Ability of Darcy’s Law for Extension in Two-Phase Flow for Sedimentary Medium in Capillary Non-equilibrium Situations

We have investigated the ability of the Darcy’s law to be extended in two-phase fluid flow through the sedimentary medium for the capillary non-equilibrium situations to calculate accurately wetting and non-wetting fluid flow rates. If the viscous and capillary forces are considered, the effective permeability requires three directions to be described in the sedimentary medium for the capillary nonequilibrium situations and Darcy’s law can’t be extended to such situations. We have used the correlated anisotropic network model to simulate the sedimentary medium and the capillary non-equilibrium situations for drainage process. There are the distinct correlations for the radii of the throats, controlling fluid flow, in the parallel and perpendicular directions to the bedding in the sedimentary medium. We have used fractional and multi-fractional Brownian motions to calculate the throat radii in the sedimentary medium. We have used a fractional Brownian motion characterized by the distinct cutoff lengths and anisotropy parameters for the parallel and perpendicular directions to the bedding and a Hurst exponent to generate the sedimentary medium. In the sedimentary medium for the capillary non-equilibrium situations, relative permeability depends on the direction of the capillary force and cannot be expressed as a scalar quantity. Therefore, the effective permeability requires three directions to be described in the sedimentary porous medium for the capillary non-equilibrium situations. Darcy’s law which is a linear relationship between the flow rate and the macroscopic pressure gradient driving force cannot be extended for two-phase flow through the sedimentary porous medium for the capillary non-equilibrium situations. We have used the thermodynamics theory of non-equilibrium process and offered the new relationships to calculate accurately wetting and non-wetting fluid flow rates through the anisotropic porous media in the capillary non-equilibrium situations. The new linear relationships have two transport coefficients which are the second rank tensor in the anisotropic porous media for the capillary non-equilibrium situations.

Author(s): Mohammad Sheikhnazari

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