Modeling of Resistivity and Acoustic Borehole Logging Measurements Using Finite Element Methods

By David Pardo, Paweł J. Matuszyk, Vladimir Puzyrev and 3 more

Modeling of Resistivity and Acoustic Borehole Logging Measurements Using Finite Element Methods provides a comprehensive review of different resistivity and sonic logging instruments used within the oil industry, along with precise and solid mathematical descriptions of the physical equations and corresponding FE formulations that govern these measurements. Additionally, the book emphasizes the main modeling considerations that one needs to incorporate into the simulations in order to obtain reliable and accurate results. Essentially, the formulations and methods described here can also be applied to simulate on-surface geophysical measurements such as seismic or marine controlled-source electromagnetic (CSEM) measurements.

Simulation results obtained using FE methods are superior. FE methods employ a mathematical terminology based on FE spaces that facilitate the design of sophisticated formulations and implementations according to the specifics of each problem. This mathematical FE framework provides a highly accurate, robust, and flexible unified environment for the solution of multi-physics problems. Thus, readers will benefit from this resource by learning how to make a variety of logging simulations using a unified FE framework.

• Provides a complete and unified finite element approach to perform borehole sonic and electromagnetic simulations
• Includes the latest research in mathematical and implementation content on Finite Element simulations of borehole logging measurements
• Features a variety of unique simulations and numerical examples that allow the reader to easily learn the main features and limitations that appear when simulating borehole resistivity measurements

• Language: English
• Publisher: Elsevier Science
• Release date: May 22, 2021
• ISBN: 9780128214657

David Pardo

David Pardo is a Research Professor at Ikerbasque, the University of the Basque Country UPV/EHU, and the Basque Center for Applied Mathematics (BCAM). He has published over 160 research articles and he has given over 260 presentations. He is now the PI of the research group on Applied Mathematical Modeling, Statistics, and Optimization (MATHMODE). His research interests include computational electromagnetics, petroleum-engineering applications (borehole simulations), adaptive finite-element and discontinuous Petrov-Galerkin methods, multigrid solvers, image restoration algorithms, and multiphysics and inverse problems.

Book preview

Preface

This book compiles our 15-years of experience simulating borehole acoustic and resistivity measurements with finite element (FE) methods. We provide a comprehensive review of different borehole resistivity and acoustic logging instruments routinely used by the oil industry, as well as a precise and solid mathematical description of the physical equations and the corresponding FE formulations that govern these measurements. Special emphasis is placed on the peculiar physical and geometrical conditions typically encountered in borehole environments. Additionally, the book describes the main modeling considerations that one needs to incorporate into the simulations to obtain stable, efficient, and accurate results. Most formulations and methods described here can also be applied to simulate surface geophysical measurements such as seismic, magnetotellurics (MT), or marine controlled-source electromagnetic (CSEM) measurements.

The target audience for this book are researchers in academia as well as in oil-production and oil-service companies who need to either design borehole resistivity and/or acoustic geophysical instruments and/or interpret geophysical measurements acquired with such instruments.

Learning about FE methods may require extra time and effort in comparison to traditional finite difference (FD) techniques. Nevertheless, we believe that the extra investment of time quickly pays off because simulation results obtained with FE methods are more flexible, stable, accurate, and efficient. FE methods employ a mathematical terminology based on FE spaces that facilitates the design of sophisticated formulations and implementations according to the specifics of each problem. Furthermore, this mathematical FE framework provides a highly-accurate, robust, and flexible unified environment for the solution of multi-physics problems. All resistivity and acoustic borehole simulations described in this book use the same FE software.

Our work was strongly influenced by the PhD dissertations of J. R. Lovell [221] and B. Anderson [11] in the area of borehole resistivity simulations, the books by F. L. Paillet and A. Cheng [265] and X.-M. Tang and A. Cheng [329] in the area of borehole acoustics, and the work of L. F. Demkowicz and his collaborators [97,99] in the area of adaptive high-order FE methods.

We are also thankful for the constructive and insightful comments received during the last fifteen years by several colleagues from academia as well from numerous researchers working with oil and oil service companies (including Baker-Hughes, Schlumberger, Halliburton, Chevron, and Shell).

The authors are particularly grateful to professors Leszek F. Demkowicz and Leonty A. Tabarovsky for sharing their vast knowledge in the area and, more importantly, for being such great colleagues.

