Resistivity Modeling:

Propagation, Laterolog and Micro-Pad Analysis

by

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

Stratamagnetic Software, LLC

Houston, Texas and Beijing, China

July 2016

Preface

Those familiar with the author’s early work in fluid mechanics will find mathematical rigor tempered with healthy skepticism in formulating and solving physical problems correctly. Validations proliferate in his books and papers. This is not unusual for engineers trained in physics and applied math, and this approach has served well as interests turned from one fluid-dynamic specialty to another; then, still more, leading to diverse activities in reservoir engineering, annular flow, formation testing, MWD design and telemetry, and so on. However the author was, for years, a "closet resistivity man" trained in electrodynamics at M.I.T.’s physics department, long a bastion of experts in astrophysics, plasma dynamics, string theory, and so on.

To this author, electromagnetic simulation for oilfield applications had always ranked high among these Herculean tasks: the dream was much too large to resist. Interestingly, understanding what *had* been done and what was *really* available actually proved to be the greater challenge. Research in the oil service industry is shrouded in secrecy. "Validations" are log examples that lead to oil discoveries and corporate revenue. Whether or not numerical models are actually consistent with Maxwell’s equations and the handful of analytical solutions developed by top classical physicists over the past century is irrelevant. Technical training in university and industry short courses simply amounts to studying marketing literature and log analysis papers focused more on differences between competitor tools than with rigorous mathematical results. All of this would not be relevant except that, after years of service to fluid mechanics, the author was asked by multiple organizations to develop suites of electromagnetic simulators that would address modern applications for hardware and interpretation development. These would be available to new competitors and old, and they would, naturally, need to be properly formulated and rigorously validated. Mathematical correctness and real equations were in demand at last.

The author’s recent book *Electromagnetic Well Logging: Models for MWD/LWD Interpretation and Tool Design*, from John Wiley & Sons in 2014, would be a first step in delivering the new models. The work provided a full three-dimensional formulation for "non-dipolar" transmitters in heterogeneous layered anisotropic media with dip. By "non-dipolar" we meant finite circular transmitters, elliptic coils, and in fact, any open or closed antenna geometry with or without embedded drill collar mandrels, plus coil sizes that might extend across multiple formation layers. Effects like charge radiation at layer interfaces, borehole invasion and eccentricity, and the like, were permitted, with algorithms running stably and rapidly converging within fifteen seconds on Intel Core i5 machines. Complementary "receiver design" methods were added to post-processing capabilities; no longer were users restricted to conventional coils wound in circular fashion – more general formulations allowed a variety of antenna designs which would ideally "see" more accurately in very realistic formations, using any range of probing frequencies from induction to dielectric.

Such problems are by no means simple. One might have thought that, since the pioneering work of Coope, Shen and Huang (1984) for axisymmetric AC analysis in vertical boreholes, numerous models would be available to study interpretation schemes or to design prototype hardware in formations with simple radial and horizontal layers. However, this is unfortunately not so. The analytical work in Coope *et al*, while correct, is highly mathematical and incomprehensible; to this author, the formulation was lacking because it could not be extended numerically to model complicated geologies, fluid invasion, plus other real-world effects. The reasons are numerous and esoteric: well known limitations of complex variables formulations, computational techniques that inaccurately modeled Dirac delta function sources, and methods that could not simulate rapidly varying fields for all frequencies. In this book, we will address complicated AC axisymmetric problems and fields very generally.

Analogous issues are found in DC laterolog applications. For instance, Li and Shen (1992) note, in their widely-read numerical analysis paper, that focusing conditions were *inferred* from the literature. Also, assumptions underlying proprietary simulators were subject to speculation – for instance, one "known" focusing model could not be disclosed because of a confidentiality agreement. But the authors’ own work was equally cryptic – their "finite element analysis" is not described at all, but presumably available only to consortium participants. The paper employed arbitrary methods. Upon convergence, the total current I_{m} = I_{0} + I_{1} + I_{1’} and the corresponding voltage V_{m} = 0.5 (V_{m0} + V_{m0’}) – 0.25 (V_{m1} + V_{m1’} + V_{m2} + V_{m2’}) are *defined* and apparent resistivity is further *defined* as R_{a} = KV_{m}/I_{m}. And, at the risk of even more *definition*, "K is a *tool constant* that will make R_{a} equal to the true formation resistivity when the tool is in a *standard medium* [our italics]." Real solutions are neither simple nor arbitrary. And of course, real formations may be anisotropic, but that’s another story – until now, anyway, secrecy has prevailed.

Direct current laterolog and pad devices are by no means simple. With modern emphases on "low resistivity pay" and anisotropy, one would expect that industry publications would address the roles of R_{v} and R_{h}. Yet, literature searches conducted as recently as 2016 disclosed few modeling results let alone basic theory. Those that were available showed current lines that were orthogonal to potential surfaces, a clear indication that isotropic media was assumed, additionally with planar flow underlying assumptions. General issues in streamline tracing should have been discussed decades ago. A current source that probes effectively in one direction may be ineffective when turned ninety degrees, and vice-versa. It is clear that interpretation in anisotropic media requires different methods in vertical versus horizontal wells – needless to say, so does tool design. "Streamline tracing," the description of paths taken by electrical current, is developed rigorously here. In the published literature, these paths are typically carelessly sketched by hand – but accurate tracing is essential to understanding which part of the formation is actually being probed, if at all. When it gets down to details, answers to critical questions are needed. Here, we develop streamfunction methods pioneered by this author in the aerospace industry to problems in resistivity logging tool design and interpretation.

