Quantitative Methods in

Reservoir Engineering

by

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

Stratamagnetic Software, LLC

Houston, Texas and Beijing, China

Preface

Most reservoir flow analysis books provide the basic equations, such as Darcy’s law, single-phase radial flow solutions, simple well test models, and the usual descriptions of relative permeability and capillary pressure, and explain elementary concepts in finite difference methods and modeling before referring readers to commercial simulators and industry case studies. These books, and the courses that promote them, are useful in introducing students to fundamental methodologies and company practices. However, few develop the physical and mathematical insight needed to create the next generation of models or to evaluate the limitations behind existing simulation tools. Many analysis techniques and computational approaches employed, in fact, are incorrect, despite their common use in reservoir evaluation. Rigor is noticeably absent.

I earned my Ph.D. at M.I.T. and an earlier M.Sc. degree from Caltech. My major areas were high-speed aerodynamics and wave propagation, which are synonymous with applied math and nonlinear differential equations – specialties that focus on rigorous solutions to practical problems. From M.I.T., I joined Boeing’s prestigious computational fluid dynamics group in Seattle and, three years later, headed up engine flow analysis at United Technologies’ Pratt and Whitney, the company that develops the world’s most powerful jet engines.

But the thrill of the hunt lost its allure, despite the thrill of being published in journals and attending high-tech conferences. Like all of you, I was attracted to the petroleum industry because of its excitement and the opportunities it offered. That was just five years into my career, as I joined a new industry undergoing rapid change – a transition requiring me to learn anew the fluid dynamics of flows as far underground as my prior learning was above ground. Since then, three decades have elapsed, in which I actively engaged in oil field research and development. In that time, for example, with leading operating and service companies like British Petroleum, Schlumberger and Halliburton, I was fortunate to have been continuously challenged by new problems both mathematical and operational.

This reservoir flow analysis and simulation book is unique because it brings several decades of perspectives and experience to the fluid mechanics of Darcy flows. Many commonly accepted "recipes" for flow evaluation are critiqued and incorrect underlying assumptions are noted. This volume aims at a rigorous and scientifically correct approach to reservoir simulation. In each of dozens of difficult problems surveyed, the state of the art is examined, and analytical or numerical solutions are offered, with the exact physical assumptions always stated precisely. Industry "common sense" approaches are avoided: once the correct model is formulated, the entire arsenal of modern analysis tools is brought to bear – we then focus on ways to extract formation information using the new solution or clever means to exploit the physics just uncovered.

Fortunately, this book does not require advanced mathematics or numerical analysis to understand. Great care was undertaken to explain and develop very advanced methods in simple terms that undergraduates can comprehend. For example, "conformal mapping" usually requires a background course in complex variables, and complementary subjects like streamfunctions and streamline tracing in homogeneous media are typically taught within this framework – one that could be intimidating and esoteric. Quite to the contrary, our special derivations require just simple calculus but more generally apply to anisotropic, heterogeneous media, and not necessarily those described using rectangular coordinate systems. This book addresses "difficult" flows, such as liquid and gas flows from fractures, general flows past shales, production from multilateral horizontal wells, multiple well interactions, rigorous approaches to effective properties, and so on, problems not often treated in the literature but relevant to modern petroleum engineering. In doing so, we strive to avoid the simplistic "recipe" approaches our industry often encourages.

Every effort is made to define and formulate the mathematical problem precisely and then to solve it as exactly as modern analysis methods will allow. These include classic differential equation models as well as modern singular integral equation approaches, all of which were unavailable to Morris Muskat when he wrote his lasting monographs on Darcy flow analysis. Our techniques go beyond purely analytical ones. For example, the problem of accurately modeling flow from interacting multilateral drain holes in anisotropic heterogeneous layered media – despite the inefficiencies imposed by non-neighboring grid point connections – is formulated and solved in Chapter 15 (the groundwork for this research won a Chairman’s Innovation Award at BP Exploration in 1991 and led to a substantial "Education for the Energy Industry" contract funded by ExxonMobil, ChevronTexaco and Shell in 2000-2003).

