Electromagnetic Well Logging E-Book

Wilson Chin

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 701

Veröffentlichungsjahr: 2014

Beschreibung

Mathematically rigorous, computationally fast, and easy to use, this new approach to electromagnetic well logging gives the reservoir engineer a new dimension to MWD/LWD interpretation and tool design Almost all publications on borehole electromagnetics deal with idealizations that are not acceptable physically. On the other hand, "exact models" are only available through detailed finite element or finite difference analysis, and more often than not, simply describe case studies for special applications. In either case, the models are not available for general use and the value of the publications is questionable. This new approach provides a rigorous, fully three-dimensional solution to the general problem, developed over almost two decades by a researcher familiar with practical applications and mathematical modeling. Completely validated against exact solutions and physics-based checks through over a hundred documented examples, the self-contained model (with special built-in matrix solvers and iteration algorithms) with a "plain English graphical user interface" has been optimized to run extremely fast - seconds per run as opposed to minutes and hours - and then automatically presents all electric and magnetic field results through integrated three-dimensional color graphics. In addition to state-of-the-art algorithms, basic "utility programs" are also developed, such as simple dipole methods, Biot-Savart large diameter models, nonlinear phase and amplitude interpolation algorithms, and so on. Incredibly useful to oilfield practitioners, this volume is a must-have for serious professionals in the field, and all the algorithms have undergone a laborious validation process with real use in the field.

Leseprobe

Contents

Cover

Half Title page

Title page

Copyright page

Preface

Acknowledgements

Chapter 1: Motivating Ideas – General Formulation and Results

1.1 Overview

1.2 Introduction

1.3 Physical Model and Numerical Formulation

1.4 Validation Methodology

1.6 Closing Remarks

1.7 References

Chapter 2: Detailed Theory and Numerical Analysis

2.1 Overview

2.2 Introduction

2.3 Preliminary Mathematical Considerations

2.4 Boundary Value Problem Formulation

2.5 Computational Issues and Strategies

2.6 Typical Simulation Results

2.7 Post-Processing and Applications

2.8 Restrictions with Fast Multi-frequency Methods

2.9 Receiver Design Philosophy

2.10 Description of Output Files

2.11 Apparent Resistivity Using Classic Dipole Solution

2.12 Coordinate Conventions for Mud and Invasion Modeling

2.13 Generalized Fourier Integral for Transient Sounding

2.14 References

Chapter 3: Validations – Qualitative Benchmarks

3.1 Overview

3.3 Advanced Problems

3.4 Sign Conventions and Validation Methodology

3.5 References

Chapter 4: Validations – Quantitative Benchmarks at 0° and 90°

4.1 Overview

4.2 Wireline Validations in Homogeneous Media

4.3 Wireline Validations in Two-Layer Inhomogeneous Media

4.4 Electric and Magnetic Field Sensitive Volume Analysis for Resistivity and NMR Applications

4.5 MWD “Steel Collar” and Wireline Computations in Homogeneous and Nonuniform Layered Dipping Media

4.6 Exact Drill Collar Validation Using Shen Analytical Solution

4.7 Dipole Interpolation Formula Validation in Farfield

4.8 Magnetic Dipole Validation in Two-Layer Formation

4.9 References

Chapter 5: Validations – Quantitative Benchmarks at Deviated Angles

5.1 Overview

5.2 Limit 1. No Collar, No Mud.

5.3 Limit 2. Collar Only, No Mud.

5.4 Limit 3. Mud Only, No Collar.

5.5 Limit 4. Collar and Mud.

Chapter 6: Validations – Quantitative Benchmarks at Deviated Angles with Borehole Mud and Eccentricity

