Determining Coefficient of Quadratic Term in Forchheimer Equation

Forchheimer equation takes non-Darcy flow effect into account in the event of high flow velocity in porous media. Its application requires both permeability, which is in linear term, and Beta factor, which is in quadratic term. Permeability and Beta factor are determined by rock type, textural of rock, effective porosity, pore throat size, geometry of the pore, and connection and distribution of pores. Beta factor comes into play when the fluid flow rate is high and the flow rate deviates from Darcy’s law. Non-Darcy flow is described by Forchheimer equation. Usually the coefficient of non-Darcy flow term is hard to be determined. Existing approaches are core measurement and empirical correlations. To the best of our knowledge there is no theoretical equation available. To get an accurate estimation of flow rate or pressure drop in the reservoir, we need a method that has solid theoretical basis. The deficiency triggered our study. Starting from multiple-capillary tubes concept, we derived a rigorous relationship between pores geometry and pressure drop required for fluid flow through the pores. Through this correlation pressure drop can be calculated from known pores geometry. Since pores geometry can be often obtained from lab experiment or well logging, the new correlation also provides a unique approach to quantify the coefficient of quadratic term in Forchheimer equation. In this study we developed a governing equation through a rigorous theoretical derivation. With this equation the non-Darcy flow coefficient in Forchheimer equation can be calculated. The required input data for the new equation are readily obtained from well log interpretation. The new equation is a powerful tool in the event of no experimental measured non-Darcy flow coefficient available. It eliminates the errors or the arbitrary content in the empirical correlations.