Fast Simulation of 2.5D LWD Resistivity Tools

As a first step towards the fast inversion of geophysical data, in this work we focus on the rapid simulations of 2.5D logging-while-drilling (LWD) borehole resistivity measurements. Given a commercial logging instrument configuration, we calibrate the FE method offline with respect to (i) the element sizes via non-uniform tensor product grids; (ii) the arbitrary polynomial order of approximation on each element; and (iii) the interpolation of certain Fourier modes. This leads to the design of proper FE discretizations to simulate measurements acquired in an arbitrary 2D formation.Numerical results show that we accurately simulate on a sequential computer any field component at a rate faster than one second per logging position.The Marie Sklodowska-Curie grant agreement No 644602 MTM2013-40824-P SEV-2013-0323 BERC 2014-201

Massive Database Generation for 2.5D Borehole Electromagnetic Measurements using Refined Isogeometric Analysis

Borehole resistivity measurements are routinely inverted in real-time during geosteering operations. The inversion process can be efficiently performed with the help of advanced artificial intelligence algorithms such as deep learning. These methods require a massive dataset that relates multiple Earth models with the corresponding borehole resistivity measurements. In here, we propose to use an advanced numerical method —refined isogeometric analysis (rIGA)— to perform rapid and accurate 2.5D simulations and generate databases when considering arbitrary 2D Earth models. Numerical results show that we can generate a meaningful synthetic database composed of 100,000 Earth models with the corresponding measurements in 56 hours using a workstation equipped with two CPUs.European POCTEFA 2014–2020 Project PIXIL (EFA362/19); The grant ‘‘Artificial Intelligence in BCAM number EXP. 2019/0043

Fast 2.5D Finite Element Simulations of Borehole Resistivity Measurements

We develop a rapid 2.5-dimensional (2.5D) finite element method for simulation of borehole resistivity measurements in transversely isotropic (TI) media. 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 analytical (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 measurement positions provides efficient scalability. Our method yields accurate borehole logging simulations for realistic synthetic examples, delivering simulations of borehole resistivity measurements at a rate faster than 0.05 s per measurement location along the well trajectory on a 96-core computer

Deep Learning for Inverting Borehole Resistivity Measurements.