Or consider boundary-conforming, curvilinear grids in Chapter 8. Fast and accurate mesh generation algorithms are developed in this book, which are cleverly applied to the solution of complicated reservoir flows. Suppose a "Houston well" produces from a "Texas-shaped" reservoir. This geometry is associated with an elementary function as unique as the logarithm is to radial coordinates. Its "analogous logarithm" permits us to instantly write the solution to all liquid and gas flows for any set of pressure-pressure and pressure-rate boundary conditions. What if the Texas-shaped reservoir is produced by a Dallas well? Ah yes, the "Dallas logarithm" – we’ll address that too. This work won a prestigious Small Business Innovation Research Award from the United States Department of Energy in 2000, the author’s first of five D.O.E. contracts.

Other areas addressed include opened fractures, curved shales, fractured holes, general heterogeneities, formation invasion, and time-lapse well logging using drilling data. In terms of techniques, we introduce modern ideas in singular integral equations, improper integrals, advanced conformal mapping, perturbation methods, numerical grid generation, artificial viscosity, moving boundary value problems, ADI and relaxation methods, and so on, developing these in context with the physics of the problem at hand. These methods, used by aerodynamicists and workers in theoretical elasticity, can be intimidating.

However, the presentation style adopted is far from difficult: while not exactly easy reading, there is nothing in this book that could not be grasped by a student who has taken basic freshman calculus. Whenever possible, Fortran source code is presented, so that students can test and evaluate ideas old and new without the trials and tribulations of debugging (free Fortran compilers are available online for unrestricted downloading from various third-party sources).

New approaches to old problems are emphasized. For example, how do mathematical aerodynamicists turned petroleum engineers view the physical world? Stare up the back end of a rocket lifting off: *Is that a fuselage with stabilizer fins, or is it a circular wellbore with radial fractures?* Pry open the maintenance box of your typical jet engine: *Are those cascades of airfoil blades, or are they distributions of stochastic shales?* *Can the solutions that describe brittle failure be repackaged to model arrays of fractures, say, the natural fracture systems that spur horizontal drilling? *Very often, the problems (inaccurately) crunched by our fastest computers can be solved (accurately) using closed-form analytical solutions found in other scientific disciplines.

In this Preface to our Second Edition, it is important to describe the additional content developed which extends the scope of the earlier work. All of the original work remains correct – additional perspectives and extensions are added from a since broader exposure to client activities. However, two subject areas are completely new, namely, "formation testing" and "multilateral well reservoir modeling." We will give a brief synopsis of this material next.

Formation testers are borehole well logging instruments that, when pressed against the sandface, extract reservoir fluids into sealed chambers for examination at the surface. During this extraction, pressure transients in the flowline are recorded for analysis. Some "testers" consist of a single source or pumping probe, while others may contain two (or more) probes. Often, these additional probes are passive observation probes. In the 1960s, interpretation methods were developed to interrogate measured pressure transients – from recorded steady-state drawdown values, effective permeabilities can be obtained from single probe tools. Later in the 1990s, dual probe tools were used to provide k_{h} and k_{v} individually, again using steady state pressure drops. These methods were popular, not only because downhole sampling provided laboratory analysis that allowed direct assessment of petroleum fluid properties, but because measurements for permeability (which affect production planning and cash flow) could be obtained as a by-product of the survey. For these reasons, formation testing remains a powerful tool in petroleum exploration, and not surprisingly, a strong source of profits for oil service companies.