This work has been possible thanks to the support provided by the University of Texas at Austin's Research Consortium on Formation Evaluation, and the USA, Australian, European, Spanish, and Basque Governments Funding Agencies. Victor Calo acknowledges the long-term support from the King Abdullah University of Science and Technology (KAUST), Curtin University, and the Commonwealth Scientific and Industrial Research Organisation (CSIRO), in particular, funding from the Center for Numerical Porous Media at KAUST, the CSIRO Professorial Chair in Computational Geoscience at Curtin University and the CSIRO Deep Earth Imaging Enterprise Future Science Platform. Myung Jin Nam thanks to KETEP of South Korea. Carlos Torres-Verdín is grateful for the financial support provided by the Brian James Jennings Memorial Endowed Chair in Petroleum and Geosystems Engineering.

Last but not least, we extend our utmost gratitude to our families for their unconditional love and support. Without them, this book would not have been possible. We dedicate this book to them.

Bibliography

[11] B.I. Anderson, Modeling and inversion methods for the interpretation of resistivity logging tool response. [PhD thesis] Delft University of Technology; 2001.

[97] L. Demkowicz, Computing with hp-Adaptive Finite Elements. Volume I: One and Two Dimensional Elliptic and Maxwell Problems. Chapman and Hall; 2006.

[99] L. Demkowicz, J. Kurtz, D. Pardo, M. Paszynski, W. Rachowicz, A. Zdunek, Computing with hp-Adaptive Finite Elements. Volume II: Frontiers: Three-Dimensional Elliptic and Maxwell Problems with Applications. Chapman and Hall; 2007.

[221] J.R. Lovell, Finite Element methods in resistivity logging. [PhD thesis] Delft University of Technology; 1993.

[265] F. Paillet, C. Cheng, Acoustic Waves in Boreholes. CRC; 1991.

[329] X.-M. Tang, A. Cheng, Quantitative Borehole Acoustic Methods. 2nd, revised and extended ed. Handbook of Geophysical Exploration. Section I. Seismic Exploration. Elsevier; 2004;vol. 24.

Chapter 1: Introduction

Abstract

This chapter describes the main features of commercial borehole resistivity and acoustic logging instruments, which we divide into two general types: wireline (WL) and logging-while-drilling (LWD) tools. In particular, acoustic logging instruments, depending on the type of acoustic source, can excite different kinds of waves in the wellbore, generally classified as body, interface, and guided waves.

The chapter also provides several modeling considerations that one should consider when performing computer simulations of borehole resistivity and acoustic measurements. We also review some of the leading publications in the area, and discuss the most relevant ones from the plethora of existing resistivity and acoustic logging simulation methods, emphasizing their main advantages and limitations. This chapter also introduces an abstract Finite Element notation that encompasses all the variational formulations we later use in the book's remainder as particular cases.

Keywords

Borehole measurements; Resistivity measurements; Acoustic measurements; Logging instruments; Finite element method; Numerical simulations; Wireline (WL) tools; Logging-while-drilling (LWD) tools

It is customary to quantify the in situ geometrical, petrophysical, fluid, and fluid-flow properties of a subsurface hydrocarbon reservoir by first measuring different physical quantities that describe the rocks. Oil companies typically acquire seismic [298] and possibly marine controlled-source electromagnetic (CSEM) measurements [87] to remotely and non-invasively detect rock properties in the subsurface. With these measurements, experts in the field appraise the expected profitability of the reservoir, possibly with the help of numerical simulators that facilitate the interpretation of the acquired geophysical, petrophysical, and production measurements.

After performing the initial static and dynamic assessment of the reservoir, various wells may be drilled into it at several locations. The well trajectories may be either predetermined based on the above characterization of the reservoir or, alternatively, logging-while-drilling (LWD) instruments may be employed to steer in real-time the trajectory of the well based on the interpretation of logging measurements acquired during the drilling process (i.e., well geosteering). Subsequently, wireline (WL) logging instruments can acquire logging measurements at various locations along the trajectory of each well before or after steel casing is set in place.