139 p.El subsuelo terrestre está formado por diferentes materiales, principalmente por rocas porosas que posiblemente contienen minerales y están rellenas de agua salada y/o hidrocarburos. Por lo general, las formaciones que crean estos materiales son irregulares y con materiales de diferentes propiedades mezclados en el mismo estrato.Uno de los principales objetivos en geofísica es determinar las propiedades petrofísicas del subsuelo de la Tierra. De este modo, las compañías pueden determinar la localización de las reservas de hidrocarburos para maximizar su producción o descubrir localizaciones óptimas para el almacenamiento de hidrógeno o el depósito de CO$_2$. Para este propósito, las compañías registran mediciones electromagnéticas utilizando herramientas de Medición Durante Perforación (LWD por sus siglas en inglés -- Logging While Drilling), las cuales son capaces de recabar datos mientras se lleva a cabo el proceso de prospección. Los datos obtenidos se procesan para producir un mapa del subsuelo de la Tierra. Basándose en el mapa generado, el operador ajusta en tiempo real la trayectoria de la herramienta de prospección para seguir explorando objetivos de explotación, incluidos los yacimientos de petróleo y gas, y maximizar la posterior productividad de las reservas disponibles. Esta técnica de ajuste en tiempo real se denomina geo-navegación.Hoy en día, la geo-navegación desempeña un papel esencial en geofísica. Sin embargo, requiere la resolución de problemas inversos en tiempo real. Esto supone un reto, ya que los problemas inversos suelen estar mal planteados.Existen múltiples métodos tradicionales para resolver los problemas inversos, principalmente, los métodos basados en el gradiente o en la estadística. Sin embargo, estos métodos tienen graves limitaciones. En particular, a menudo necesitan calcular el problema inverso cientos de veces para cada conjunto de mediciones, lo que es computacionalmente caro en problemas tridimensionales (3D).Para superar estas limitaciones, proponemos el uso de técnicas de Aprendizaje Profundo (DL por sus siglas en inglés -- Deep Learning) para resolver los problemas inversos. Aunque la etapa de entrenamiento de una Red Neuronal Profunda (DNN por sus siglas en inglés Deep Neural Network) puede requerir mucho tiempo, una vez que la red está correctamente entrenada puede predecir la solución en una fracción de segundo, facilitando las operaciones de geo-navegación en tiempo real. En la primera parte de esta tesis, investigamos las funciones de pérdida apropiadas para entrenar una DNN cuando se trata de un problema inverso.Además, para entrenar adecuadamente una DNN que se aproxime a la solución inversa, necesitamos un gran conjunto de datos que contenga la solución del problema directo para muchos modelos terrestres diferentes. Para crear dicho conjunto de datos, necesitamos resolver una Ecuación en Derivadas Parciales (PDE por sus siglas en inglés -- Partial Differential Equation) miles de veces. La creación de un conjunto de datos puede llevar mucho tiempo, especialmente para los problemas bidimensionales y tridimensionales, ya que la resolución de la PDE mediante métodos tradicionales, como el Método de Elementos Finitos (FEM por sus siglas en inglés -- Finite Element Method), es computacionalmente caro. Por lo tanto, queremos reducir el coste computacional de la construcción de la base de datos necesaria para entrenar la DNN. Para ello, proponemos el uso de métodos de Análisis Isogeométrico refinado (rIGA por sus siglas en inglés -- refined Isogeometric Analysis).Además, exploramos la posibilidad de utilizar técnicas de DL para resolver PDE, que es la limitación computacional principal al resolver problemas inversos. Nuestro objetivo principal es desarrollar un simulador rápido para resolver PDE paramétricas. Como primer paso, en esta tesis analizamos los problemas de cuadratura que aparecen al resolver PDE utilizando DNN y proponemos diferentes métodos de integración para superar estas limitacionesbca

Deep Learning for Inverting Borehole Resistivity Measurements

There exist multiple traditional methods to solve inverse problems, mainly, gradient-based or statistics-based methods. However, these methods have severe limitations. In particular, they often need to compute the forward problem hundreds of times, which is computationally expensive in three-dimensional (3D) problems. In this dissertation, we propose the use of Deep Learning (DL) techniques to solve inverse problems. Although the training stage of a Deep Neural Network (DNN) may be time-consuming, after the network is properly trained it can forecast the solution in a fraction of a second, facilitating real-time operations. In the first part of this dissertation, we investigate appropriate loss functions to train a DNN when dealing with an inverse problem. Additionally, to properly train a DNN that approximates the inverse solution, we require a large dataset containing the solution of the forward problem. To create such dataset, we need to solve aPartial Differential Equation (PDE) thousands of times. Building a dataset may be time-consuming, especially for two and three-dimensional problems since solving PDEs using traditional methods, such as the Finite Element Method (FEM), is computationally expensive. Thus, we want to reduce the computational cost of building the database needed to train the DNN. For this, we propose the use of rIGA methods. In addition, we explore the possibility of using DL techniques to solve PDEs, which is the main computational bottleneck when solving inverse problems. Our main goal is to develop a fast forward simulator for solving parametric PDEs. As a first step, in this dissertation we analyze the quadrature problems that appear while solving PDEs using DNNs and propose different integration methods to overcome these limitations

