custEM: Customizable finite-element simulation of complex controlled-source electromagnetic data

We have developed the open-source toolbox custEM (customizable electromagnetic modeling) for the simulation of complex 3D controlled-source electromagnetic (CSEM) problems. It is based on the open-source finite-element library FEniCS, which supports tetrahedral meshes, multiprocessing, higher order polynomials, and anisotropy. We use multiple finite-element approaches to solve the time-harmonic Maxwell equations, which are based on total or secondary electric field and gauged potential formulations. In addition, we develop a secondary magnetic field formulation, showing superior performance if only magnetic fields are required. Using Nédélec basis functions, we robustly incorporate the current density on the edges of the mesh for the total field formulations. The latter enable modeling of CSEM problems taking topography into account. We evaluate semianalytical 1D layered-earth solutions with the pyhed library, supporting arbitrary configurations of dipole or loop sources for secondary field calculations. All system matrices have been modified to be symmetric and solved in parallel with the direct solver MUMPS. Aside from the finite-element kernel, mesh generation, interpolation, and visualization modules have been implemented to simplify and automate the modeling workflow. We prove the capability of custEM, including validation against analytic-solutions, crossvalidation of all implemented approaches, and results for a model with 3D topography with four examples. The object-oriented implementation allows for customizable modifications and additions or to use only submodules designed for special tasks, such as mesh generation or matrix assembly. Therefore, the toolbox is suitable for crossvalidation with other codes and as the basis for developing 3D inversion routines

PETGEM: A parallel code for 3D CSEM forward modeling using edge finite elements

We present the capabilities and results of the Parallel Edge-based Tool for Geophysical Electromagnetic modeling (PETGEM), as well as the physical and numerical foundations upon which it has been developed. PETGEM is an open-source and distributed parallel Python code for fast and highly accurate modeling of 3D marine controlled-source electromagnetic (3D CSEM) problems. We employ the N\'ed\'elec Edge Finite Element Method (EFEM) which offers a good trade-off between accuracy and number of degrees of freedom, while naturally supporting unstructured tetrahedral meshes. We have particularised this new modeling tool to the 3D CSEM problem for infinitesimal point dipoles asumming arbitrarily isotropic media for low-frequencies approximations. In order to avoid source-singularities, PETGEM solves the frequency-domain Maxwell's equations of the secondary electric field, and the primary electric field is calculated analytically for homogeneous background media. We assess the PETGEM accuracy using classical tests with known analytical solutions as well as recent published data of real life geological scenarios. This assessment proves that this new modeling tool reproduces expected accurate solutions in the former tests, and its flexibility on realistic 3D electromagnetic problems. Furthermore, an automatic mesh adaptation strategy for a given frequency and specific source position is presented. We also include a scalability study based on fundamental metrics for high-performance computing (HPC) architectures.Comment: \c{opyright} 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/ This project has received funding from the EC-H2020 under the Marie Sklodowska-Curie grant agreement No. 644202, and from the EC-H2020 under the HPC4E Project, grant agreement No. 68977

Evaluation of three approaches for simulating 3D time-domain electromagnetic data

We implemented and compared the implicit Euler time-stepping approach, the inverse Fourier Transform-based approach, and the Rational Arnoldi method for simulating 3D transient electromagnetic data. We utilize the finite-element method with unstructured tetrahedral meshes for the spatial discretization supporting irregular survey geometries and anisotropic material parameters. Both, switch-on and switch-off current waveforms, can be used in combination with direct current solutions of Poisson problems as initial conditions. Moreover, we address important topics such as the incorporation of source currents and opportunities to simulate impulse as well as step response magnetic field data with all approaches for supporting a great variety of applications. Three examples ranging from simple to complex real-world geometries and validations against external codes provide insight into the numerical accuracy, computational performance, and unique characteristics of the three applied methods. We further present an application of logarithmic Fourier transforms to convert transient data into the frequency domain. We made all approaches available in the open-source Python toolbox custEM, which previously supported only frequency-domain electromagnetic data. The object-oriented software implementation is suited for further elaboration on distinct modeling topics and the presented examples can serve for benchmarking other codes