Measurements acquired with both WL and LWD instruments provide a detailed description of the subsurface along the well trajectory, and thus improve the description and characterization of the reservoir. Such borehole measurements are governed by different physical phenomena such as electromagnetic or acoustic wave propagation, natural or induced nuclear radioactivity, neutron scattering and absorption, or nuclear magnetic resonance, among others. Accordingly, different logging instruments have been designed to assess different material properties of the subsurface along the well trajectory.

In the case of resistivity logging instruments, several transmitters operating at various frequencies (typically between one and seven) excite an electromagnetic (EM) field. The emitted wave interacts with the mandrel, the borehole, the formation, and possibly the casing. Several receivers placed at specific locations along the logging instrument record the resulting signal. A carefully crafted combination of EM fields and voltages measured at the receivers estimates the average resistivity of the formation in the vicinity of the transmitters/receivers. These models traditionally assume that the EM properties of the borehole, mud cake, and formation are within certain validity ranges. Mathematically speaking, resistivity logging instruments are tuned to solve (or at least approximate) the inverse problem at the hardware level for a certain range of parameters and material properties. Unfortunately, such ideal designs are rarely achieved for real borehole logging conditions. Therefore, in practice, further post-processing of the recorded data seeks to improve the estimation of the petrophysical properties of the near-well rocks. In particular, oil companies use correction charts to post-process the measurements. These charts account for borehole-eccentered tool effects, borehole rugosity, shoulder-bed effects, mud-filtrate invasion, layers dipping at a non-negligible angle to the well trajectory, etcetera. Nevertheless, even after incorporating these ad hoc corrections, results are often inaccurate and thus, unreliable. An accurate numerical inversion workflow is necessary to estimate actual rock properties.

In the case of acoustic logging instruments, acoustic sources excite a pulse within a certain range of frequencies by generating a pressure disturbance in the surrounding borehole fluid. The highest excited frequency by this disturbance is typically below 25 kHz except for ultrasonic devices. This excitation wave interacts with the neighboring materials and with several (e.g., 8 to 13) equally-spaced receivers (acoustic array). This array records the acoustic pressure as a function of time. Then additional post-processing using inversion methods is necessary. In the frequency domain, standard methods are dispersion-processing techniques (e.g., matrix-pencil [115], variants of Prony's method [95], weighted spectral semblance [259], forward–backward amplitude and phase estimation method [214]). Alternatively, in the time domain, the slowness-time coherence (STC) technique [193] directly processes the acoustic waveforms. Consequently, one obtains information about properties of different propagating modes (phase and group velocities, attenuation, and reflection coefficients) that can be further reinterpreted in terms of mechanical, elastic, and petrophysical properties of the surrounding rock formations to improve the description of the hydrocarbon reservoir.

For the proper design of borehole logging instruments as well as for the subsequent interpretation of the recorded logging measurements, the use of reliable and accurate simulation methods is essential. For instance, numerical simulation software can also generate the correction charts that mitigate borehole effects on the measurements. Corrections for more complex effects such as bed thickness, mud-filtrate invasion, and bed dip, are difficult to capture with charts; rapid modeling is necessary to perform these corrections in real-time, especially for well geosteering operations.

In the last few years, the community devoted a significant amount of effort to providing a unique interpretation of multi-physics measurements by employing combined inversion frameworks rather than by using multiple single-physics inversion software. This multi-physics inversion is performed using advanced numerical methods that require simulation software for both resistivity and acoustic borehole measurements, for example. The ability to perform multi-physics simulations within the same unified finite element (FE) framework highly facilitates this endeavor.

Before delving into the details of the finite element method and its use to simulate borehole geophysical measurements, we conclude this introduction with an overview of the different existing borehole resistivity instruments and mention the most prominent numerical methods employed during the last decades to simulate the measurements acquired with them. Next, we provide a similar description for borehole acoustic measurements. Finally, we describe various FE methods that can successfully perform the above simulations, including the self-adaptive hp-FE software that we consistently utilize for this purpose.

1.1 Borehole resistivity measurements

We first describe the main features of the different types of commercial borehole resistivity logging instruments. Next, we provide several modeling considerations that one should take into account when performing computer simulations of borehole resistivity measurements. We conclude with a brief review of some of the leading work in the area.