Fast One-dimensional Finite Element Approximation of Geophysical Measurements

There exist a wide variety of geophysical prospection methods. In this work, we focus on resistivity methods. We categorize these resistivity prospection methods according to their acquisition location as (a) on the surface, such as the ones obtained using Controlled Source Electromagnetics (CSEM) and magnetotelluric, and (b) in the borehole, such as the ones obtained using Logging-While-Drilling (LWD) devices. LWD devices are useful both for reservoir characterization and geosteering purposes, which is the act of adjusting the tool direction to travel within a specific zone. When inverting LWD resistivity measurements, it is a common practice to consider a one-dimensional (1D) layered media to reduce the problem dimensionality using a Hankel transform. Using orthogonality of Bessel functions, we arrive at a system of Ordinary Differential Equations (ODEs); one system of ODEs per Hankel mode. The dimensionality of the resulting problem is referred to as 1.5D since the computational cost to resolve it is in between that needed to solve a 1D problem and a 2D problem. When material properties (namely, resistivity, permittivity, and magnetic permeability) are piecewise-constant, we can solve the resulting ODEs either (a) analytically, which leads to a so-called semi-analytic method after performing a numerical inverse Hankel transform or (b) numerically. Semi-analytic methods are faster, but they also have important limitations, for example, (a) the analytical solution can only account for piecewise constant material properties, and other resistivity distributions cannot be solved analytically, which prevents to accurately model, for example, an Oil-Water Transition (OWT) zone when fluids are considered to be immiscible; (b) a specific set of cumbersome formulas has to be derived for each physical process (e.g., electromagnetism, elasticity, etc.), anisotropy type, etc.; (c) analytical derivatives of specific models (e.g., cross-bedded formations, or derivatives with respect to the bed boundary positions) are often difficult to obtain and have not been published to the best of our knowledge. In view of the above limitations, we propose to solve our forward problems using a numerical solver. A traditional Finite Element Method (FEM) is slow, which makes it unfeasible for our application. To achieve high performance, we developed a multiscale FEM that pre-computes a set of optimal local basis functions that are used at all logging positions. The resulting method is slow when compared to a semi-analytic approach for a single logging position, but it becomes highly competitive for a large number of logging positions, as needed for LWD geosteering applications. Moreover, we can compute the derivatives using an adjoint state method at almost zero additional cost in time. We describe an adjoint-based formulation for computing the derivatives of the electromagnetic fields with respect to the bed boundary positions. The key idea to obtain this adjoint-based formulation is to separate the tangential and normal components of the field, and treat them differently. We then apply this method to a 1.5D borehole resistivity problem. Moreover, we compute the adjoint-state formulation to compute the derivative of the magnetic field with respect to the resistivity value of each layer. We verify the accuracy of our formulations via synthetic examples. When simulating borehole resistivity measurements in a reservoir, it is common to consider an Oil-Water Contact (OWC) planar interface. However, this consideration can lead to an unrealistic model since, in the presence of capillary pressure, the mix of two immiscible fluids (oil and water) often appears as an OWT zone. These transition zones may be large in the vertical direction (20 meters or above), and in context of geosteering, an efficient method to simulate an OWT zone can maximize the production of an oil reservoir. In this work, we prove that by using our proposed 1.5D numerical method, we can easily consider arbitrary resistivity distributions in the vertical direction, as it occurs in an OWT zone. Numerical results on synthetic examples demonstrate significant differences between the results recorded by a geosteering device when considering a realistic OWT zone vs. an OWC sharp interface. As an additional piece of work of this Ph.D. Dissertation, we explore the possibility of using a Deep Neural Network (DNN) to perform a rapid inversion of borehole resistivity measurements. Herein, we build a DNN that approximates the following inverse problem: given a set of borehole resistivity measurements, the DNN is designed to deliver a physically meaningful and data-consistent piecewise one-dimensional layered model of the surrounding subsurface. Once the DNN is built, the actual inversion of the field measurements is efficiently performed in real time. We illustrate the performance of a DNN designed to invert LWD measurements acquired on high-angle wells via synthetic examples

