Conservation of Mass

The concept of conservation of mass holds a fundamental role in most problems in fluid physics. For a given problem this concept is cast in the form of an equation of continuity. Such an equation describes a condition—conservation of mass—that must be satisfied in any formal analysis of a problem. Thus an equation of continuity often is one of several complementary equations that are solved simultaneously to arrive at a solution to a flow problem, for example, the flow velocity as a function of coordinate position in a flow field. (Typically these complementary equations, as we will see in later chapters, involve conservation of momentum or energy, or both.) Although we did not explicitly use this idea in analyzing the one-dimensional flow problems at the end of Chapter 3, it turns out that continuity was implicitly satisfied in setting up each problem. We will return to these problems to illustrate this point. We will develop equations of continuity for three general cases: purely fluid flow, saturated single-phase flow in porous media, and unsaturated flow in porous media. The most general of the three equations is that for unsaturated flow, where pores are partially filled with the fluid phase of interest, such that the degree of saturation with respect to that phase is less than one. We will then show that this equation reduces, in the special case in which the degree of saturation equals one, to a simpler form appropriate for saturated single-phase flow. Then, this equation for saturated flow could be reduced further, in the special case in which the porosity equals one, to a form appropriate for purely fluid flow. For pedagogical reasons, however, we shall reverse this order and consider purely fluid flow first. In addition we will consider conservation of a solid or gas dissolved in a liquid, and take this opportunity to introduce Fick’s law for molecular diffusion. For simplicity we will consider only species that do not react chemically with the liquid, nor with the solid phases of a porous medium. Most of the derivations below are based on the idea of a small control volume of specified dimensions embedded within a fluid or porous medium.

Fluid Statics and Buoyancy

Fluid statics concerns the behavior of fluids that possess no linear acceleration within a global (Earth) coordinate system. This includes fluids at rest as well as fluids possessing steady motion such that no net forces exist. Such motions may include steady linear motion within the global coordinate system as well as rotation with constant angular velocity about a fixed vertical axis. In this latter case, centrifugal forces must be balanced by centripetal forces (which arise, for example, from a pressure gradient acting toward the axis of rotation). Moreover, we assert that no relative motion between adjacent fluid elements exists. Fluid motion, if present, is therefore like that of a rigid body. In addition, we neglect molecular motions that lead to mass transport by diffusion. Thus, the idea of a static fluid is a macroscopic one. The developments in this chapter clarify how pressure varies with coordinate position in a static fluid. Both compressible and incompressible fluids are treated. In the simplest case in which the density of a fluid is constant, we will see that pressure varies linearly with vertical position in the fluid according to the hydrostatic equation. In addition, we will consider the possibility that fluid density is not constant. Then, variations in density must be taken into account when computing the pressure at a given position in a fluid column; the pressure arising from the weight of the overlying fluid no longer varies linearly with depth. In the case of an isothermal fluid, whose temperature is constant throughout, any variation in density must arise purely from the compressible behavior of the fluid in response to variations in pressure. In the case where temperature varies with position, fluid density may vary with both pressure and temperature. We will in this regard consider the case of a thermally stratified fluid whose temperature varies only with the vertical coordinate direction. Because fluid statics requires treating how fluid temperature, pressure, and density are related, the developments below make use of thermodynamical principles developed in Chapter 4.

Thermodynamic Properties of Fluids

Fluid behavior in many geological problems is strongly influenced by extant thermal conditions and flow of heat. Recall, for example, that the coefficient A in Glen’s law for ice (3.40) varies over three orders of magnitude with a change in temperature of 50 °C. The effect of this is to strongly modulate the rate of ice deformation for a given level of stress. Recall further that we introduced several fluid properties—fluid compressibility, for example—where we asserted that our purely mechanical developments were incomplete inasmuch as they did not treat effects of varying temperature. The reasons for this will become clear in this chapter, including why it is difficult to maintain isothermal conditions when the pressure of a fluid is changing. In addition, many geological problems involve fluid flows that are induced by effects of variations in thermal conditions over time and space. These include buoyancy-driven convective motions that arise from variations in fluid density associated with variations in temperature (Chapter 16). Specific examples include convective overturning in a magma chamber, which can significantly influence how crystallizing minerals are distributed; convective circulations of water and chemical solutions in a sedimentary basin, which can influence where rock materials are dissolved and where they are precipitated as cements within pores; and convective circulation of water within the active layer above seasonally frozen ground, which may influence where patterned ground develops in periglacial environments. These processes, and viscous flows in general, invariably involve conversions of mechanical energy to heat, or vice versa. So in considering problems involving heat energy, we should recall from introductory chemistry and physics that such conversions can involve work performed on the fluid or its surroundings, and anticipate that the effects of this ought be manifest in fluid behavior. This chapter, then, is concerned with fluid pressure, temperature, and density, and how these variables are related to heat, mechanical energy, and work. We will note in digressions how these macroscopic concepts, like fluid viscosity, often have clear interpretations at a molecular scale based on kinetic theory of matter.

