Appendix I

Appendix I
Elements of Fluid Mechanics

The analysis of groundwater flow requires an understanding of the elements of fluid mechanics. Albertson and Simons (1964) provide a useful short review; Streeter (1962) and Vennard (1961) are standard texts. Our purpose here is simply to a standtroduce the basic fluid properties: mass density, weight density, compressibility, and viscosity; and to examine the concepts of fluid pressure and pressure head.

An examination of the principles of fluid mechanics must begin with a review of the mechanics of materials in general. Table A1.1 provides a list of the basic mechanical properties of matter, together with their dimensions and units in the SI metric system. The SI system has as its basic dimensions: mass, length, and time; with basic SI units: kilogram (kg), meter (m), and second (s). All other properties are measured in units that are derived from this basic set. Some of these properties are so widely encountered, and their basic dimensions so complex, that special SI names have been coined for their derived units. As noted on Table A1.1, force and weight are measured in newtons (N), pressure and stress in N/m2 or pascals (Pa), and work and energy in joules (J).

Table A1.1 Definitions, Dimensions, and SI Units for Basic Mechanical Properties

 Dimension of unit Property Symbol Definition SI unit SI symbol Derived Basic Mass M kilogram kg kg Length l meter m m Time t second s s Area A A = l2 m2 Volume V V = l3 m3 Velocity v v = l/t m/s Acceleration a a = l/t2 m/s2 Force F F = Ma newton N N kg•m/s2 Weight w W = Mg newton N N kg•m/s2 Pressure p P = F/A pascal Pa N/m2 kg/m•s2 Work W W = Fl joule J N•m kg•m2/s2 Energy Work done joule J N•m kg•m2/s2 Mass density = M/V kg/m3 Weight density = w/V N/m3 kg/m2•s2 Stress , Internal response to external p pascal Pa N/m2 kg/m•s2 Strain = V/V Dimensionless Young’s modulus E Hooke’s law N/m2 kg/m•s2

Table A1.2 provides an SI analysis of certain fluid properties and groundwater terms that occur in this text. Each is more fully described in Chapter 2.

Much of the technology associated with groundwater resource development in North America is still married to the FPS (foot-pound-second) system of units.

Table A1.2 Definitions, Dimensions, and SI Units for fluid Properties and Groundwater Terms

 Dimensions of unit Property Symbol Definition SI unit SI symbol Derived Basic Volume V V = l3 liter (= m3 × 10-3) m3 Discharge Q Q = l3/t /s m3/s Fluid pressure p p = F/A pascal Pa N/m2 kg/m•s2 Head h m Mass density = M/V kg/m­3 Dynamic viscosity Newton’s law centipoise (= N•s/m2 × 10-6) cP cP, N•s/m2 kg/m•s Kinematic viscosity v v = centistoke (=m2/s × 10-6) cSt cSt m2/s Compressibility , = 1/E m2/N Hydraulic conductivity K Darcy’s law cm/s Permeability k k = K/pg cm2 m2 Porosity n Dimensionless Specific storage SS = pg( + n) 1/m Storativity S S = SSb* Dimensionless Transmissivity T T = Kb* m2/s
*b, thickness of confined aquifer (see Section 2.10).

Table A1.3 Conversion Factors FPS (foot-pound-second)
System of Units to SI Units