However, as new exploration fields continually decreased in mobility, the times required to achieve steady-state pressures increased substantially. Quite often, wait times of hours and sometimes days instead of minutes were required. This increased drilling and logging costs, and also, raised the risk of stuck tools lost in the well. Thus, improvements to formation testing methodologies were clearly needed. In 2004, the United States Department of Energy, through its Small Business Innovation Research (SBIR) program, awarded about two hundred contracts for all fields of energy, e.g., wind, ocean, geothermal, air conditioning, plasma and nuclear physics, and so on. Four awards were allotted to fossil fuels. Of these four, two were won by our Stratamagnetic Software LLC and both in the area of formation testing. Since then, this author has published two books in formation testing, and developed (patent pending) methods to provide k_{h} and k_{v} in real-time using early-time (seconds) data collected from low to very-low mobility formations. These methods, which have reduced both risks and costs, were developed by extending the cylindrical flow principles described in our prior book edition to the ellipsoidal flows now encountered in pumping from transversely isotropic reservoirs.

In Chapter 15 of the first edition, problems and limitations associated with horizontal and multilateral well analysis were discussed. A general and very versatile formulation was provided and successful but preliminary numerical results were described. Since then, extensive improvements allow both rapid computing speed on low-end hardware and excellent numerical stability. The methodology applies to general well systems in layered, anisotropic and heterogeneous media producing liquids or gases, where wells can act as producers or injectors or change roles or shut-in during simulation. Importantly, pressure and rate constraints can be altered during simulation, and complicated wells can be added to the reservoir if desired. "Drilling while simulating" is introduced for the first time.

In 2000, three major operating companies funded Houston’s well known "E.E.I." or "Education for the Energy Industry" program to introduce petroleum focused careers to inner city students from Kindergarten to Grade 12 at the Aldine Independent School District. Stratamagnetic Software LLC won the competitive bidding over eighty more established companies with a simple but irresistible proposition: develop a learning platform and curriculum allowing teachers and students to understand horizontal and multilateral well production and simulate the most complicated flows using our custom-developed software. Our reservoir flow models have since been commercialized by several organizations. The additional content provided in this Second Edition summarizes key capabilities of the new approach.

More efficient technologies are now more important than ever in the present economic environment – companies are downsizing, research budgets are being reduced, and experienced staff are driven to focus on more practical and immediate activities. The author hopes that the new work will help the industry explore more efficiently and cost-effectively, and that it will convince new engineers joining the petroleum engineering industry that innovation continues and thrives despite the doom and the gloom. Our new approaches to horizontal and multilateral well modeling will be essential to economic planning for unconventional resources; our new approaches to formation tester pressure transient interpretation will hopefully pave the way for further development in petrophysics. And we trust that our original contributions in fracture modeling will enable more efficient fracturing programs and reduced needs for water resources. In short, we’re here for the long haul and we won’t be "turning out the lights" any time soon.

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

Houston, Texas and Beijing, China

Email: wilsonchin@aol.com

Cell: (832) 483-6899

Acknowledgements

The author wishes to express his appreciation to his colleagues at Schlumberger, Halliburton, BakerHughes, British Petroleum, GE Oil & Gas and other companies for their insights and suggestions over the years and for shaping his approach to reservoir engineering and simulator development. He is especially grateful to Katie Hammon, Senior Acquisitions Editor, for providing him the platform to communicate important ideas in a highly technical volume. Here content is developed that enables users to solve important and modern problems using the latest modeling technologies available. Moreover, the methods are not "me too" in nature; they are unique, e.g., singular integral equation, advanced finite difference methods, specialized conformal mappings, boundary-conforming grid generation techniques, and so on, but no attempt is made to be all things to all people. It is often said that "one does not understand, until one can compute it." Few recent oilfield achievements have approached areas like horizontal and multilateral well design, formation testing, and fracture flow heterogeneous reservoirs, in economic and practical significance, the subject of this Second Edition – and with the methodologies presented here, we believe that we have closed an important gap in petroleum engineering by simultaneously advancing simulation methods while reducing planning and modeling costs by several orders of magnitude. For the opportunity to present these ideas, many new and developed only within the past several years, to a broad audience, I thank Elsevier Science, and again, Katie. And I look forward to constructive feedback from readers both young and experienced, important feedback necessary to improve upon our methods and which all new technologies thrive upon.