6.1 Overview

6.2 Repeat Validations

6.3 References

Chapter 7: Validations – Receiver Voltage Response and Apparent Resistivity

7.1 Overview

7.2 Focused Studies

7.3 General Transmitter Design Philosophy

7.4 General Receiver Design Philosophy

7.5 Apparent Resistivity Estimation from Classic Dipole Model

Chapter 8: Simulator Overview and Feature Summary

8.1 Overview

8.2 Simulator Comparisons

8.3 Technical Specifications

8.4 Advanced Logging Applications

8.5 Formulation Features

8.6 Computational Technology

8.7 User Interface

8.8 Integrated Utility Programs

8.9 Detailed Output and Integrated Graphics

8.10 System Requirements

8.11 Validation Approach

8.12 Simulator Speed Analysis

8.13 Sample User Interface Screens

8.14 Transmitter and Receiver Design Interface

Chapter 9: Simulator Tutorials and Validation Problems

9.1 Problem Set 1. Dipole and Biot-Savart Model Consistency – Validating Magnetic Fields.

9.2 Problem Set 2. Validating Farfield Phase Predictions.

9.3 Problem Set 3. Drill Collar Model Consistency – Exact Drill Collar Validation Using Shen Analytical Solution.

9.4 Problem Set 4. Magnetic Dipole in Two-Layer Formation.

9.5 Problem Set 5. Effects of Eccentricity and Invasion.

9.6 Problem Set 6. A Complicated Horizontal Well Geology.

9.7 Problem Set 7. Effects of Layering, Anisotropy and Dip.

9.8 Problem Set 8. Transmitter and Receiver Design.

9.9 Problem Set 9. Apparent Anisotropic Resistivities for Electromagnetic Logging Tools in Horizontal Wells.

9.10 Problem Set 10. Borehole Effects – Invasion and Eccentricity.

Cumulative References

Index

About the Author

Electromagnetic Well Logging

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Library of Congress Cataloging-in-Publication Data:

ISBN 978-1-118-83103-8

Preface

Electromagnetic wave resistivity methods in Measurement-While-Drilling and Logging-While-Drilling applications, or simply MWD/LWD, are now approaching their fourth decade of practice. They are instrumental in anisotropy determination, dip angle analysis, bed boundary detection, fluid identification, and so on, and are important to economic analysis, stimulation planning, geosteering, unconventional resources and numerous exploration challenges. Essentially, phase delays and amplitude changes measured at (one or more) coil receivers relative to (one or more) transmitters are interpreted using Maxwell’s equations to provide clues related to vertical and horizontal resistivities Ry and Rh. That said, the objectives are well-defined and easily understood. However, the general modeling problem is difficult and mathematical challenges persist.

Fifty years ago, induction logging practice and interpretation were straightforward. Formations were thick and homogeneous. Wells were vertical. Tools were concentrically placed. Azimuthal symmetry was the rule. Coils wound around fiberglass mandrels, with their planes perpendicular to the axis, implied that only Rh was available from measurements. But that was fine – fluid flowed only radially toward the well so only horizontal (or radial) properties mattered. Like everything else back then, life was simple in the slow lane, and well logging and math modeling were no exception. The simple dipole model taught in physics sufficed for most purposes and log analysis was elementary.

Deviated and horizontal well drilling have redefined the problem. Coils are now wrapped around steel mandrels whose planes need not intersect tool axes at right angles. Diameters are typically several inches, greater than the thin layer thicknesses they were designed to evaluate. Drill collars navigate through narrow pay zones bounded by beds with contrasting electrical properties. Charges (acting as secondary transmitters that are responsible for polarization horns) are induced at their interfaces whose strengths depend on conductivity differences, frequency, coil orientation and dip. Transmitters and receivers are closely situated. Needless to say, the dipole model as generations of practitioners have appreciated, is history, at least in MWD/LWD applications. A completely different approach is required. But even in recent wireline triaxial induction applications, which pose less of a challenge, dipole models may apply but not without major reformulation. Complications due to dip, layering and anisotropy still impose limits on rigor, accuracy and speed. But without good math models for these new physical phenomena, well logs cannot be properly interpreted and hardware improvements will remain on the sidelines.

