Chapter 1
Fundamental Ideas and Background

As suggested in our title Modern Borehole Analytics for Annular Flow, Hole Cleaning and Pressure Control and in our Preface, this book deals generally with the subject of borehole flow modeling. We build upon original research efforts documented in the author's earlier monographs, (i) Borehole Flow Modeling in Horizontal, Deviated and Vertical Wells (Gulf Publishing, 1992), (ii) Computational Rheology for Pipeline and Annular Flow (Elsevier, 2001), and (iii) Managed Pressure Drilling: Modeling, Strategy and Planning (Elsevier, 2012).

The last book, which was translated into Chinese in 2016, presents major research results completed under Contract No. 08121-2502-01, sponsored by the United States Department of Energy - 2009 Research Partnership to Secure Energy for America (RPSEA), Ultra-Deepwater Exploration Program, for "Advanced Steady-State and Transient, Three-Dimensional, Single and Multi-phase, non-Newtonian Simulation System for Managed Pressure Drilling."

The foregoing "MPD book" supersedes the prior two and focuses on validated analytical and mathematical models. As such, it does not discuss experimental results in detail, such as those cited in its references. Nor does it address the subjects of mudcake characterization and growth, which are considered in (i) Quantitative Methods in Reservoir Engineering, 2nd Edition - with New Topics in Formation Testing and Multilateral Well Flow Analysis (Elsevier, 2017) for single-phase flows and (ii) Formation Testing: Low Mobility Pressure Transient Analysis (with CNOOC, John Wiley, 2015) for multiphase flows.

The subject of formation permeability and pore pressure prediction, which is very relevant to mudcake growth and coupling to the formation, especially tight formations, is also omitted from the MPD reference. It was largely developed in the context of formation tester pressure transient and contamination modeling, treated extensively in two books, (i) Formation Testing Pressure Transient and Contamination Analysis (with CNOOC, John Wiley, 2014) and (ii) Formation Testing: Low Mobility Pressure Transient Analysis (with CNOOC, John Wiley, 2015).

As explained in our Preface, the present volume focuses on practical applications, and not theory, whose inclusion would have made this work unwieldy and difficult to read. The complete picture for borehole annulus, mudcake and formation is considered here. It goes without saying that modern algorithms are sophisticated and output intensive. Gone are the days of simple engineering models and algebraic formulas designed for "back of the envelope" answers. Real solutions now require complicated partial differential equation formulations, whose field solutions demand computer menus offering different numerical options, outputs with three-dimensional color graphics, and varied post-processing utilities. With the exception of Chapter 6, which deals with mudcake growth in single-phase flow, together with formulas and source code, all of our models are hosted by advanced software. However, our software models, validated and in use at major service companies, are affordable, easy to use, and aimed at mainstream audiences.

In this first chapter, we will outline the basic problems solved - for details, the reader is referred to the foregoing cited book references. Our capabilities are described in terms of specific problems and their solutions. To ensure clarity, we described the formulations in terms of input menus and our results in terms of output data listings and color graphics. Users desiring further explanation or examples are encouraged to consult our references, or even better, replicate and extend our computed results. Our explanations below, while oriented to laymen and non-specialists, are nonetheless rigorous and scientifically correct.

1.1 Background, industry challenges and frustrations.

In the following sections, we introduce annular flow modeling (subject of Chapters 2, 3 and 4), mudcake dynamics (Chapters 5 and 6), and permeability and pore pressure prediction (Chapters 7 and 8). Only brief overviews of the problems are provided, as details are available in the referenced books. Applications are considered in specific chapters.

1.1.1 Annular flow modeling issues and problem definition.

The fundamental problem in downhole applications is borehole flow modeling in the annulus. Real annuli are typified by varied geometries, e.g., refer to those sketched in Figure 1-1.

Figure 1.1. Real and idealized annular geometry models.

Figure 1-1c represents flow in a circular pipe. For many steady-state non-Newtonian flows, pipe solutions are available analytically, including closed form representations for the circular cross-section "plug flow" found at the center of the pipe in the case of yield stress fluids (plugs move as solid bodies and plug flows are convected downstream with constant speed). Some approaches to annular flow employ somewhat dubious notions related to "equivalent hydraulic radius," where flow rates for given pressure gradients are computed from an "equivalent" pipe flow - a somewhat questionable and ill-defined concept at best. For concentric annuli, e.g., Figure 1-1b, numerical solutions are available for Power law fluids only; in the case of Bingham plastics and Herschel-Bulkley fluids, a concentric "ring plug" wraps around the inner body - here. concentricity arises from symmetry considerations, but simple solutions do not appear to be available. Real annuli are highly eccentric, as in Figure 1-1a, and numerical solutions for non-yield cases are available in bipolar coordinates. Very often, simpler "pie-slice" models (see Figure 1-1e) are used, consisting of crude solution "slices" extracted from concentric solutions. When eccentricity is small, the annulus is often "unwrapped" as in Figure 1-1d, resulting in multiple "slot flows" solved by simpler rectangular flow formulas.

Of course, the general problem is represented by Figure 1-1f, where a highly eccentric annulus is shown, which may possess non-flat cuttings beds, irregularly shaped washouts, and so on. This general problem, and all of the simpler prior flows, have been solved by the author and are documented in his three annular flow books for Newtonian, Power law, Bingham plastic and Herschel-Bulkley fluids, for example, as schematically described by Figure 1-2 in terms of constitutive relations.

Figure 1.2. Constitutive relations for basic rheologies.

Plug flows, as we have noted, arise from yield stress effects; in a circular pipe, the plug is always circular and situated at the center of the pipe. For concentric annuli, by virtue of symmetry considerations, the plug is a concentric ring that wraps around the centerbody. Plug flows introduce nontrivial changes to velocity and stress patterns in the annular cross-section, and are associated with dynamic attributes important in hole cleaning and mud displacement in cementing applications.

For the general annulus in Figure 1-1f, the shape, size and location of the plug have long represented unresolved modeling challenges. Authors typically assume that a plug ring exists which wraps around the centerbody or drillpipe, although it will not form a perfect circle. A macroscopic "pie slice" view of the annulus is taken, and within each slice of the pie, a plug segment roughly parallel to the local outer annular contour is assumed. The cumulative effect of all such slices is a "wrap around plug ring" with variable azimuthal thickness. This seems to be reasonable, providing an implementable "recipe" or algorithm.

However, the logic is flawed. Consider a highly eccentric example where the inner pipe diameter is continuously reduced. At some point, one expects to find an oval or elliptical plug in the wide part of the annulus, as in the far right of Figure 1-3 - much like that of a circular pipe, although it will neither be circular in cross-section nor centered (however, the left two plug flows are reasonable). How its shape, size and location vary with geometric details, and in fact, with flow rate and non-Newtonian rheology, have been open questions until now. The problem is solved numerically in Managed Pressure Drilling (2012) and we refer readers to the book for the detailed theory and applications.

Figure 1.3. Different plug zone configurations.

The general borehole flow problem considered in the present book is defined in part by Figure 1-4. Here we have an arbitrary pumping schedule where different non-Newtonian fluids are pumped at different volume flow rates for different time durations down a circular drillpipe (or casing), through the drillbit, and finally, up the annulus. The annular geometry may be quite general, as noted earlier; in addition, the borehole axis may be curved (so that centrifugal forces enter the flow description). Furthermore, the pipe (or casing) may rotate and move axially as arbitrary functions of time, to be defined through computer menus to the user's discretion. Finally, the pump pressure gradient may be completely transient. In a typically eccentric annulus, plug flows are accurately calculated as noted above. This general annulus flow problem is treated in...