As mentioned at the outset of this book, arriving at practical solutions to many problems in geomechanics requires knowledge of the magnitude and orientation of all three principal stresses. This is well illustrated by the range of geomechanical topics and case studies presented in Chapters 10–12. The first subject discussed in this chapter is the magnitude of the least principal stress, S3, as obtained by hydraulic fracturing, specifically mini-frac (or micro-frac) tests done specifically for the purpose of measuring stress. As discussed at length below, because hydraulic fracturing frequently occurs during leak-off tests (LOT's) and especially extended leak-off tests (XLOT's), these tests also can be used to determine S3. In normal and strike-slip faulting environments,S3 is equivalent to Shmin. In reverse faulting environments, S3 is equivalent to Sv. Methods for determination of Shmin from Poisson's ratio (obtained from P- and S-wave sonic logs) are based on questionable physical and geologic assumptions. These methods will be discussed briefly in Chapter 9. Suffice it to say at this point that direct measurement of the least principal stress through some form of hydraulic fracturing is the only reliable method known that is practical to use in wells and boreholes at any appreciable depth.

One can determine the magnitude of the least principal stress from a micro-frac, a very small-scale hydraulic fracture induced only to measure stress at a particular depth, usually at a specific depth through perforations in cemented casing.

In this chapter I address a number of problems related to wellbore stability that I illustrate through case studies drawn from a variety of sedimentary basins around the world. In this chapter we focus on wellbore stability problems associated with mechanical failure of the formations surrounding a wellbore. Failure exacerbated by chemical reactions between the drilling mud and the formation is addressed only briefly. I make no attempt to discuss a number of critically important issues related to successful drilling such as hole cleaning, wellbore hydraulics, mechanical vibrations of the drilling equipment, etc. and refer readers to excellent texts such as that of Bourgoyne Jr., Millheim et al. (2003).

In each case study considered in this chapter, a comprehensive geomechanical model was developed utilizing the techniques described in previous chapters. The problems addressed fall into two general categories: Preventing significant wellbore instability during drilling and limiting failure of the formation surrounding the wellbore during production. The latter problem is sometimes referred to as sand (or solids) production as significant formation failure during production results in fragments of the formation being produced from the well along with hydrocarbons. Another aspect of wellbore failure with production, the collapse of well casings due to depletion-induced compaction and/or the shearing of wells by faults through injection- (or depletion-) induced faulting, will be discussed in Chapter 12.

In this chapter I review a number of fundamental principles of rock failure in compression, tension and shear that provide a foundation for many of the topics addressed in the chapters that follow. The first subject addressed below is the classical subject of rock strength in compression. While much has been written about this, it is important to review basic types of strength tests, the use of Mohr failure envelopes to represent rock failure as a function of confining stress and the ranges of strength values found for the rock types of interest here. I also discuss the relationship between rock strength and effective stress as well as a number of the strength criteria that have been proposed over the years to describe rock strength under different loading conditions. I briefly consider rock strength anisotropy resulting from the presence of weak bedding planes in rock, which can be an important factor when addressing problems of wellbore instability. This is discussed in the context of two specific case studies in Chapter 10.

In this chapter I also discuss empirical techniques for estimating rock strength from elastic moduli and porosity data obtained from geophysical logs. In practice, this is often the only way to estimate strength in many situations due to the absence of core for laboratory tests. This topic will be of appreciable interest in Chapter 10 when I address issues related to wellbore stability during drilling.

