Some Representative Flow Fields

we consider two important, frequently encountered flow fields shear flow field and elongational flow field. They will be used throughout this chapter and in later 

---

On any arbitrary surface dA, the resultant stress can be represented as a vector r. The velocity at the surface is represented as a vector V. At any point in the flow field, the stress state is represented by a second-order tensor T. On a surface, which may represent some portion of the control surface that bounds a control volume, the stress is represented as a vector. The relationship between the stress tensor at a point T and the stress vector r on a particular surface that passes through the point is given as... 

Alternatives to batch testing include the use of diffusion cells or flowthrough columns. Diffusion cells are easier to operate, but are less representative of field conditions where some advection may occur. However, operation of columns at very low flow rates is difficult and subject to artifacts. To minimize possible wall effects associated with shrink/swell behavior of low-permeability clay materials, several researchers have utilized column devices that provide a confining pressure, such as flexible wall permeameters (e.g., Acar and Haider, 1990 Smith and Jaffe, 1994 Shackelford and Redmond, 1995 Khandelwal et al., 1998 Khandelwal and Rabideau, 2000). 

All models possess empirical features and can be of questionable reliability under some release conditions. The Gaussian models are simple and valid for releases of neutrally or positively buoyant materials in a uniform flow field with no downwind obstacles. The box models represent a macroscopic approach to heavy-gas dispersion. 

In order to determine the distributions of pressure, velocity, and temperature the principles of conservation of mass, conservation of momentum (Newton s Law) and conservation of energy (first law of Thermodynamics) are applied. These conservation principles represent empirical models of the behavior of the physical world. They do not, of course, always apply, e.g., there can be a conversion of mass into energy in some circumstances, but they are adequate for the analysis of the vast majority of engineering problems. These conservation principles lead to the so-called Continuity, Navier-Stokes and Energy equations respectively. These equations involve, beside the basic variables mentioned above, certain fluid properties, e.g., density, p viscosity, p conductivity, k and specific heat, cp. Therefore, to obtain the solution to the equations, the relations between these properties and the pressure and temperature have to be known. (Non-Newtonian fluids in which p depends on the velocity field are not considered here.) As discussed in the previous chapter, there are, however, many practical problems in which the variation of these properties across the flow field can be ignored, i.e., in which the fluid properties can be assumed to be constant in obtaining fire solution. Such solutions are termed constant... 

Numerical models have also been applied to simulate macrodispersion in spatially variable ksat fields (e.g., Thompson Gelhar, 1990 Moissis Wheeler, 1990 Wheatcraft et al., 1991). This Darcian approach requires averaging of flow over some Representative Elementary Volume (REV). 

We considered two approximate treatments of the DC field, i.e., one where we only included Z of Fig. 5 and equations (48)-(50), and another where the full sawtooth curve z was included. Some representative results are shown in Figs 7 and 8. Since the Wannier functions can be ascribed to individual unit cells, we show in Fig. 7 the number of electrons (relative to the number, 8, for the undistorted system) of each unit cell in the case that the field operator has the symmetry of z of Fig. 5. Not surprisingly, the electrons do show an asymmetric distribution, although the flow from one end of the Born von Karman zone to the other is small. The number of electrons inside the muffin-tin spheres also gives information on the electron redistributions. Thus, for e-E = 0.0002 hartree these numbers are 3.2403 and 3.2413 for the two carbon atoms per unit cell for the operator zi of Fig. 5, and 3.2217 and 3.2575 for the operator z- Here we also see a larger effect for z than for z However, for the z all atomic spheres show the same numbers, so that the charge redistribution of Fig. 6 is restricted to the interstitial region. 

There will be some uncertainty as to the well initials, since the exploration and appraisal wells may not have been completed optimally, and their locations may not be representative of the whole of the field. A range of well initials should therefore be used to generate a range of number of wells required. The individual well performance will depend upon the fluid flow near the wellbore, the type of well (vertical, deviated or horizontal), the completion type and any artificial lift techniques used. These factors will be considered in this section. 

The Mechanism of Electrical Conduction. Let us first give some description of electrical conduction in terms of this random motion that must exist in the absence of an electric field. Since in electrolytic conduction the drift of ions of either sign is quite similar to the drift of electrons in metallic conduction, we may first briefly discuss the latter, where we have to deal with only one species of moving particle. Consider, for example, a metallic bar whose cross section is 1 cm2, and along which a small steady uniform electric current is flowing, because of the presence of a weak electric field along the axis of the bar. Let the bar be vertical and in Fig. 16 let AB represent any plane perpendicular to the axis of the bar, that is to say, perpendicular to the direction of the cuirent. 

---