Macroscopic dispersion is the mixing, on the scale of several hundreds of grain diameters, at a point in a permeable medium that is free of boundary effects. Megascopic dispersion is the one-dimensional (lD) dispersion derived by averaging across an entire cross section. This work investigates how both dispersions vary with heterogeneity, aspect ratio, diffusion coefficient, and autocorrelation. The theoretical results are compared to existing field and laboratory data and to existing theories for limiting cases.The degree of autocorrelation in the medium determines whether or not megascopic dispersivity (dispersion coefficient divided by velocity) is uniquely defined. Large correlation distances (with respect to the medium dimensions) imply a dispersivity that grows with distance traveled. Small correlation distances imply a dispersivity that is eventually stabilized at some constant value. This value is related to the heterogeneity of the medium. On the field scale, diffusion is insignificant, but on a laboratory scale, it can stabilize the dispersivity even if the medium is correlated. Macroscopic dispersivity is sensitive to diffusion in both the laboratory and field scale. It is smaller than or equal to megascopic dispersivity, also in conformance with experimental data, and comparable to laboratory-measured dispersivity.

Introduction During the last decade the interest in two-phase flow through horizontal and inclined pipes has increased considerably. Design engineers are primarily concerned with the calculation of pressure drop in two-phase lines and prediction of liquid holdup (in-situ liquid volume fraction), in prediction of liquid holdup (in-situ liquid volume fraction), in order to size flow lines and calculate slug catcher size. These calculations are also important for the design of surface facilities such as pumps, compressors and separators. Due to the complex nature of the different now patterns which exist in two-phase flow, very few theoretical patterns which exist in two-phase flow, very few theoretical mechanistic models have been developed. However, a large number of empirical correlations are available in the literature to calculate pressure drop and liquid holdup. Of this large number, only a few pressure drop correlations compute holdup, friction and acceleration gradients based on the flow patterns within the pipe. Once the correct flow pattern is determined at a specified flow rate, pressure and temperature, the appropriate holdup and pressure and temperature, the appropriate holdup and friction correlations are evaluated for the corresponding flow regime. The method for calculating the acceleration pressure gradient also depends on flow regime in some pressure gradient also depends on flow regime in some cases. LIQUID HOLDUP AND PRESSURE DROP CORRELATIONS Equations describing the pressure drop of a two-phase system are generally developed based on a single flow equation representing homogeneous mixture properties. The general pressure gradient equation is given as: (1) where: And HL is the result of empirical correlation. (2) The definition of the density term used in the friction (pf) and acceleration (P a) gradients varies with different investigators. From the form of Equation (1), it can be easily seen that the total pressure gradient is made up of three individual gradients (1) gravitational, (2) friction and (3) acceleration. Thus Equation (1) can be written as: (3) Many correlations have been developed for predicting liquid holdup and pressure-drop. Different investigators used different assumptions to develop these correlations. Some assumed that the gas and liquid phases travel at the same velocity (no slippage between phases), and evaluated only friction factor empirically. Others developed methods for calculating both liquid holdup and friction factor. The third group of investigators divided the flow conditions into different flow patterns and developed separate correlations for each flow pattern. All the correlations tested in this study belong to the third group. These correlations first predict the flow pattern in the pipeline at the specified physical conditions pattern in the pipeline at the specified physical conditions with the help of a flow pattern map. Some of them use the same map for all pipe inclinations, while the others have different maps for different pipe inclinations. After determining the flow pattern, the corresponding holdup and friction factor correlations are used to calculate pressure drop. Three correlations were evaluated in this study. These are briefly described in the following sections. BEGGS AND BRILL (BB) CORRELATION The Beggs and grill correlation is based on extensive air-water data in 1 and 1.5 in pipes. The correlation uses a base of 58 data points for horizontal flow, along with a large number of points for inclined flow. A flow regime is first determined from a log-log plot of input liquid content and the mixture Froude number. Liquid holdup is then calculated from correlations for the flow regime existing in pipes only.

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