Many readers know of me as a researcher with broad interests in managed pressure drilling, MWD design and telemetry, formation testing, annular flow for drilling and completions, reservoir flow analysis, and other areas related to fluid mechanics. As an engineer, I have been challenged by “things that I can see,” and this prior work has led to nine books, over forty domestic and international patents, and about one hundred papers. After all, I earned my Doctorate at the Massachusetts of Institute of Technology in aerospace engineering, and its flying vehicles and robots personified everything that an engineer would and should dream about. But on finishing my thesis and happily preparing for my grand exit, I was asked that fateful day, “What about your minor?” My minor? I thought it was Applied Math. “No, an M.I.T. education means broadening yourself. You can’t do that with something you’re good in.”

And with that comment, my Committee had me enroll in the school’s Course 8, its reputable but notoriously difficult Physics Department, one known for Nobel Prize winners, string theorists, relativity and quantum physicists, people responsible for things that I could neither see nor feel. I studied electrodynamics and I was challenged. I dreamed electric and magnetic fields instead of fluid streamlines. I thought the Navier-Stokes equations were bad, but Maxwell’s equations were worse. Nonetheless, I survived, and lived to join Boeing, where I worked in Aerodynamics Research. And thank goodness, no more electrodynamics. But the company’s powerful tools and their connection to “e/m” would lay dormant until, like sleeping giants, they would awaken and change my world and the way I thought. All of which goes to show how life works in strange ways. Nothing is predictable, but at least electrodynamics is.

In the early days of aerodynamics, point vortexes were used to model lifting airfoils. Faster flow on top meant lower pressure per Bernoulli’s equation; slower flow beneath meant higher pressure, hence net lift. These simple models eventually gave way to distributions of vortexes, sources, sinks and other singularities. These were in turn supplemented by numerical methods solving partial differential equations, initially using staircase grids which modeled wing sweep, and later, less noisy boundary conforming mesh systems.

My interest in borehole electromagnetics was sparked by the plethora of methods that acquiesced to the demands of the general MWD/LWD problem. Models with respectable names, e.g., Born approximation, hybrid method, integral equations, magnetic dipole and geometric factor, lent an air of credibility, but nonetheless conveyed the impossibility for modeling the physical problem in its reality on its terms. About a decade ago, I observed parallels with aerodynamics methods. Why not replace point dipole models with distributions of current source singularities? Why not replace the staircase grids used to model dipping bed interfaces with boundary conforming meshes? Why not replace the industry’s simulators for B and E, which gave way to nightmares associated with fictitious currents and “staggered grids,” with simpler equivalent Poisson models for vector and scalar potentials A and V used in aerodynamics?

The strategy was two-fold: improve geometric description, while utilizing “off the shelf” partial differential equation solvers that were sophisticated, available and highly validated. The idea was more than just practical. Nobel Prize winner Richard Feynman, at Caltech where I studied earlier, had asked why one would employ B and E models when A and V seemed more intuitive. And as it would turn out, when transmitter coils are excited harmonically, the equations for the transformed variables would turn out simpler and look just like the complex Helmholtz equations Boeing solved to model unsteady flows!

There was, however, one catch. One reputable geophysicist had attempted a similar approach to obtain unphysical results. The problem turned out to be inappropriate use of finite difference formulas. In physics, a property may be continuous and its normal derivative not, and conversely. For instance, for heat transfer in a two-medium system, temperature and heat flux continuity at the interface implies that the derivative is double-valued. In Darcy flows past thin shales, the normal derivative is continuous but the pressure is not. When discontinuities are properly modeled, and stable iterative “relaxation” methods are used to solve the transformed Maxwell equations, the key physical features inherent in borehole electrodynamics are all accounted for. In this book, we develop our methods from first principles and validate our algorithms with every model accessible in the literature to demonstrate physical consistency.