Fast one-dimensional finite element approximation of geophysical measurements

135 p.When inverting Logging-While-Drilling (LWD) resistivity measurements, it is a common practice to consider a one-dimensional (1D) layered media to reduce the problem dimensionality using a Hankel transform. Using orthogonality of Bessel functions, we arrive at a system of Ordinary Differential Equations (ODEs); one systema of ODEs per Hankel mode. The dimensionality of the resulting problem is referred to as 1.5D since the computational cost to resolve it is in between that needed to solve a 1D problema and a 2D problem. When material properties are piecewise-constant, we can solve the resulting ODEs either (a) analytically, which leads to a so-called semi-analytic method, or (b) numerically. Semi-analytic methods are faster, but they also have important limitations, for example, (a) the analytical solution can only account for piecewise constant material properties, and other resistivity distributions cannot be solved analytically, which prevents to accurately model, for example, and OWT zone when fluids are considered to be inmiscible; (b) a specific set of cumbersome formulas has to be derived for each physical process (e.g. electromagnetism, elasticity, etc.), anisotropy type, etc.; (c) analytical derivatives of specific models (e.g. cross-bedded formations, or derivatives with respect to the bed boundary positios) are often diffcult to obtain and have not been published to the best of our knowledge

Theoretical Developments in Electromagnetic Induction Geophysics with Selected Applications in the Near Surface

Near-surface applied electromagnetic geophysics is experiencing an explosive period of growth with many innovative techniques and applications presently emergent and others certain to be forthcoming. An attempt is made here to bring together and describe some of the most notable advances. This is a difficult task since papers describing electromagnetic induction methods are widely dispersed throughout the scientific literature. The traditional topics discussed herein include modeling, inversion, heterogeneity, anisotropy, target recognition, logging, and airborne electromagnetics (EM). Several new or emerging techniques are introduced including landmine detection, biogeophysics, interferometry, shallow-water electromagnetics, radiomagnetotellurics, and airborne unexploded ordnance (UXO) discrimination. Representative case histories that illustrate the range of exciting new geoscience that has been enabled by the developing techniques are presented from important application areas such as hydrogeology, contamination, UXO and landmines, soils and agriculture, archeology, and hazards and climat

Goal-Oriented Adaptivity using Unconventional Error Representations

In Goal-Oriented Adaptivity (GOA), the error in a Quantity of Interest (QoI) is represented using global error functions of the direct and adjoint problems. This error representation is subsequently bounded above by element-wise error indicators that are used to drive optimal refinements. In this work, we propose to replace, in the error representation, the adjoint problem by an alternative operator. The main advantage of the proposed approach is that, when judiciously selecting such alternative operator, the corresponding upper bound of the error representation becomes sharper, leading to a more efficient GOA. These representations can be employed to design novel h, p, and hp energy-norm and goal-oriented adaptive algorithms. While the method can be applied to a variety of problems, in this Dissertation we first focus on one-dimensional (1D) problems, including Helmholtz and steady-state convection-dominated diffusion problems. Numerical results in 1D show that for the Helmholtz problem, it is advantageous to select the Laplace operator for the alternative error representation. Specifically, the upper bounds of the new error representation are sharper than the classical ones used in both energy-norm and goal-oriented adaptive methods, especially when the dispersion (pollution) error is significant. The 1D steady-state convection-dominated diffusion problem with homogeneous Dirichlet boundary conditions exhibits a boundary layer that produces a loss of numerical stability. The new error representation based on the Laplace operator delivers sharper error upper bounds. When applied to a p-GOA, the alternative error representation captures earlier the boundary layer, despite the existing spurious numerical oscillations. We then focus on the two- and three-dimensional (2D and 3D) Helmholtz equation. We show via extensive numerical experimentation that the upper bounds provided by the alternative error representations are sharper than the classical ones. When using the alternative error indicators, a naive p-adaptive process converges, whereas under the same conditions, the classical method fails and requires the use of the so-called Projection Based Interpolation (PBI) operator or some other technique to regain convergence. We also provide guidelines for finding operators delivering sharp error representation upper bounds.Basque Government Consolidated Research Group Grant IT649-1