Heat as a proxy to image dynamic processes with 4D electrical resistivity tomography

Since salt cannot always be used as a geophysical tracer (because it may pollute the aquifer with the mass that is necessary to induce a geophysical contrast), and since in many contaminated aquifer salts (e.g., chloride) already constitute the main contaminants, another geophysical tracer is needed to force a contrast in the subsurface that can be detected from surface geophysical measurements. In this context, we used heat as a proxy to image and monitor groundwater flow and solute transport in a shallow alluvial aquifer (< 10 m deep) with the help of electrical resistivity tomography (ERT). The goal of our study is to demonstrate the feasibility of such methodology in the context of the validation of the efficiency of a hydraulic barrier that confines a chloride contamination to its source. To do so, we combined a heat tracer push/pull test with time-lapse 3D ERT and classical hydrogeological measurements in wells and piezometers. Our results show that heat can be an excellent salt substitution tracer for geophysical monitoring studies, both qualitatively and semi-quantitatively. Our methodology, based on 3D surface ERT, allows to visually prove that a hydraulic barrier works efficiently and could be used as an assessment of such installations

2.5-D Deep Learning Inversion of LWD and Deep-Sensing em Measurements Across Formations with Dipping Faults

Deep learning (DL) inversion of induction logging measurements is used in well geosteering for real-time imaging of the distribution of subsurface electrical conductivity. We develop a DL inversion workflow to solve 2.5-D inverse problems arising in well geosteering. The inversion workflow employs three DL modules: a 'look-around' fault detection module and two inversion modules for reconstructing anisotropic resistivity models in the presence or absence of fault planes, respectively. Our DL approach is capable of detecting and quantifying arbitrary dipping fault planes in real time. We compare inversion performance considering only short logging-while-drilling (LWD) measurements versus using both short LWD and deep-sensing measurements. The latter measurements provide enhanced depth-of-investigation while minimizing uncertainty. We also obtain improved results when using multidimensional inversion, especially nearby fault planes. This study verifies the applicability of real-time 2.5-D DL inversion across arbitrary faulted formations for well geosteering

Error Control and Loss Functions for the Deep Learning Inversion of Borehole Resistivity Measurements

Deep learning (DL) is a numerical method that approximates functions. Recently, its use has become attractive for the simulation and inversion of multiple problems in computational mechanics, including the inversion of borehole logging measurements for oil and gas applications. In this context, DL methods exhibit two key attractive features: a) once trained, they enable to solve an inverse problem in a fraction of a second, which is convenient for borehole geosteering operations as well as in other real-time inversion applications. b) DL methods exhibit a superior capability for approximating highly-complex functions across different areas of knowledge. Nevertheless, as it occurs with most numerical methods, DL also relies on expert design decisions that are problem specific to achieve reliable and robust results. Herein, we investigate two key aspects of deep neural networks (DNNs) when applied to the inversion of borehole resistivity measurements: error control and adequate selection of the loss function. As we illustrate via theoretical considerations and extensive numerical experiments, these interrelated aspects are critical to recover accurate inversion results

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

Physics-guided deep-learning inversion method for the interpretation of noisy logging-while-drilling resistivity measurements

Deep learning (DL) inversion is a promising method for real-Time interpretation of logging-while-drilling (LWD) resistivity measurements for well-navigation applications. In this context, measurement noise may significantly affect inversion results. Existing publications examining the effects of measurement noise on DL inversion results are scarce. We develop a method to generate training data sets and construct DL architectures that enhance the robustness of DL inversion methods in the presence of noisy LWD resistivity measurements. We use two synthetic resistivity models to test the three approaches that explicitly consider the presence of noise: (1) adding noise to the measurements in the training set, (2) augmenting the training set by replicating it and adding varying noise realizations and (3) adding a noise layer in the DL architecture. Numerical results confirm that each of the three approaches enhances the noise-robustness of the trained DL inversion modules, yielding better inversion results-in both the predicted earth model and measurements-compared to the basic DL inversion and also to traditional gradient-based inversion results. A combination of the second and third approaches delivers the best results

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

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