Engineering correctness is paramount, but without rapid computing and numerical stability, the best of methods are not practical. As recently as last year, one consortium known for its three-dimensional models reported efficiency gains that reduced computing times from three hours to two! We have done much better. Our calculations require just ten seconds on typical Intel Core i5 systems and at most one minute for difficult problems. We have used every possible means to reduce our need for computing resources. For instance, variable grids mean low memory requirements, smart “in place” relaxation methods eliminate many array access issues, “finite radius coils” imply less singular fields (than point dipoles) and are associated with faster convergence, and direct zeroing of electric fields at drill collar nodes when applicable eliminates needless equation access and solution. Our algorithms, which also target thinly laminated sand-shale sequences or potential laminated pay reservoirs, are optimized for stable and fast convergence for high Rv/Rh.

To this, we added automated three-dimensional color graphics to display all coordinate components of real and imaginary quantities, for all E, B, A and V fields, plus interfacial surface charge when dealing with deviated and horizontal wells that penetrate layered media. We have provided “point summaries” in both rectangular (geology focused) and cylindrical (tool-oriented) coordinates for logging and hardware design applications. We’ve developed simple dipole, Biot-Savart, interpolation and apparent resistivity “apps” for fast comparisons, log analysis and validation. Our powerful but portable numerical engine is written in Fortran and is easily ported to other operating environments.

But through it all, we have not lost sight of the physics and the need for new hardware in a downhole environment that continually seeks greater challenges. We’ve avoided “canned” voltage formulas and opted for more general ∫ab E • dl approaches to facilitate innovative receiver design. We’ve provided voltage responses automatically in our post-processing and included receiver design interfaces allowing the user to design his own antennas. And our transmitter coils need not be circular; for example, they may be oriented at any angle relative to the tool axis. Our discrete current source approach, in fact, supports alternative antenna concepts, e.g., elliptic coils, open coils and nonplanar coils which do not necessarily wrap around the collar.

Our methodology need not represent the final product, but instead, provides the highly documented foundation for more powerful and versatile tools for borehole electrodynamic analysis. However, the software in its present form is intended for petrophysicists who wish to acquire more detailed perspectives about their logging runs. Readers anxious for “hands on” results are encouraged to browse through Chapters 8 and 9 first, written to convey ideas rapidly and to facilitate applications; all of the examples shown, in fact, were completed and documented in a single work day, with all calculations running quickly and stably the first and every time. Efficiency is enhanced by a user-friendly graphical Windows interface designed about typical petroleum workflows. A quick perusal of Chapter 9, in fact, may be useful in understanding how easily the detailed numerical results of Chapters 1–7 were created and how our claims for rapid simulation are realized in practice.

Stratamagnetic Software, LLC, was formed in 1999 to develop and commercialize this approach, “strata” conveying the subtleties associated with layering and “magnetic,” well, recalling my dreaded minor in graduate school. But as luck would have it, we worked for more than a decade in other interesting fluid-dynamics areas, e.g., formation testing, annular flow, MWD telemetry, and so on, engineering challenges that literally paid the bills. However, our vision and obsession to develop the general borehole model presented in this book have never faltered. With fast and accurate logging interpretation demand driving offshore evaluation, rapid geosteering and the hunt for unconventional energy resources, and with fluids modeling (I think, for the time being) finally behind us, the time for uncompromised borehole electrodynamics is now … and the simulator and its complete underlying technology are yours.

Wilson C. Chin, Ph.D., M.I.T. Email: [email protected] Phone: [phone]

Acknowledgements

Our novel approach to “general three-dimensional electromagnetic models for non-dipolar transmitters in layered anisotropic media with dip,” first published in Well Logging Technology Journal, Xi’an, China, August 2000 more than a decade ago, was subject to more than the usual reviews. Wondering whether the problem I had addressed was so trivial that no one cared, or too difficult, that others would not consider it, I turned to two well known M.I.T. physicists adept at the subject.

