The beta factor, or Non-Darcy flow coefficient is the critical parameter in determining the performance of propped hydraulic fractures in gas wells. In addition, non-Darcy effects enter into the analysis of laboratory-derived gas permeametry data, of reservoir core, and its extrapolation to the field.
In this paper, both the beta factor and the Klinkenberg slip constant are investigated. Flow tests under carefully controlled conditions have been completed in consolidated and unconsolidated media. Anomalies in the beta factor can be readily dealt with as far as proppant packs are concerned, but their implications for gas permeametry in consolidated core may be more serious.
The flow of gas in consolidated and unconsolidated porous media is important to the petroleum industry for many reasons. Laboratory gas flow tests are used to determine reservoir core permeability. These results are frequently extrapolated to higher 'reservoir' pressures. In propped hydraulic fracturing of gas wells, the conductivity of the proppant pack to gas flowing at high velocity will determine the economic benefits of fracturing the well.
It is generally accepted that Darcy's empirical relation (1856), which asserts a linear relationship between the pressure drop across a porous medium, and the velocity of fluid flowing through that medium is only valid over a limited range of Reynolds number. Defining Reynolds number:
where, V, are the fluid density, superficial velocity (flow rate/bulk cross sectional area), viscosity respectively, and d is some characteristic length, often taken as the mean grain diameter, then for Re~1-10 Darcy's rule fails. In fact, the pressure drop increases more rapidly than predicted by Darcy. Forchheimer (1901) modified Darcy's relation to account for these additional pressure losses seen in experiment, and Cornell and Katz (1953) reformulated the Forchheimer equation:
where P/L is the pressure drop per unit length, K is the absolute or Darcy permeability, is the density, and is the coefficient of inertial resistance, or simply the beta factor.
Now, the permeability and the beta factor arc considered to be properties of the porous medium, and to be independent of fluid type.