Solving for voltage distributions and current paths in fields with prescribed resistivity is one thing. But understanding what constitutes resistivity is another – an issue that raises more profound questions. What is resistivity? A simple analogy highlights the subtleties. Draw two "dots" on a solid surface an inch apart. Now measure that distance with a standard ruler – the answer, of course, is one inch. Repeat the measurement with a ruler, say, 10^{-100} inch long – because you’ll traverse every mountain and valley about every electron and proton, your answer might be, well, a thousand times that of the original. A similar situation arises, for example, with cross-bedded sands, which are treated in Chapter 6. Rock grains may be isotropic in a microscopic sense, but taken in the aggregate over multiple dipping layers, a direct current measurement may perceive anisotropy. An alternating current device may "see" events differently, e.g., are six-inch receiver spacings inherently different from thirty-inch spacings" tools? Quite clearly, the resistivity found depends on the "ruler" used.

Archie, of "Archie’s law" renown, long ago postulated an empirical relationship connecting resistivity to water saturation. Its application is universal and simple: determine farfield "virgin" resistivity from electrical measurements and his well known law gives saturation immediately. This recipe has dominated log analysis and reserves estimation for decades but it is overly simplified. All petrophysicists are familiar with the classic Schlumberger sketch for axisymmetric resistivity problems showing borehole fluid, mudcake, invaded zone (with spatially varying properties) and virgin rock. Correction charts proliferate which allow users to adjust predictions to account for idealizations that do not apply. But all of this is now unnecessary and antiquated given recent advances in resistivity and fluid-dynamical simulation.

Our approach is simple. The spatially variable water saturation field, which also evolves in time, is one that is easily calculated and found independently of resistivity. This fluid distribution depends on mudcake properties, which control invasion rates by virtue of extremely low cake permeabilities, wellbore and reservoir pressures, and relative permeability and capillary pressure (in the case of immiscible displacements) and molecular diffusion (for miscible flow). Now imagine that we have calculated S_{w}(r,t) in its entirety. Then, via Archie’s law, the corresponding resistivity distribution R{S_{w}(r,t)} is available for "plug in" to any of the general resistivity codes developed here and in Chin (2014) for various tools. Receiver responses are calculated. But, naturally, they are unlikely to agree with measured values. Of course, we recognize that multiphase properties are typically unknowns subject to guess work and refinement, so parameters related to, say, diffusion or relative permeability, are adjusted. Resistivity calculations are performed again and the process repeated until a parameter set consistent with receiver data is found.

This type of iterative analysis is no different from "history matching" in well testing (which matches to pressure transient response) or reservoir engineering (which utilizes production rate to gauge correctness). Our approach differs from the conventional use of Archie’s law in one significant detail: *distributions* of resistivity are used for history matching rather than *single* values. This topic is introduced in Chapter 7 by way of a simple example, but clearly, other permutations and possibilities quickly suggest themselves. Finally, Chapter 8 examines more sophisticated examples for "simpler, plug flow" fluid-dynamics models using algebraic as opposed to differential equations. These approaches will be useful in future developments of the "time lapse logging" methods introduced in Chapter 7 and in Chin *et al* (1986).

So, it is with personal satisfaction that the author has solved, and has disclosed in this third volume of John Wiley & Sons’ new *Advances in Petroleum Engineering* series, those difficult resistivity problems not considered in Chin (2014). The process of "telling all" is not without risk – one wrong claim or equation can derail a consulting practice built over perspiration and time. The validations presented here reduce this risk. Furthermore, they are designed to encourage acceptance by an industry accustomed to endorsing marketing claims with minimal justification. Why is one coil configuration better than another? Why are certain (arbitrary) depth of investigation definitions used? Why use "apparent resistivities" related to fictitious isotropic reference media when real formations are anisotropic with R_{v} >> R_{h}? And why should amplitude and phase resistivities "see" different depths of investigation even though their coupled solution follows from a single formulation? Are there better ways to use Archie’s law? Can we find improved methods that couple electromagnetic and fluid analyses which create additional value to petroleum engineering?

This book provides tools which facilitate research and software design. It raises questions. It promotes an understanding of the physics and an appreciation for mathematics with all its limitations. Finally it hopes, through a number of new ideas introduced, to elevate what has been a profession dominated by empirical service company equations and borehole correction charts into a scientific discipline that nurtures even more principled approaches. The research in this volume sets the stage for more comprehensive integration between electromagnetic analysis and fluid-dynamics in future publications – a work in progress that will continue despite the oil economy.

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

Houston, Texas and Beijing, China

Email: wilsonchin@aol.com

Phone: (832) 483-6899

Acknowledgements

The author gratefully acknowledges the efforts of several generations of petroleum physicists who have endeavored to bring rigor and understanding to very complicated geological applications of modern electromagnetism. Also, many of the problems successfully addressed here and in *Electromagnetic Well Logging: Models for MWD/LWD Interpretation and Tool Design* could not have been were it not for the Boeing Commercial Airplane Company in Seattle, Washington. It was here, during the author’s formative years just out of M.I.T., where exciting ideas related to complex Helmholtz partial differential equations, distributed sources, sinks and vortexes, three-dimensional streamline tracing, functions with discontinuous values or derivatives, and so on, were discussed and debated with enthusiasm and turned into software productively used to design modern aircraft. Many thanks go to Boeing, and in particular, to Paul Rubbert, Edward Ehlers, Donald Rizzetta and other colleagues. As usual, the author is indebted to Phil Carmical, Acquisitions Editor and Publisher, for his support and encouragement in disseminating his highly technical research monographs, together with equations, cryptic Greek symbols, formal algorithms and more. In times of uncertainty, such as the economic turmoil now facing all of us, it is even more important to "solve problems right" and work more productively. What our industry needs is more math and not less, more questioning and less acceptance, and it is through this latest volume that the author hopes to stimulate thought and continuing research in an important engineering endeavor central to modern exploration for oil and gas.