Turbulent Boundary-Layer Shear Flows

Turbulent shear flows next to solid boundaries are one of the most important types of flow in geology. In such flows, turbulence is generated primarily by boundary effects; vorticity originates near a boundary in association with the velocity gradients that arise from the no-slip condition at the boundary. Such gradients provide a ready source of vorticity for eddies and eddy-like structures to develop in response to the destabilizing effects of inertial forces, and then move outward into the adjacent flow. Eddies are also generated within the wakes of bumps that comprise boundary roughness, for example, sediment particles on the bed of a stream channel (Example Problem 11.4.2). As we have seen in Chapter 14, the fluctuating motions of turbulence involve, over any elementary area, fluxes of fluid momentum that are manifest as apparent (Reynolds) stresses. In addition, the complex motions of eddies and eddy-like structures efficiently advect heat and solutes from one place to another within a turbulent flow, and thereby facilitate mixing of heat and solutes throughout the fluid. For similar reasons, turbulent motions are responsible for lofting fine sediment into the fluid column of a stream channel and in the atmosphere. We will concentrate in this chapter on steady unidirectional flows where the mean streamwise velocity varies only in the coordinate direction normal to a boundary and the mean velocity normal to the boundary is zero. We also will adopt a classic treatment of turbulent boundary flow in developing the idea of L. Prandtl’s mixing-length hypothesis, from which we will obtain the logarithmic velocity law, a function that describes how the mean streamwise velocity varies in the coordinate direction normal to a boundary. In developing Prandtl’s hypothesis, we will see that the presence of apparent stresses associated with fluctuating motions leads to the idea of an eddy viscosity or apparent viscosity. Unlike the Newtonian viscosity, the eddy viscosity is a function of the mean velocity, and therefore coordinate position. This means that the eddy viscosity cannot, in general, be removed from stress terms involving spatial derivatives, as we previously did with the Newtonian viscosity in simplifying the Navier–Stokes equations.

Turbulent Flows

Many geological flows involve turbulence, wherein the velocity field involves complex, fluctuating motions superimposed on a mean motion. Flows in natural river channels are virtually always turbulent. Magma flow in dikes and sills, and lava flows, can be turbulent. Atmospheric flows involving eolian transport are turbulent. The complex, convective overturning of fluid in a magma chamber or geyser is a form of turbulence. Thus, a description of the basic qualities of these complex flows is essential for understanding many geological flow phenomena. Turbulent flows generally are associated with large Reynolds numbers. Recall from Chapter 5 that the Reynolds number Re is a measure of the ratio of inertial to viscous forces acting on a fluid element, . . . Re = ρUL/μ . . . . . . (14.1) . . . where the characteristic velocity U and length L are defined in terms of the particular flow system. Thus, turbulence is typically associated, for given fluid density ρ and viscosity μ, with high-speed flows (although we must be careful in applying this generality to thermally driven convective motions; see Chapter 16). A simple, visual illustration of this occurs when smoke rises from a cigar within otherwise calm, surrounding air. The smoke acts as a flow tracer. Smoke molecules at the cigar tip start from rest, since they are initially attached to the cigar. Upward fluid motion, as traced by the smoke, initially is of low speed, and viscous forces have a relatively important influence on its behavior. The flow is laminar; smoke streaklines are smooth and locally parallel. But as the flow accelerates upward, it typically reaches a point where viscous forces are no longer sufficient to damp out destabilizing effects of growing inertial forces, and the flow becomes turbulent, manifest as whirling, swirling fluid motions (see Tolkien [1937]). Throughout this chapter we will consider only incompressible Newtonian fluids. Unfortunately, the complexity of turbulent fluid motions precludes directly using the Navier–Stokes equations to describe them. Instead, we will adopt a procedure whereby the Navier–Stokes equations are recast in terms of temporally averaged or spatially averaged values of velocity and pressure, and fluctuations about these averages.