Meshing strategies for 3d geo-electromagnetic modeling in the presence of metallic infrastructure

In 3D geo-electromagnetic modeling, an adequate discretisation of the modeling domain is crucial to obtain accurate forward responses and reliable inversion results while reducing the computational cost. This paper investigates the mesh design for subsurface models, including steel-cased wells, which is relevant for many exploration settings but still remains a numerically challenging task. Applying a goal-oriented mesh refinement technique and subsequent calculations with the high-order edge finite element method, simulations of 3D controlled-source electromagnetic models in the presence of metallic infrastructure are performed. Two test models are considered, each needing a distinct version of approximation methods to incorporate the conductive steel casings of the included wells. The influence of mesh quality, goal-oriented meshing, and high-order approximations on problem sizes, computational cost, and accuracy of electromagnetic responses is investigated. The main insights of our work are: (a) the applied numerical schemes can mitigate the computational burden of geo-electromagnetic modeling in the presence of steel artifacts; (b) investigating the processes driving the meshing of models with embedded metallic infrastructures can lead to adequate strategies to deal with the inversion of such electromagnetic data sets. Based on the modeling results and analyses conducted, general recommendations for modeling strategies are proposed when performing simulations for challenging steel infrastructure scenarios.Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. The work of O.C-R. has received funding from the Ministerio de Educación y Ciencia (Spain) under Project TED2021-131882B-C42.The code development of P.R. has been financed by the Smart Exploration project. Smart Exploration has received funding from the European Union’s Horizon 2020 Framework Programme under grant agreement N∘ 775971. The computations were enabled by resources provided by the Swedish National Infrastructure for Computing (SNIC) at UPPMAX partially funded by the Swedish Research Council through grant agreement N∘ SNIC 2021/22-883.Peer ReviewedPostprint (published version

Land CSEM Simulations and Experimental Test Using Metallic Casing in a Geothermal Exploration Context: Vallès Basin (NE Spain) Case Study

Controlled-source electromagnetic (CSEM) measurements are complementary data for magnetotelluric (MT) characterization although its methodology on land is not sufficiently developed and tested as in marine environments. Acquiring expertise in CSEM is crucial for surveys in places where MT cannot be performed due to high levels of cultural noise. To acquire that expertise, we perform CSEM experiments in the Vallès fault [Northeast (NE), Spain], where MT results have been satisfactory and allow us to verify the CSEM results. The Vallès basin is relevant for potential heat generation because of the presence of several geothermal anomalies and its nearby location in urban areas. In this article, we present the experimental setup for that region, a 2-D joint MT+CSEM inverse model, several 3-D CSEM simulations in the presence of metallic casing, and its comparison with real data measurements. We employ a parallel and high-order vector finite element algorithm to discretize the governing equations. By using an adapted meshing strategy, different scenarios are simulated to study the influence of the source position/direction and the conductivity model in a metallic casing presence. An excellent agreement between the simulated data and analytical/real field data demonstrates the feasibility of study metallic structures in realistic configurations. Our numerical results confirm that metallic casing strongly influences electromagnetic (EM) responses, making surface measurements more sensitive to resistivity variations near the metallic structure. It could be beneficial getting higher signal-to-noise ratios and sensitivity to deep targets. However, such a casing effect depends on the input model (e.g., conductivity contrasts, frequency, and geometry)

A multi-domain decomposition-based Fourier finite element method for the simulation of 3D marine CSEM measurements

We introduce a multi-domain decomposition Fourier finite element (MDDFFE) method for the simulation of three-dimensional (3D) marine controlled source electromagnetic measurement (CSEM). The method combines a 2D finite element (FE) method in two spatial dimensions with a hybrid discretization based on a Fourier FE method along the third dimension. The method employs a secondary field formulation rather than the total field formulation. We apply the MDDFFE method to several synthetic marine CSEM examples exhibiting bathymetry and/or multiple 3D subdomains. Numerical results show that the use of the MDDFFE method reduces the problem size by as much as 87 % in terms of the number of unknowns, without any sacrifice in accuracy