1.1.1 Borehole resistivity instruments

The design of borehole resistivity instruments seeks to quantify the spatial distribution of the electrical resistivity of rock formations penetrated by a well. This material property is primarily sensitive to the porosity and water saturation of the formation via Archie's equation [15,113,114] (and its many variants for the case of shaly rocks). Therefore, electrical resistivity helps to determine the location and volume of hydrocarbons present in a reservoir. Resistivity measurements can detect variations in lithology, for instance detect variations between sandstones, shales, and carbonates, and can differentiate between aquifers and hydrocarbon-saturated rocks. In hydro-geophysical applications, resistivity measurements are used to quantify the salinity of aquifers.

There also exist high-frequency electromagnetic logging instruments that determine the spatial distribution of the dielectric permittivity in the vicinity of the receivers. This permittivity variation facilitates the identification of conductive and polarizable minerals present in rocks, and variations of water salinity; in some special cases, they can also quantify rock pore connectivity. The high frequency of operation needed to measure the dielectric permittivity of rocks and the lossy nature of the subsurface significantly reduce the depth of investigation to a few centimeters or less due to the skin depth effect of these measurements (see Fig. 1.1). Therefore, dielectric measurements are often less relevant from the petrophysical point of view than those intended to estimate the resistivity/conductivity of the formation.

Figure 1.1 Depth of Investigation (DOI) of various resistivity logging instruments.

Numerous resistivity logging instruments exist, which we classify according to the frequency of operation as:

•  Zero-frequency Spontaneous Potential (SP) instruments[101]. The SP instrument is a passive direct current (DC) measurement; that is, there is no active external electrical or magnetic source. Spontaneous potentials arise in the borehole as a result of (a) gradients in electrolyte concentration between the borehole and formation water and (b) differential mobility between ions and cations present in aqueous solution. Two connected electrodes by an insulated cable can measure these potentials: the first one is located at the surface while the second one moves along the borehole. For each logging position, measurements record the potential difference between both electrodes. Borehole SP measurements can only be acquired in the presence of water-base drilling muds.

•  Low-frequency galvanic instruments. These tools (a.k.a. laterolog resistivity instruments) include voltage and current electrodes operating at low frequencies, typically between 10 Hz and 2 kHz. Ohm's law is the guiding design principle behind these logging instruments, which states that the resistance R of a circuit is the ratio , where I is the electrical current driving the circuit, and V is the difference of electrical potential measured between two points in that circuit.

Historically, the Schlumberger brothers designed the first borehole resistivity instruments using these ideas. They adapted a similar galvanic measurement system already in use for surface geophysical prospection of ore deposits to the borehole environment. Early designs used four electrodes centered in the borehole (see Fig. 1.2). Two electrodes (labeled A and B) were spaced some distance apart to inject electrical current into the surrounding rock formation; these are the current electrodes. Two additional electrodes (labeled M and N), located between the current electrodes A and B, were used as voltage electrodes. The system measures a normalized voltage between electrodes M and N, scaled by the current injected between electrodes A and B. To estimate an apparent resistivity, the method assumes an equivalent circuit resistance model to explain these normalized measurements. The method scales the injected current with the normalized voltage originated from the electrical conduction across a hypothetical homogeneous and isotropic rock. The apparent resistivity calculation also accounts for the distance between electrodes (A to B and M to N) (i.e., accounts for the geometrical effect of the measurement). An alternative system design could be implemented to maintain a constant potential difference between electrodes A and B and measure the corresponding electrical current between electrodes M and N. Nevertheless, practical engineering considerations dictate that controlling the amount of current injected into electrodes A and B is more effective than controlling the corresponding voltage difference.

Figure 1.2 Illustration of a laterolog logging device with two electrodes that inject electrical current (A and B) and two current electrodes (M and N).

The physics of laterolog resistivity instruments requires the borehole to be electrically conductive, which limits their use with oil-base drilling muds. Indeed, several factors affect the acquisition and processing of laterolog measurements, for example, the presence of borehole mud, tool mandrel, and casing during logging operations. Several techniques exist to ameliorate these undesired borehole-tool interactions. One method uses another pair of monitoring electrodes to actively quantify the amount of electrical current traversing through the borehole from electrode A to B. The measurement quality degrades when the amount of electrical current traversing through the borehole is higher than the current traversing through the surrounding rock formation. To correct this adversity in real-time, a second pair of voltage electrodes maintains the borehole at a constant electrical potential, therefore mitigating electrical currents traversing the borehole. The system uses real-time active feedback control to immediately null that potential difference within the borehole. The control system uses the voltage measured by the monitoring electrodes to drive the active electrodes. Finally, the system records the potential between electrodes M and N.