Porous Media Flows

So far our treatment of fluid motions has not emphasized the behavior of fluids residing within porous geological materials. Let us now turn to this topic and, in doing so, make use of our insight regarding purely fluid flows. The general topic of fluid behavior within porous geological materials is an extensive one, forming the heart of such fields as groundwater hydrology, soils physics, and petroleum-reservoir dynamics. In addition, this topic is an essential ingredient in studies concerning the physical and chemical evolution of sedimentary basins, and the dynamics of accretionary prisms at convergent plate margins. In view of the breadth of these topics, the objective of this chapter is to introduce essential ingredients of fluid flow and transport within porous materials that are common to these topics. Our first task is to examine the physical basis of Darcy’s law, and to generalize this law to a form that can be used with an arbitrary orientation of the working coordinate system relative to the intrinsic coordinates of a geological unit that are associated with its anisotropic properties. We will likewise examine the basis of transport of solutes and heat in porous materials. We will then develop the equations of motion for the general case of saturated flow in a deformable medium. In this regard, several of the Example Problems highlight interactions between flow and strain of geological materials during loading, because this interaction bears on many geological processes. Examples include consolidation of sediments during loading, and responses of aquifers to loading by oceanic and Earth tides, and seismic stresses. We will concentrate on the description of diffuse flows within the interstitial pores of granular materials, as opposed to flows within materials containing dual, or multiple, pore systems such as karstic media, or media containing both interstitial and fracture porosities. We will consider unsaturated, as well as saturated, conditions. For simplicity, the subscript h is omitted from the notation of quantities such as specific discharge q and hydraulic conductivity K.

Fluid Kinematics

Let us momentarily recall an elementary problem from physics: describing the motion of a ballistic particle. To do this in a formal way first required developing expressions for the geometry of motion, independently of any treatment of the forces producing the motion. One may recall that this task involved defining expressions for the speed, velocity, and acceleration of the particle with respect to a specified coordinate system. In fact, a clear geometrical description of particle motion was essential for understanding its cause. We require similar, explicit descriptions for fluid motion to later relate this motion to the forces involved. Fluid kinematics thus involves the description of flow without explicit treatment of the forces producing motion. In this regard, our treatment of fluid kinematics actually is a pedagogical step toward the topic of fluid dynamics, wherein forces are explicitly treated. Our essential objective is to derive an expression that describes how the velocity of a fluid is changing. A change in velocity implies that the fluid is accelerating and, therefore, that a net force is acting on the fluid. Consider Newton’s second law, F = ma, which states that the net force F acting on a particle of mass m equals the product of this mass and the acceleration a. We thus seek an explicit expression for the acceleration a. We will supplement this later with an expression for F to obtain a full dynamical description of fluid motion. This chapter will introduce the idea of a substantive derivative, which will be used numerous times in subsequent chapters. In developing this idea, we will distinguish between Eulerian and Lagrangian views of fluid motion, and introduce the important concept of a convective acceleration. In most situations of interest in geology, convective accelerations arise when a real boundary induces a change in the direction or magnitude of the velocity of nearby flow; convective accelerations therefore typically involve converging or diverging flow. A good example is flow in a river channel whose bed is irregular due to bedform topography, for example, due to a point bar in a meander bend.