Edge-elements formulation of 3D CSEM in geophysics : a parallel approach

Electromagnetic methods (EM) are an invaluable research tool in geophysics whose relevance has increased rapidly in recent years due to its wide industrial adoption. In particular, the forward modelling of three-dimensional marine controlled-source electromagnetics (3D CSEM FM) has become an important technique for reducing ambiguities in the interpretation of geophysical datasets through mapping conductivity variations in the subsurface. As a consequence, the 3D CSEM FM has application in many areas such as hydrocarbon/mineral exploration, reservoir monitoring, CO2 storage characterization, geothermal reservoir imaging and many others due to there quantities often displaying conductivity contrasts with respect to their surrounding sediments. However, the 3D CSEM FM at real scale implies a numerical challenge that requires an important computational effort, often too high for modest multicore computing architectures, especially if it fuels an inversion process. On the other hand, although the HPC code development is dominated by compiled languages, the popularity of high-level languages for scientific computations has increased considerably. Among all of them, Python is probably the language that has shown more interest, mainly because of flexibility and its simple and clean syntax. However, its use for HPC geophysical applications is still limited, which suggests a path for research, development and improvement. Therefore, this thesis reports the attempts at designing and implementing a methodology that has not been systematically applied for solving 3D CSEM FM with an HPC application baked upon Python. The net contribution of this effort is the development and documentation of a new open-source modelling code for 3D CSEM FM in geophysics, namely, the Parallel Edge-based Tool for Geophysical Electromagnetic Modelling (PETGEM). The importance of having this modelling tools lies in the fact that they provide synthetic results that can be compared with real data which has a practical use both in the industry and academia. Still, available 3D CSEM FM codes are usually written in low-level languages whose implemented methods are often innaccessible to the scientific community since they are commercial. PETGEM is written mostly in Python and relies on mpi4py and petsc4py packages for parallel computations. Other scientific Python packages used include Numpy andScipy. This code is designed to cope with the main challenges encountered within the numerical simulation of the problem under consideration: tackle realistic problems with accuracy, efficiency and flexibility. It uses the Nédélec Edge Finite Element Method (EFEM) as discretisation technique because its divergence-free basis is very well suited for solving Maxwell¿s equations. Furthermore, it supports completely unstructured tetrahedral meshes which allows the representation of complex geometries and local refinement, positively impacting the accuracy of the solution. The parallel implementation of the code using shared/distributed-memory architectures is investigated and described throughout this document. In addition, the thesis deals with the numerical and physical challenges of the 3D CSEM FM problem. Through this work, frequency-domain Maxwell's equations have been discretised using EFEM and validated by comparison with analytical solutions and published data, proving that modelling results are highly accurate. Moreover, this work discusses an automatic mesh adaptation strategy and the convergence rate of the iterative solvers that are widely used in the literature for solving the EM problem is presented. In summary, this thesis shows that it is possible to integrate Python and HPC for the solution of 3D CSEM FM at large scale in an effective way. The new modelling tool is easy to use and the adopted algorithms are not only accurate and efficient but also have the possibility to easily add or remove components without having to rewrite large sections of the code.Los métodos electromagnéticos (EM) son una herramienta de investigación inestimable en geofísica, cuya relevancia ha aumentado rápidamente en los últimos años debido a su amplia adopción industrial. En particular, el modelado electromagnético de fuente controlada (3D CSEM FM) se ha convertido en una técnica importante para reducir las ambigüedades en la interpretación de datos geofísicos a través del mapeo de las variaciones de conductividad en el subsuelo. Como resultado, el 3D CSEM FM tiene aplicación en muchas áreas como la exploración de hidrocarburos/minerales, monitoreo de yacimientos, caracterización de almacenamiento de CO2, imágenes de yacimientos geotérmicos, entre otros, debido a que éstos muestran contrastes de conductividad con respecto a sus sedimentos circundantes. Sin embargo, el 3D CSEM FM a escala real implica un desafío numérico que requiere un esfuerzo computacional importante, a menudo demasiado exigente para arquitecturas multicore modestas, especialmente si éste forma parte de un proceso de inversión. Por otra parte, aunque el desarrollo aplicaciones HPC está dominado por lenguajes compilados, la popularidad de los lenguajes de alto nivel para cómputo científico ha aumentado considerablemente. Entre todos ellos, Python es probablemente el idioma que ha mostrado más interés, principalmente a su flexibilidad y sintaxis simple. Sin embargo, su uso para geocómputo con HPC sigue siendo limitado, lo que sugiere un camino para la investigación, el desarrollo y la mejora. Por lo tanto, esta tesis describe el diseño e implementación de una metodología que hasta ña fecha no se ha aplicado sistemáticamente para resolver el 3D CSEM FM con una aplicación HPC basada en Python. La contribución neta de este esfuerzo es el desarrollo y documentación de un nuevo código open-source para el modelado 3D CSEM FM en geofísica, es decir, Parallel Edge-based Tool for Geophysical Electromagnetic Modelling (PETGEM). La importancia del desarrollo de estas herramientas radica en el hecho de que proporcionan resultados sintéticos que pueden ser comparados con datos reales, lo cual tiene un uso práctico en la industria y el mundo académico. A pesar de ello, los códigos disponibles para 3D CSEM FM suelen estar escritos en lenguajes de bajo nivel, y en muchos casos sus métodos no son accesibles a la comunidad científica ya que son comerciales. PETGEM ha sido principalmente escrito en Python y se basa en paquetes mpi4py y petsc4py para cálculos paralelos. El código está diseñado para hacer frente a los principales desafíos que se encuentran en la simulación numérica del problema en cuestión: abordar problemas realistas con precisión, eficiencia y flexibilidad. Además, utiliza el Método de Elementos Finitos de Borde (EFEM) como técnica de discretización ya que sus bases son muy adecuadas para resolver las ecuaciones de Maxwell. Además, soporta mallas tetraédricas no estructuradas que permiten la representación de geometrías complejas y refinamiento local, impactando positivamente la precisión de la solución. A lo largo del documento se investiga la implementación paralela en arquitecturas de memoria compartida/distribuida. Además, la tesis revisa los desafíos numéricos y físicos del problema 3D CSEM FM. A través de este trabajo, las ecuaciones de Maxwell en el dominio de la frecuencia se han discretizado utilizando EFEM y validado contra soluciones analíticas y datos previamente publicados, lo que demuestra que los resultados del modelado son precisos. Por otra parte, este trabajo discute una estrategia de adaptación automática de malla y la tasa de convergencia de los solvers iterativos que se utilizan ampliamente en la literatura. En resumen, esta tesis muestra que es posible integrar Python y HPC para la solución de 3D CSEM FM a gran escala de una manera efectiva. La nueva herramienta de modelado es fácil de usar y los algoritmos adoptados no sólo son precisos y eficientes, sino también flexibles

