Is Poiseuille's Law Valid in Rough Pores of Natural Porous Media?.

Is Poiseuille's law valid in rough pores of natural porous media?

 

Behzad Ghanbarian and Allen G. Hunt

Abstract

Poiseuille’s law is a fundamental equation in modeling single- and multi-phase flow in porous media. It is valid for incompressible, viscous and Newtonian fluids movement through a tube with constant circular cross-section area where the flow is laminar. However, in natural porous media like soils pores are neither perfectly circular nor uniform. Nor are they straight and smooth. Experimental evidence of Arya et al. (1999) implies that the exponent of the Poiseuille’s equation (Q ∝ r⁴/l; where Q is the volume flow rate, r is the pore radius, and l is its length) is not necessarily 4; instead they found a range 2.664 to 4.714 for r⁴/l term using 16 soil samples with different textures. In this study, we propose a general form of Poiseuille’s law for pores with rough cross-sectional area. In our theory, the exponent is 2(3-D_s) where D_s is the pore-solid interface fractal dimension in 2D ranging from 1 to 2. Thus, we find a range of 2 to 4 for the exponent in our generalized Poiseuille’s law instead of 4 in the traditional Poiseuille law. Then, we propose a relation between average radius and length for rough cylindrical pores without converging-diverging geometry. Taking that relationship into account for anisotropic porous media, we find that the exponent of r^2(3-Ds)/l term in the Poiseuille equation changes between 1 and 6 which is consistent with the results of Arya et al. (1999) derived from hydraulic conductivity measurements and the numerical simulations of Zhang et al. (1996) who reported exponents greater than 5. Further numerical simulations are needed to evaluate the applicability of our generalized Poiseuille’s law in porous media.