Addressing problems associated with the deformation and changes of stress within and surrounding depleting reservoirs is important for many reasons. Most well known are the problems associated with casing collapse and surface subsidence that create substantial difficulties in some oil and gas reservoirs due to compaction in weak formations. The significant stress changes that occur in highly depleted reservoirs (e.g. Figure 2.10a) can make drilling a new well to deeper targets quite problematic due to the need to lower mud weights in depleted formations to avoid lost circulation. Depletion also has the potential to induce faulting, both within and outside reservoirs in some geologic environments. While these problems can be formidable in some reservoirs, depletion can also have beneficial impact on reservoir performance. For example, hydraulic fracturing can be more effective in depleted reservoirs than in the same reservoirs prior to depletion. In some weak reservoirs, compaction drive is an effective mechanism for enhancing the total amount of hydrocarbons recovered, especially if the permeability changes accompanying compaction are not severe.

To address these issues in a comprehensive manner, this chapter is organized in three sections. In the first section I consider processes accompanying depletion within reservoirs and focus initially on the stress changes associated with depletion. We begin by discussing reservoir stress paths, the reduction of horizontal stress magnitude within the reservoir resulting from the decrease in pore pressure associated with depletion.

The subject of hydrodynamic stability theory is concerned with the response of a fluid system to random disturbances. The word “hydrodynamic” is used in two ways here. First, we may be concerned with a stationary system in which flow is the result of an instability. An example is a stationary layer of fluid that is heated from below. When the rate of heating reaches a critical point, there is a spontaneous transition in which the layer begins to undergo a steady convection motion. The role of hydrodynamic stability theory for this type of problem is to predict the conditions when this transition occurs. The second class of problems is concerned with the possible transition of one flow to a second, more complicated flow, caused by perturbations to the initial flow field. In the case of pressure-driven flow between two plane boundaries (Chap. 3), experimental observation shows that there is a critical flow rate beyond which the steady laminar flow that we studied in Chap. 3 undergoes a transition that ultimately leads to a turbulent velocity field. Hydrodynamic stability theory is then concerned with determining the critical conditions for this transition.

For both types of problem, we can view the mathematical problem as one of determining the consequence of adding an initial perturbation in the velocity, pressure, temperature, or solute concentration fields to a basic unperturbed state. If the perturbation grows in time, the original unperturbed state is said to be unstable.

In the preceding chapters, we focused mainly on fluid dynamics problems, with only an occasional problem involving heat or mass transfer. In this chapter, we change our focus to problems of heat (or single-solute mass) transfer. Specifically, we address the problem of heat (or mass) transfer from a finite body to a surrounding fluid that is moving relative to the body. In this chapter, we concentrate on problems in which the fluid motion is viscous in nature, and thus is “known” (or can be calculated) from creeping-flow theory. Later, after we have considered flows at nonzero Reynolds number, we will also consider heat (or mass) transfer for this situation.

In all of the fluid mechanics problems that we have considered until now, the nonlinear inertia terms in the equations of motion were either identically zero or small compared with the viscous terms. We begin this chapter by considering the corresponding heat (or mass) transfer problem, in which the fluid motion is “slow” in a sense to be described shortly, so that convection effects are weak and the transport process is dominated by conduction. When convection terms in the thermal energy equation can be neglected altogether, the resulting pure conduction problem is mathematically and physically analogous to the creeping flows that we have been studying in the preceding two chapters. The transport of heat is purely “diffusive” in this limit, i.e., conduction, just as the transport of momentum (or vorticity) in a creeping flow is also “diffusive.”

Although the application of the “thin-film” approximation to analyze lubrication problems is one of its most important successes, there is an even larger body of problems in which the thin-film approximation can still be applied but in which the upper surface (or in some cases both surfaces of the thin film) is an interface. Examples include such diverse applications as gravity currents in geological phenomena, such as the gravitationally driven spread of molten lava; the dynamics of foams and or emulsions for which the thin films between bubbles (or drops) play a critical role in the dynamics; the dynamics of thin films in coating operations, and a variety of other materials processing applications; and thin films in biological systems, such as the coatings of the lung. Not only are the areas of application very diverse, but such films can and do display an astonishing array of complex phenomena, in spite of the limitations inherent in the thin-film assumptions. In part this is a consequence of the wide variety of physical effects that can play a role, including the capillary and Marangoni phenomena associated with surface tension, the possibility of a significant role for nonhydrodynamic effects such as van der Waals forces across the thin film and the possibility of transport processes such as evaporation/condensation.