Synthetic modelling study of marine controlled-source electromagnetic data for hydrocarbon exploration

The marine controlled-source electromagnetic method (CSEM) is a geophysical technique for mapping subsurface electrical resistivity structure in the offshore environment. It has gained ground in recent years as a tool for remote detection and mapping of hydrocarbon reservoirs as it serves as an independent yet complementary method to seismic acquisition. While CSEM data contains useful information about the subsurface, modelling and inversion are required to convert data into interpretable resistivity images. Improvement of modelling tools will assist in closing the gap between acquisition and interpretation of CSEM data. The primary focus of this study was to explore the limits of our present modelling capabilities in the context of marine electromagnetic scenarios. Software based on the three-dimensional CSEM finite-element forward code CSEM3DFWD (Ansari and Farquharson, 2014; Ansari et al., 2015) was employed in this study. While testing of this software had been expanded to models of relevance to mineral exploration, its performance for models which are representative of marine geologic environments, in particular those which are encountered in offshore oil and gas exploration, had not yet been investigated. In this study, marine models of increasing complexity were built and tested, with the ultimate goal of synthesizing marine CSEM data for three-dimensional earth models which were complete in their description of the subsurface. Computed responses were compared to results existing in the literature, when available. To investigate the capability of the code in modelling realistic scenarios, forward solutions were computed for a marine reservoir model based on the real-life North Amethyst oil field, located in the Jeanne d’Arc Basin, offshore Newfoundland. When the capability of modelling realistic earth models is fully realized, forward modelling may be used to assess the utility of the marine CSEM method as a tool for hydrocarbon detection and delineation in specific offshore scenarios