I expressed this concern to Professor John Belcher, my former electromagnetics teacher, and he honestly replied, “To me it sounds like a very difficult problem that I would have no idea of how to approach.” That, coming from a Professor of Astrophysics, the Principal Investigator for the Voyager Plasma Science Experiment, a two-time winner of NASA’s Exceptional Scientific Achievement Medal, plus other well-deserved honors, was unsettling as it attested to the difficulty of this innocuously looking problem.

Professor Belcher would refer me to another M.I.T. colleague, Markus Zahn, Professor of Electrical Engineering, affiliated with the school’s prestigious Laboratory for Electromagnetic and Electronic Systems, and author of the classic book Electromagnetic Field Theory: A Problem Solving Approach (John Wiley & Sons, 1979). Professor Zahn’s reply is reproduced below.

“I enjoyed reading your paper because as far as I could tell everything was correct in it. By the way depending on the reciprocal frequency with respect to the dielectric relaxation time, ε/σ, or the magnetic diffusion time, σμL2, the problem can be considered electro-quasistatic or magneto-quasistatic and decouples the vector and scalar potentials, generally allowing a simpler set of approximate Maxwell equations to be solved.

About fifteen years ago I did a similar but simpler analysis for Teleco using a Fourier series method under magneto-quasistatic conditions to develop a downhole method for transmitting measurable signals to the surface. This was to be an electromagnetic replacement for the pressure pulse method. Your numerical method lets you treat great complexities in geometry.”

These comments, in Clint Eastwood’s words, would “make my day.” The method was designed to handle geometric complexity and it did: general coil and antenna topologies, arbitrary layers at dip, interfacial charge, the complete frequency spectrum, plus steel mandrels, all without the “decoupling” that Professor Zahn alluded to.

The paper was later submitted to Petrophysics (Society of Professional Well Log Analysts) and critically reviewed by David Kennedy, who suggested numerous changes to style and focus, and then, to a senior Schlumberger colleague and friend for his expert insights on borehole electromagnetics. Confident the approach would prove useful to the industry, I formed Stratamagnetic Software, LLC to commercialize the method, but would delay publication until all of the theory, numerics, validations and software could be documented. This process, given intervening work in drilling, cementing, formation testing, MWD telemetry and other areas, consumed more than ten years but would offer the challenge of producing a unique and usable product.

With deep offshore exploration becoming routine but nonetheless more challenging by the day, and with real-time, three-dimensional imaging, and difficulties with low resistivity pay and anisotropy dominating the well logging agenda, publication of this wide body of work is now timely indeed. The author is indebted to Professors John Belcher and Markus Zahn, to SPWLA President David Kennedy, and to my Schlumberger colleague and friend, for their encouragement, support and votes of confidence. He is also grateful to his doctoral thesis advisor Professor Marten Landahl, the aerospace pioneer, for suggesting an electrodynamics minor, a critical decision that would be crucial to important methods integrating fluid mechanics and resistivity logging, to appear.

Scientific progress requires more than cursory knowledge of industry models, typically presented in advertising, and more often than not, “validated” by field usage and payzone discoveries. Until companies share their methods through unrestricted technical exchanges, true progress will not be possible. Without equations, detailed math formulations and open access to software, engineers and petrophysicists remain dependent on input and output devices. The author is especially indebted, in this regard, to Phil Carmical, Acquisitions Editor and Publisher, not just for his interest in this book and other works in progress, but for his continuing support and willingness in reporting the mundane but important technical details that really matter.

