interact and affect each other at local and regional scales, conforming complex ecological networks permeable to external forcing. Existing approaches to capture and model such complexity typically make unrealistic or excessively restrictive assumptions. Here we use concepts from open dynamical systems and metacommunity theory to develop a framework in which the system dynamics is a function of both interspecific interactions in the focal system (e.g., a local community of coexisting species) and unobserved biotic and abiotic interactions with the local and regional environment (e.g., the metacommunity). Species in the focal system interact through direct exchanges of biomass (i.e., trophic interactions), as well as through altering the acquisition and/or transformation of biomass by other species (nontrophic interactions). Interactions are affected by environmental fluctuations and by migration and emigration processes, which can take place at different time scales and can be modeled by stochastic differential equations driven by a mixture of continuous and discontinuous processes. In this manner, the proposed framework provides a wider and more flexible representation of the complexity of ecological systems, in comparison with the closed-system paradigm that isolates the system from the environment. Because the core model explicitly recognizes the existence of local and regional processes, it is also a natural starting point to examine spatially structured networks.