Dimensional Analysis and Similitude

Some fluid flow problems are sufficiently simple that they can be treated mathematically in a straightforward way, making use of definitions of physical quantities, and taking into account initial and boundary conditions. For example, our derivation of the average velocity in a conduit with parallel walls (Example Problem 3.7.1) was obtained in a straightforward way once we specified the geometry of the problem, then made use of the definition of a Newtonian fluid and the no-slip condition. Whereas this type of analysis may work for some problems, it would be misleading to think that such direct approaches to solving problems are, in principle, always possible, hinging only on one’s mathematical skills and adeptness in specifying the geometry of a problem. Herein arise two noteworthy points. First, when initially examining a problem, one can sometimes obtain a clear idea of the desired solution before attempting a formal mathematical analysis. The means to do this, as we shall see below, is supplied by dimensional analysis, and it is a strategy that ought to be adopted in many circumstances. In fact, it is worth noting that dimensional analysis underlies many of the problems presented in this text. The advantage of knowing the form of a desired solution, of course, is that one has a clear target to guide the subsequent mathematical analysis. Indeed, this is the vantage point from which many classic problems, for example Stokes’s law for settling spheres, were initially examined. Second, a complete mathematical formulation of a problem may not be possible, due to the complexity of the problem, or due to absence of information required to constrain the mathematics of the problem. As a simple example, suppose that we were unaware of the no-slip condition in our analysis of the conduit-flow problem (Example Problem 3.7.1). Our analysis in this case would have essentially ended with (3.70), with the constant of integration C undetermined. Nevertheless, we could get close to our result (3.75) for the average velocity by another way.

Vorticity and Fluid Strain

Many flows involve a sense of rotation. Clear examples include cyclones, whirlpools, and eddies. Less apparent, perhaps, is the interesting result that a one-dimensional shearing flow—for example, Couette flow—possesses a rotational component. As we will see below, the idea of fluid vorticity provides a way to characterize the rotational qualities of such flows. In addition, our treatment of vorticity will provide a way to distinguish between simple shear and pure shear of a fluid. Because shearing motions involve viscous dissipation of energy in real fluids, our descriptions of vorticity and shear will form an important part of the development of dynamical equations for flows that involve viscous forces (Chapter 12). The idea of vorticity also is useful in visualizing the onset of flow separation (Example Problem 11.4.2), very viscous flow behavior (Example Problem 12.6.5), and certain aspects of turbulence (Chapters 14 and 15). Beyond this, our treatment of vorticity is not emphasized. Let us envision a vorticity meter made of two small orthogonal vanes, with the end of one vane marked for easy identification. (Such meters can readily be constructed and used as described next.) Consider placing this meter at some position within a fluid that is rotating like a rigid body. The vorticity meter in this case rotates with the fluid in such a way that its orientation relative to the axis of rotation of the fluid remains fixed. As we will see below, the fluid possesses a definite vorticity that is reflected in the observation that the vorticity meter rotates with respect to its own axis. In this regard, we also may observe that the angular velocity of this local rotation of the meter is the same regardless of its distance from the fluid axis. Now consider a one-dimensional (Couette) shear flow. A vorticity meter placed at any position within this flow also rotates about its center due to the streamwise velocity differential over the span of the meter. Judging from the behavior of the meter, this flow also possesses a definite vorticity. We also may envision that the rate of rotation varies directly with the velocity gradient du/dy.

Mechanical Properties of Fluids and Porous Media

Fluids possess characteristic physical properties that govern how they behave when forces are applied to them. Of particular interest are properties that govern fluid responses to ordinary mechanical forces; thus we will not consider electrical and magnetic phenomena, and we will defer treating thermodynamical properties of fluids to the next chapter. Further, we will adopt a classic treatment of bulk behavior, as opposed to fluid behavior below the continuum scale. For example, we will examine how a Newtonian fluid deforms (in bulk) in a systematic way when it is subjected to shear stress, such that the rate of strain is proportional to the stress. In contrast, the same nominal fluid very close to a rock surface within a small pore may exhibit a much more complicated rheological behavior, from which we can infer that surface phenomena fundamentally alter the behavior of the fluid from that observed in bulk. We also will see how different types of real fluids respond differently to an applied shear stress, including how the rate of strain of certain fluids varies with the duration of stress as well as other factors. In this regard we will distinguish between time-independent and time-dependent behaviors; in the latter case the behavior of a fluid depends on its history of strain. These basic ideas of fluid behavior are a foundation for most of the material covered in the remainder of the text. Nonetheless, this initial treatment of fluid properties will provide sufficient insight to fluid behavior to begin using simple mathematical analyses to examine several important fluid-flow problems. Likewise, the description here of mechanical properties of porous media also is a foundation for later use. Coverage of this topic, however, is restricted to items that will be useful to understanding fluid behavior. Certain mechanical properties of fluids are defined in terms of the forces acting on them. In this context, it is useful to distinguish two types of forces: body forces and surface forces. A body force is one whose magnitude is proportional to the volume or mass of a fluid parcel.