Abstract
Using the creeping Navier Stokes equation within a spherical cavity and the Darcy equation in the surrounding homogeneous and isotropic porous medium, the flow field in the entire system is evaluated. Applying this result to a representative generalizing model of a uniformly vuggy, homogeneous and isotropic porous medium, an engineering estimation of the interdependence of the matrix permeability km, the vug porosity permeability km, the vug porositytotal volume of vug space 0v = ----------------------------total volume of sample
and the system permeability ks of the vuggy porous medium is derived. This interdependence can be expressed by the formula:
Introduction
The objective of this study is the derivation of an engineering formula that shows the interdependence of matrix permeability, km, vug porosity, 0 v, and system permeability, ks, of a uniformly vuggy porous medium. In the first section, with the above porous medium. In the first section, with the above goal in mind and to satisfy more general interests, we shall study and predict the flow field within a single cavity bounded by a sphere, of radius R, and in the surrounding homogeneous and isotropic porous medium. In the second section, we shall porous medium. In the second section, we shall suggest as a generalizing model of a uniformly vuggy, homogeneous and isotropic porous medium a regular cubic array of monosized spherical cavities. Applying the formula for the pressure field near a single spherical cavity, we shall then develop the sought engineering formula. To describe the creeping flow of the incompressible liquid of viscosity, in the spherical cavity, we shall employ the creeping Navier Stokes equation,
.............................(1)
The Darcy equation,
,...........................(2)
will be used to describe the flow of this liquid in the porous medium of permeability k that fills the space outside the cavity. p designates the liquid pressure referred to datum, denotes the flow pressure referred to datum, denotes the flow vector, and * is used to indicate macroscopically averaged quantities pertaining specifically to a porous medium. porous medium. In hydrodynamics, one generally requests continuity of the pressure, of the flow vector, and of the shear tensor throughout the fundamental domain of the problem - in particular, along the boundary surfaces, which separate subdomains. When applying these principles to this problem, one would impose at the spherical boundary that separates the cavity from the porous medium:continuity of the pressure,continuity of the component of u that is orthogonal to the surface,continuity of the other component of u that is tangential to the surface,continuity of the shear component tangential to the surface.
Arguments of this nature have lead to the suggestion of a generalization of the Darcy equation, namely, the Brinkman equation,
...............(3)
However, both the necessity and the validity of this generalization have been challenged; indeed, it has been shown that a mathematically consistent solution of our problem may be obtained, using Eqs. 1 and 2 within the respective subdomains, provided one abandons the request for continuity of the shear at the wall of the cavity (compare Boundary Condition d above).**
SPEJ
P. 69
Stokes Equation in a Semi-Infinite Region: Generalization of the Lamb Solution and Applications to Marangoni Flows