In this chapter, we derive the governing equations for this class of thin films and show how they can be modified to account for the presence or absence of the various physical phenomena that were mentioned above.

We are now in a position to begin to consider the solution of heat transfer and fluid mechanics problems by using the equations of motion, continuity, and thermal energy, plus the boundary conditions that were given in the preceding chapter. Before embarking on this task, it is worthwhile to examine the nature of the mathematical problems that are inherent in these equations. For this purpose, it is sufficient to consider the case of an incompressible Newtonian fluid, in which the equations simplify to the forms (2–20), (2–88) with the last term set equal to zero, and (2–93).

The first thing to note is that this set of equations is highly nonlinear. This can clearly be seen in the term u · grad u in (2–88). However, because the material properties such as ρ, Cp, and k are all functions of the temperature θ, and the latter is a function of the velocity u through the convected derivative on the left-hand side of (2–93), it can be seen that almost every term of (2–88) and (2–93) involves a product of at least two unknowns either explicitly or implicitly. In contrast, all of the classical analytic methods of solving partial differential equations (PDEs) (for example, eigenfunction expansions by means of separation of variables, or Laplace and Fourier transforms) require that the equation(s) be linear. This is because they rely on the construction of general solutions as sums of simpler, fundamental solutions of the DEs.

In Chap. 9 we considered strong-convection effects in heat (or mass) transfer problems at low Reynolds numbers. The most important findings were the existence of a thermal boundary layer for open-streamline flows at high Peclet numbers and the fundamental distinction between open- and closed-streamline flows for heat or mass transfer processes at high Peclet numbers. An important conclusion in each of these cases is that conduction (or diffusion) plays a critical role in the transport process, even though Pe → ∞. In open-streamline flows, this occurs because the temperature field develops increasingly large gradients near the body surface as Pe → ∞. For closed-streamline flows, on the other hand, the temperature gradients are O(1) – except possibly during some initial transient period – and conduction is important because it has an indefinite time to act.

In this chapter we continue the development of these ideas by considering their application to the approximate solution of fluid mechanics problems in the asymptotic limit Re → ∞, with a particular emphasis on problems in which boundary layers play a key role. Before embarking on this program, however, it is useful to highlight the expected goals and limitations of the analysis in which we formally require Re → ∞ but still assume that the flow remains laminar. In practice, of course, most flows will become unstable at a large, but finite, value of Reynolds number and eventually undergo a transition to turbulence, and this is the flow we will see in the lab.

This book represents a major revision of my book Laminar Flow and Convective Transport Processes that was published in 1992 by Butterworth-Heinemann. As was the case with the previous book, it is about fluid mechanics and the convective transport of heat (or any passive scalar quantity) for simple Newtonian, incompressible fluids, treated from the point of view of classical continuum mechanics. It is intended for a graduate-level course that introduces students to fundamental aspects of fluid mechanics and convective transport processes (mainly heat transfer and some single solute mass transfer) in a context that is relevant to applications that are likely to arise in research or industrial applications. In view of the current emphasis on small-scale systems, biological problems, and materials, rather than large-scale classical industrial problems, the book is focused more on viscous phenomena, thin films, interfacial phenomena, and related topics than was true 14 years ago, though there is still significant coverage of high-Reynolds-number and high-Peclet-number boundary layers in the second half of the book. It also incorporates an entirely new chapter on linear stability theory for many of the problems of greatest interest to chemical engineers.

The material in this book is the basis of an introductory (two-term) graduate course on transport phenomena. It starts with a derivation of all of the necessary governing equations and boundary conditions in a context that is intended to focus on the underlying fundamental principles and the connections between this topic and other topics in continuum physics and thermodynamics.