In addition to including a hardware-driven feedback control system to mitigate electrical currents traversing through the borehole, modern laterolog instruments include multiple sets of voltage electrodes to effectively control the radial depth of investigation of the measurements; the longer the spacing between the voltage electrodes the deeper the radial depth of investigation. The number and spacing of current-injection and voltage-measurement electrodes, as well as the selected linear combination of voltages, are determined by solving an optimization problem. This problem uses the analytical solutions (Green's functions) corresponding to simple borehole conditions. Under real logging conditions, the simple models employed for tool design no longer hold and results can be affected by borehole rugosity, borehole size and resistivity, tool eccentricity, mud-filtrate invasion into permeable formations, and other factors. The post-processing of the recorded measurements significantly reduces undesired effects (i.e., minimizes effects unrelated to the electrical resistivity of the surrounding rocks as encountered at pre-drill conditions) using tool-specific correction charts that are generated via numerical simulations.

Traditionally, many different models are used when designing tools. These models consider a wide range of material descriptions that deliver different analytic expressions for the most sensitive measurements. The large number of combinations and measurement objectives gives rise to many different types of logging instruments. Such instruments include Microlog [102], Laterolog [103], Microlaterolog [104], Spherically Focused Log (SFL) [312], and Dual Laterolog [325], among others.

For cased wells, the presence of a metallic pipe (casing) surrounding the borehole significantly influences the measurements when acquired with commercial logging instruments. The casing masks the response from formation resistivity behind it. Casing effects on electrical resistivity measurements are so substantial that a correction chart cannot address them. In 1990, Kaufman [191] developed an alternative measurement system that is sensitive to the formation resistivity behind steel casing. He used elementary circuit principles to design a measurement acquisition system that cancels the primary (strongest) field produced by the steel casing, thereby emphasizing the secondary field that is highly sensitive to the formation resistivity behind casing.

The particular frequency of operation employed in galvanic logging instruments has a limited effect since, at such low frequencies, the behavior of the EM fields is almost frequency independent. Indeed, galvanic logging instruments are often simulated at zero frequency (DC). However, actual logging instruments do not operate at such frequency due to deleterious electrode contact effects of electrochemical nature and the natural spontaneous potentials that can significantly increase the noise-to-signal ratio. Under certain borehole conditions, the use of a non-zero frequency of operation may produce abnormally large effects (e.g., Groningen [205,371] and Delaware [313]).

Galvanic logging instruments can also be used to acquire cross-well and borehole-to-surface measurements. In the latter cases, both the background conductivity of the formation and the frequency of operation determine the maximum allowable distance between transmitters and receivers for acceptable noise-to-signal ratios. Above that threshold distance (which can be easily estimated based on the skin depth and other factors), noise effects dominate, and measurements become unreliable.

•  Medium-frequency induction instruments[106]. These tools have solenoidal or toroidal coils that operate at frequencies that range between 10 kHz and 10 MHz. Because these instruments do no longer employ electrodes, they provide reliable measurements in the presence of oil-base drilling muds as well as in air-filled boreholes.

The design of induction instruments uses analytical solutions for simple models to determine the number of transmitters and receivers and their locations, as well as the number of turns of each receiver coil, and the frequencies of operation. After measurement acquisition, post-processing of these data is implemented to reduce borehole effects via correction charts.

Contrary to the case of galvanic measurements, the particular frequency of operation plays a determinant role in electromagnetic induction logging. Indeed, the frequency of operation determines the validity range of the post-processed apparent resistivity measurements. The adjustment of operating frequencies depends on the spacing between the transmitter and receivers, which, together with the electrical resistivity of the rock formation surrounding the borehole, determine the resolution and depth-of-investigation of the acquired measurements. In the presence of a metallic mandrel, such as in logging-while-drilling operations, induction devices should operate at sufficiently high frequencies to mitigate mandrel effects on the recorded measurements. On the other side, when frequencies are high (typically, above 1 MHz) and the rock formation is resistive, rock dielectric effects may severely affect the measurements. More precisely, if the product of the frequency of operation, dielectric constant, and resistivity exceeds the value 0.05, then rock dielectric effects may influence the recorded measurements significantly. This behavior increases the difficulties of simulating and interpreting the recorded borehole resistivity logging measurements. Because of that, logging-while-drilling induction instruments deliver both amplitude- and phase-related measurements.