 Multiply By To obtain Length ft 3.048 × 10–1 m ft 3.048 × 10 cm ft 3.048 × 10–4 km mile 1.069 × 103 m mile 1.069 km Area ft2 9.290 × 10–2 m2 mi2 2.590 km2 acre 4.407 × 103 m2 acre 4.407 × 10–3 km2 Volume ft3 2.832 × 10–2 m3 U.S. gal 3.785 × 10–3 m3 U.K. gal 4.546 × 10–3 m3 ft3 2.832 × 10 U.S. gal 3.785 U.K. gal 4.546 Velocity ft/s 3.048 × 10–1 m/s ft/s 3.048 × 10 cm/s mi/h 4.470 × 10–1 m/s mi/h 1.609 km/h Acceleration ft/s2 3.048 × 10–1 m/s2 Mass lbm* 4.536 × 10–1 kg slug* 1.459 × 10 kg ton 1.016 × 103 kg Force and weight lbf* 4.448 N poundal 1.383 × 10–1 N Pressure and stress psi 6.895 × 103 Pa or N/m2 lbf/ft2 4.788 × 10–1 Pa poundal/ft2 1.488 Pa atm 1.013 × 105 Pa in Hg 3.386 × 103 Pa mb 1.000 × 102 Pa Work and energy ft-lbf 1.356 J ft-poundal 4.214 × 10–2 J Btu 1.055 × 10–3 J calorie 4.187 J Mass density lb/ft3 1.602 × 10 kg/m3 slug/ft3 5.154 × 102 kg/m3 Weight density lbf/ft3 1.571 × 102 N/m3 Discharge ft3/s 2.832 × 10–2 m3/s ft3/s 2.832 × 10 /s U.S. gal/min 6.309 × 10–5 m3/s U.K. gal/min 7.576 × 10–5 m3/s U.S. gal/min 6.309 × 10–2 /s U.K. gal/min 7.576 × 10–2 /s Hydraulic conductivity (see also Table 2.3) ft/s 3.048 × 10–1 m/s U.S. gal/day/ft2 4.720 × 10–7 m/s Transmissivity ft2/s 9.290 × 10–2 m2/s U.S. gal/day/ft 1.438 × 10–7 m2/s
*A body whose mass is 1 lb mass (lbm) has a weight of 1 lb force (lbf). 1 lbf is the force required to accelerate a body of 1 lbm to an acceleration of  = 32.2 ft/s2. A slug is the unit of mass which, when acted upon by a force of 1 lbf, acquires an acceleration of 1 ft/s2.

Table A1.3 provides a set of conversion factors for converting FPS units to SI units.

The mass density (or simply, density)  of a fluid is defined as its mass per unit volume (Table A1.1). The weight density (or specific weight, or unit weight)  of a fluid is defined as its weight per unit volume. The two parameters are related by

(A1.1)

For water,  = 1.0 g/cm3 = 1000 kg/m3;  = 9.8 × 103 N/m3. In the FPS system,  = 62.4 lbf/ft3.

The specific gravity of any material is the ratio of its density (or specific weight) to that of water. For water,  = 1.0; for most soils and rocks, G 2.65.

The viscosity of a fluid is the property that allows fluids to resist relative motion and shear deformation during flow. The more viscous the fluid, the greater the shear stress at any given velocity gradient. According to Newton’s law of viscosity,

(A1.2)

where  is the shear stress, dv/dy the velocity gradient, and  the viscosity, or dynamic viscosity. The kinematic viscosity v is given by

(A1.3)

where  is the fluid density.

The compressibility of a fluid reflects its stress-strain properties. Stress is the internal response of a material to an external pressure. For fluids, stress is imparted through the fluid pressure. Strain is a measure of the linear or volumetric deformation of a stressed material. For fluids, it takes the form of reduced volume (and increased density) under increasing fluid pressures. The compressibility of water  is fully discussed in Section 2.9. It is defined by Eq. (2.44).

The density, viscosity, and compressibility of water are functions of temperature and pressure (Dorsey, 1940; Weast, 1972). In general, their variation is not great, and for the range of pressures and temperatures that occur in most ground-water applications, it is common to consider them as constants. At 15.5°C,  = 1.0 g/cm3,  = 1.124 cP, and  = 4.4 × 10–2 m2/N.

The fluid pressure p at any point in a standing body of water is the force per unit area which acts at that point. Under hydrostatic conditions, the fluid pressure at a point reflects the weight of the column of water overlying a unit cross-sectional area around the point. It is possible to express pressure relative to absolute zero pressure, but more commonly it is expressed relative to atmospheric pressure. In the latter case it is called gage pressure, as this is the pressure reading that is obtained on gages zeroed to the atmosphere.

The pressure head  at a point in a fluid is the height that a column of water would attain in a manometer placed at that point. In a standing body of water,  is equal to the depth of the point of measurement below the surface. If p is expressed as a gage pressure,  is defined by the relationship

(A1.4)

In effect, the pressure head  is a measurement of the fluid pressure p.

Fluid pressures are also developed in groundwater as it flows through porous geological formations and soils. In Section 2.2, the elements of fluid mechanics presented in this appendix are applied in the development of groundwater flow theory.