FAST AND AUTOMATIC INVERSION OF LWD RESISTIVITY MEASUREMENTS FOR PETROPHYSICAL INTERPRETATION

This paper describes an extension of a recently developed fast inversion method (Pardo and Torres-VerdÍn (2015)) for estimating a layer-by-layer electric resistivity distribution from logging-whiledrilling (LWD) electromagnetic induction measurements. The well trajectory is arbitrary and the developed method is suitable for any commercial logging device with known antennae configurations, including tri-axial instruments. There are two key novel contributions in this work: First, the three-dimensional (3D) transversely isotropic (TI) formation is now approximated by a sequence of various "stitched" one-dimensional (1D) sections rather than by a single 1D reduced model. This provides added flexibility in order to approximate complex 3D formations. Second, we introduce the concept of "negative apparent resistivities" in the inversion method. By using the values of attenuation and phase differences that correspond to a "negative" resistivity in a homogeneous formation, the amount of data lost when converting magnetic fields into apparent resistivities is minimized, thus leading to a more robust inversion method that also convergences faster. The developed inversion method can be used to interpret LWD resistivity measurements and to adjust the well trajectory in real (logging) time. Numerical inversion results of challenging synthetic and actual field measurements confirm the high stability and superior approximation properties of the developed inversion algorithm. Because of the efficiency, flexibility, and stability of the inversion algorithm, formation-evaluation specialists can readily employ it for routine petrophysical interpretation and appraisal of complex LWD and wireline resistivity measurements acquired under general geometrical and geological constraints.BERC 2014-2017 SEV-2013-0323 GEAGAM (644602) MTM2013-40824-

A numerical 1.5D method for the rapid simulation of geophysical resistivity measurements

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

Fast 1D Inversion of Logging-While-Drilling Resistivity Measurements for Improved Estimation of Formation Resistivity in High-Angle and Horizontal Wells

We have developed an efficient inversion method to estimate layer-by-layer electric resistivity from loggingwhile-drilling electromagnetic induction measurements. The method assumes a 1D model based on planarly layered transversely isotropic formations with known bed boundaries, penetrated by arbitrary well trajectories. Forward simulations are based on a 1D reduced model in which borehole and mandrel effects are assumed to be negligible. The stopping criteria, regularization term, regularization parameter, the Jacobian matrix, and inversion variables are automatically estimated by the inversion algorithm, thereby minimizing the required user interaction and expertise. Numerical inversion results of challenging synthetic and field data confirmed the high stability and superior approximation properties of the developed inversion algorithm. We evaluated results indicating that triaxial induction measurements provide significantly more stable inversion results than conventional coaxial measurements, especially in the presence of anisotropic formations.RISE Horizon 2020 European Project GEAGAM (644602) MTM2013-40824-P SEV-2013-0323 CYTED 2011 project 712RT0449 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 inversion of logging-while-drilling resistivity measurements acquired in multiple wells

This paper introduces a new method for the fast inversion of borehole resistivity measurements acquired in multiple wells using logging-while-drilling (LWD) instruments. There are two key novel contributions. First, we approximate general three-dimensional (3D) transversely isotropic (TI) formations with a sequence of several \stitched" one-dimensional (1D) planarly layered TI sections. This allows us to approximate the solution of rather complex 3D formations using only 1.5D simulations. Second, the developed method supports the simultaneous inversion of measurements acquired in different neighboring wells and/or with different logging instruments. Numerical experiments performed with realistic 3D synthetic formations confirm the flexibility of the method and the reliability of inversion products. The method yields relative errors below 5% on the model space, and it enables the interpretation of resistivity measurements acquired in multiple wells (e.g., an exploratory, an offset, and a geosteering well) and with any combination of co-axial and/or tri-axial commercial logging measurements acquired with known antennae configurations. Numerical results also indicate that thinly-bedded resistive formations are very sensitive to the presence of noise on the measurements and/or to possible errors on bed boundary locations, while conductive layers are only weakly sensitive to those effects

A summary of my twenty years of research according to Google Scholars

I am David Pardo, a researcher from Spain working mainly on numerical analysis applied to geophysics. I am 40 years old, and over a decade ago, I realized that my performance as a researcher was mainly evaluated based on a number called \h-index". This single number contains simultaneously information about the number of publications and received citations. However, dif- ferent h-indices associated to my name appeared in di erent webpages. A quick search allowed me to nd the most convenient (largest) h-index in my case. It corresponded to Google Scholars. In this work, I naively analyze a few curious facts I found about my Google Scholars and, at the same time, this manuscript serves as an experiment to see if it may serve to increase my Google Scholars h-index

A summary of my twenty years of research according to Google Scholars

I am David Pardo, a researcher from Spain working mainly on numerical analysis applied to geophysics. I am 40 years old, and over a decade ago, I realized that my performance as a researcher was mainly evaluated based on a number called \h-index". This single number contains simultaneously information about the number of publications and received citations. However, dif- ferent h-indices associated to my name appeared in di erent webpages. A quick search allowed me to nd the most convenient (largest) h-index in my case. It corresponded to Google Scholars. In this work, I naively analyze a few curious facts I found about my Google Scholars and, at the same time, this manuscript serves as an experiment to see if it may serve to increase my Google Scholars h-index

Stability of finite difference numerical simulations of acoustic logging-while-drilling with different perfectly matched layer schemes

In acoustic logging-while-drilling (ALWD) finite difference in time domain (FDTD) simulations, large drill collar occupies, most of the fluid-filled borehole and divides the borehole fluid into two thin fluid columns (radius ∼27 mm). Fine grids and large computational models are required to model the thin fluid region between the tool and the formation. As a result, small time step and more iterations are needed, which increases the cumulative numerical error. Furthermore, due to high impedance contrast between the drill collar and fluid in the borehole (the difference is >30 times), the stability and efficiency of the perfectly matched layer (PML) scheme is critical to simulate complicated wave modes accurately. In this paper, we compared four different PML implementations in a staggered grid finite difference in time domain (FDTD) in the ALWD simulation, including field-splitting PML (SPML), multiaxial PML(MPML), non-splitting PML (NPML), and complex frequency-shifted PML (CFS-PML). The comparison indicated that NPML and CFS-PML can absorb the guided wave reflection from the computational boundaries more efficiently than SPML and M-PML. For large simulation time, SPML, M-PML, and NPML are numerically unstable. However, the stability of M-PML can be improved further to some extent. Based on the analysis, we proposed that the CFS-PML method is used in FDTD to eliminate the numerical instability and to improve the efficiency of absorption in the PML layers for LWD modeling. The optimal values of CFS-PML parameters in the LWD simulation were investigated based on thousands of 3D simulations. For typical LWD cases, the best maximum value of the quadratic damping profile was obtained using one d[subscript 0]. The optimal parameter space for the maximum value of the linear frequency-shifted factor (α[subscript 0]) and the scaling factor (β[subscript 0]) depended on the thickness of the PML layer. For typical formations, if the PML thickness is 10 grid points, the global error can be reduced to <1% using the optimal PML parameters, and the error will decrease as the PML thickness increases.National Natural Science Foundation (China) (Grant 41174118)Postdoctoral Fellowship of China (Grant 2013M530106)China Scholarship Council (Grant 2010644006)Major State S&T Special Project (Grant 2008ZX05020-004

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