There exist a large variety of induction instruments that are sensitive to distinct volumes of the rock formations surrounding the borehole. Because of their electromagnetic induction principle, borehole induction resistivity measurements exhibit enhanced sensitivity to low-resistivity (i.e., highly conductive) rock formations. Modern induction instruments vary the operating frequencies to measure the electrical resistivity at different radial distances away from the borehole. The penetration distance varies from a few centimeters to over two meters depending on the electrical resistivity of the rocks. The greater the electrical resistivity the longer the radial length of investigation.

Moreover, triaxial induction instruments [10] incorporate tilted coils to estimate the full resistivity tensor. This tensor detects the presence of electrically anisotropic rock formations (for instance, in the case of sandstone-shale laminated formations and cross-bedded sandstones). Triaxial induction resistivity measurements detect electrically anisotropic rock formations, quantify dominant dip and azimuth angles (i.e., estimate the preferential orientation of the sandstone-shale laminations with respect the borehole axis), and determine anisotropic electrical properties of the reservoir. In the case of LWD instruments, directional measurements obtained with induction logging instruments facilitate the estimation of the distance from the tool to the oil-to-water and shale-sandstone interfaces, for instance. Directional resistivity measurements have a significantly larger depth-of-investigation (several meters) than conventional resistivity logging measurements.

Oil-service companies recently commercialized deep and extra-deep azimuthal resistivity logging instruments with transmitter-receiver spacings of up to 30 m (see e.g., [111,261,262,314,326]). This technology, often referred to as reservoir mapping-while-drilling (MWD or RMWD), exceeds several times the formation coverage of conventional LWD-sensing technologies and allows operators to optimize the well trajectory (well geosteering) using real-time information, thus reducing drilling risk, maximizing reservoir exposure, and increasing production potential. Therefore, the reliable and accurate simulation and interpretation of the measurements acquired with these devices is of paramount importance due to the enhanced economic benefits of developing geosteering operations [50].

•  High-frequency (dielectric) measurements. These instruments estimate the electrical conductivity and dielectric permittivity of the formation as a function of frequency. They operate at high frequencies (over 10 MHz), such that rock electrical permittivity effects are noticeable in the measurements [317]. The relatively high-frequency band and the lossy nature of rock formations imply that the depth of investigation of dielectric measurements is just a few centimeters or less. This depth restriction presents some difficulties when combining the corresponding measurements with low-frequency resistivity measurements, especially in the presence of mud-filtrate invasion and/or spatially heterogeneous rocks.

Modern commercially-available borehole dielectric instruments operate with multiple transmitter and receiver antennas and acquire measurements at several frequencies over a wide frequency range. The availability of multi-frequency measurements enables the assessment of various petrophysical properties of interest at once. These properties include abnormal mineralogy, pore connectivity, formation water resistivity, Archie's parameters, and hydrocarbon saturation, among others. The measurements can be complemented with magnetic resonance measurements, for instance, to enhance quantitative interpretations.

We refer to the books [11,19,163,221,313] for the reader interested in further details as regards specific and commercially available borehole resistivity logging instruments.

1.1.2 Main simulation challenges

When performing numerical simulations of Maxwell's equations, one should first take into account the frequency of operation or, more precisely, the number of wavelengths that one needs to consider in the computational domain, also known as the electrical size of the domain. In the case of borehole resistivity measurements, EM fields quickly decay in amplitude as we move away from the transmitter due to the lossy nature of the subsurface. As a result, the size of the computational domain is always a small fraction of a wavelength, even for the highest-frequency logging instruments. This behavior facilitates computations because accurate simulations of high-frequency wave-propagation problems are challenging to perform. Nevertheless, resistivity logging instruments have a unique feature that severely complicates their simulation. Namely, they measure a secondary field that can be several orders of magnitude smaller than the primary field. Thus, the numerical method should reproduce the same cancelations as those occurring in nature with the primary field. These cancelations have a physical justification and meaning. As a consequence, relatively small cancelation errors in the numerical representation of the primary field produce large errors in the secondary field representation, which is the one measured by logging