Chapter 1

Motivating Ideas – General Formulation and Results

1.1 Overview

The general, three-dimensional, electromagnetic problem in layered anisotropic media with dip is solved using a full finite difference, frequency domain solution to Maxwell’s equations that does not bear the inherent limitations behind Born, geometric factor, hybrid and linearized integral equation approaches. Several important physical capabilities are introduced. First, transmitter coils, no longer represented by point dipoles, are modeled using eight azimuthally equidistant nodes where complex currents are prescribed. The coil may reside across multiple beds, a feature useful in modeling responses from thinly laminated zones; the transmitter operates in wireline “coil alone” or Measurement-While-Drilling (MWD) “steel collar” modes, with or without conductive mud or anisotropic invasion, and with or without borehole eccentricity. Because coil size and near-field details are explicitly considered, accurate simulation of charge radiation from bed interfaces (responsible for polarization horns) and Nuclear Magnetic Resonance (NMR) sensitive volume size and orientation in layered media are both assured.

Second, dipping interfaces are importantly oriented along coordinate planes, eliminating well known numerical noise effects associated with “staircase grids.” Transmitter and layer-conforming variable mesh systems, which expand in the farfield to reduce computational overhead, are automatically generated by the simulator. Third, costly performance penalties incurred by anisotropic “staggered grid” formulations are avoided in the vector and scalar potential method, where all complex Helmholtz equations are solved by modern matrix inversion algorithms that intelligently seek high gradient fields, relaxing and suppressing their numerical residuals. Fourth, rapid computing speeds, e.g., seconds to a minute on typical personal computers, make the approach invaluable for array deconvolution, NMR applications, and rigsite log and geosteering analysis. The availability of a single, self-consistent, open-source model eliminates the uncertainties associated with different proprietary formulations solved by different methodologies at different organizations.

Benchmark studies show excellent agreement with analytical dipole solutions in uniform and layered media and with classical Biot-Savart responses for finite loop coils. Suites of results are described, for responses in complicated media, with and without steel mandrels, invasion and borehole eccentricity, for a range of dip angles. Depth-of-penetration simulations, for electric and magnetic fields, are offered, with a view towards integrated resistivity and NMR formation evaluation. The new algorithm, which is extremely stable, fast and robust, is highly automated and does not require user mathematical expertise or intervention. It is hosted by user-friendly Windows interfaces that support approximately thirty complete simulations every hour. Fully integrated three-dimensional, color graphics algorithms display electromagnetic field solutions on convergence. Receiver voltage responses are given along tool axes, together with circumferential contributions in separate plots; detailed tabulated field results are reported in both geologically-focused rectangular and tool-oriented cylindrical coordinates. Features useful to modern logging instrument design and interpretation are available. For example, users may reconfigure transmitter coils to noncircular oblique geometries “on the fly” (results for elliptical cross-sections and linear geometries used in existing resistivity designs are given later). In addition, users may dynamically “rewire” nodal outputs in order to experiment with novel transmitter, receiver and formation evaluation concepts or to interrogate problem geologies for additional formation properties.

1.2 Introduction

The interpretation of borehole resistivity logs in layered media with dip is complicated by anisotropy, low-to-high mandrel conductivities, nonzero transmitter coil diameter, borehole eccentricity, multiple wave scattering, and polarized interfaces and charge radiation, interacting effects which cannot be studied using simplifying dipole, geometric factor, hybrid or linearized integral equation models. These approaches restrict the physics for mathematical expediency. As such, they address only specific and narrow aspects of the complete problem, e.g., purely planar layering, axisymmetric analysis, dipping bed effects modeled by vertical and horizontal dipoles, and so on.

A comprehensive model encompassing all of the above effects has been elusive. For example, real formations are not isotropic, but an anisotropic formulation covering the complete frequency spectrum plus real layering effects is not available. Moran and Gianzero (1979), for instance, deal with the induction limit only, and do not address dipping beds and the problems associated with interfacial surface charge. Howard and Chew (1992) tackle these issues, but the numerically intensive isotropic model invokes geometric factor and Born-type assumptions.

Most computational algorithms do not converge for wide ranges of frequency or resistivity and anisotropy contrast. While models are available for induction sondes with non-conducting mandrels, specialized codes are required to handle the high collar conductivities typical in MWD applications. It is usually not possible to simulate induction and MWD runs in the same formation with the same model, thus complicating interpretation and tool design.