Doctor of Philosophy

dissertationThe motivation for this work is the forward and inverse problem for magnetotellurics, a frequency domain electromagnetic remote-sensing geophysical method used in mineral, geothermal, and groundwater exploration. The dissertation consists of four papers. In the first paper, we prove the existence and uniqueness of a representation of any vector field in H(curl) by a vector lying in H(curl) and H(div). It allows us to represent electric or magnetic fields by another vector field, for which nodal finite element approximation may be used in the case of non-constant electromagnetic properties. With this approach, the system matrix does not become ill-posed for lowfrequency In the second paper, we consider hexahedral finite element approximation of an electric field for the magnetotelluric forward problem. The near-null space of the system matrix for low frequencies makes the numerical solution unstable in the air. We show that the proper solution may obtained by applying a correction on the null space of the curl. It is done by solving a Poisson equation using discrete Helmholtz decomposition. We parallelize the forward code on multicore workstation with large RAM. In the next paper, we use the forward code in the inversion. Regularization of the inversion is done by using the second norm of the logarithm of conductivity. The data space Gauss-Newton approach allows for significant savings in memory and computational time. We show the efficiency of the method by considering a number of synthetic inversions and we apply it to real data collected in Cascade Mountains. The last paper considers a cross-frequency interpolation of the forward response as well as the Jacobian. We consider Pade approximation through model order reduction and rational Krylov subspace. The interpolating frequencies are chosen adaptively in order to minimize the maximum error of interpolation. Two error indicator functions are compared. We prove a theorem of almost always lucky failure in the case of the right hand analytically dependent on frequency. The operator's null space is treated by decomposing the solution into the part in the null space and orthogonal to it

Three-dimensional modelling and inversion of controlled source electromagnetic data

The marine Controlled Source Electromagnetic (CSEM) method is an important and almost self-contained discipline in the toolkit of methods used by geophysicists for probing the earth. It has increasingly attracted attention from industry during the past decade due to its potential in detecting valuable natural resources such as oil and gas. A method for three-dimensional CSEM modelling in the frequency domain is presented. The electric field is decomposed in primary and secondary components, as this leads to a more stable solution near the source position. The primary field is computed using a resistivity model for which a closed form of solution exists, for example a homogeneous or layered resistivity model. The secondary electric field is computed by discretizing a second order partial differential equation for the electric field, also referred in the literature as the vector Helmholtz equation, using the edge finite element method. A range of methods for the solution of the linear system derived from the edge finite element discretization are investigated. The magnetic field is computed subsequently, from the solution for the electric field, using a local finite difference approximation of Faraday’s law and an interpolation method. Tests, that compare the solution obtained using the presented method with the solution computed using alternative codes for 1D and 3D synthetic models, show that the implemented approach is suitable for CSEM forward modelling and is an alternative to existing codes. An algorithm for 3D inversion of CSEM data in the frequency domain was developed and implemented. The inverse problem is solved using the L-BFGS method and is regularized with a smoothing constraint. The inversion algorithm uses the presented forward modelling scheme for the computation of the field responses and the adjoint field for the computation of the gradient of the misfit function. The presented algorithm was tested for a synthetic example, showing that it is capable of reconstructing a resistivity model which fits the synthetic data and is close to the original resistivity model in the least-squares sense. Inversion of CSEM data is known to lead to images with low spatial resolution. It is well known that integration with complementary data sets mitigates this problem. It is presented an algorithm for the integration of an acoustic velocity model, which is known a priori, in the inversion scheme. The algorithm was tested in a synthetic example and the results demonstrate that the presented methodology is promising for the improvement of resistivity models obtained from CSEM data