In the preceding chapters, a number of asymptotic methods were introduced for the approximate solution of nonlinear flow problems. In many of the cases considered so far, including all of the problems of the two preceding chapters, the asymptotic limiting process produced a simplification of the full nonlinear problem by restricting the geometry of the flow domain to one in which certain terms in the equations could be neglected because the length scales in some direction (or directions) become very large compared with the length scales in other directions. In retrospect, even the exact unidirectional flow problems of Chap. 3 can often be regarded as a first approximation of some more general problem in which the geometry reduces to a unidirectional form in the limit as a ratio of two length scales vanishes, e.g., the “Dean” problem of Chap. 4, which reduces to the unidirectional Poiseuille flow problem in the limit as the ratio of the tube radius to the radius of curvature of the tube in the axial direction goes to zero. In some cases, this disparate ratio of length scales was true everywhere in the flow domain, and then the asymptotic solution was found to be “regular”; e.g., the Dean problem or the eccentric cylinder problem of Chap. 5. In others, the region with a small length-scale ratio was restricted to a local part of the overall flow domain, and in these cases, the asymptotic approximation was of the “singular” type, in which the simplified form of the governing equations is valid only locally, and the resulting approximate solution must be “matched” to a solution of the unapproximated equations that are valid elsewhere.

Although the full Navier–Stokes equations are nonlinear, we have studied a number of problems in Chap. 3 in which the flow was either unidirectional so that the nonlinear terms u · ∇u were identically equal to zero or else appeared only in an equation for the cross-stream pressure gradient, which was decoupled from the primary linear flow equation, as in the 1D analog of circular Couette flow. This class of flow problems is unusual in the sense that exact solutions could be obtained by use of standard methods of analysis for linear PDEs. In virtually all circumstances besides the special class of flows described in Chap. 3, we must utilize the original, nonlinear Navier–Stokes equations. In such cases, the analytic methods of the preceding chapter do not apply because they rely explicitly on the so-called superposition principle, according to which a sum of solutions of linear equations is still a solution. In fact, no generally applicable analytic method exists for the exact solution of nonlinear PDEs.

The question then is whether methods exist to achieve approximate solutions for such problems. In fluid mechanics and in convective transport problems there are three possible approaches to obtaining approximate results from the nonlinear Navier–Stokes equations and boundary conditions.

First, we may discretize the DEs and boundary conditions, using such formalisms as finite-difference, finite-element, or related approximations, and thus convert them to a large but finite set of nonlinear algebraic equations that is suitable for attack by means of numerical (or computational) methods.

A BRIEF HISTORICAL PERSPECTIVE OF TRANSPORT PHENOMENA IN CHEMICAL ENGINEERING

“Transport phenomena” is the name used by chemical engineers to describe the subjects of fluid mechanics and heat and mass transfer. The earliest step toward the inclusion of specialized courses in fluid mechanics and heat or mass transfer processes within the chemical engineering curriculum probably occurred with the publication in 1923 of the pioneering text Principles of Chemical Engineering by Walker, Lewis, and McAdams. This was the first major departure from curricula that regarded the techniques involved in the production of specific products as largely unique, to a formal recognition of the fact that certain physical or chemical processes, and corresponding fundamental principles, are common to many widely differing industrial technologies.

A natural outgrowth of this radical new view was the gradual appearance of fluid mechanics and transport in both teaching and research as the underlying basis for many of the unit operations. Of course, many of the most important unit operations take place in equipment of complicated geometry, with strongly coupled combinations of heat and mass transfer, fluid mechanics, and chemical reaction, so that the exact equations could not be solved in a context of any direct relevance to the process of interest. Hence, insofar as the large-scale industrial processes of chemical technology were concerned, even at the unit operations level, the impact of fundamental studies of fluid mechanics or transport phenomena was certainly less important than a well-developed empirical approach (and this remains true today in many cases).