Electromagnetic Well Logging:

Models for MWD/LWD Interpretation and Tool Design

Wilson C. Chin, Ph.D., M.I.T.

Stratamagnetic Software, LLC

Houston, Texas and Beijing, China

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 R_{v} and R_{h}. 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 R_{h} 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 R_{v}/R_{h}.

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 ò_{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 non-planar 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: wilsonchin@aol.com

Phone: (832) 483-6899

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 schools’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, e/s, or the magnetic diffusion time, smL^{2}, 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.