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, where Q is the volume flow rate) is not necessarily 4; instead they found a range 2.664 to 4.714 for 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 which is reasonably consistent with the results of Arya et al. (1999) derived from hydraulic conductivity measurements. However, their values greater than 4 (e.g., 4.072, 4.095, 4.139, 4.344, 4.471, 4.714) for 6 out of 16 samples requires pore-solid fractal dimension, D_s, less than 1 which is not supported in our theory. This inconsistency with our prediction might be due to assumptions invoked in the extraction of the power by by Arya et al. (1999). Further numerical simulations are needed to evaluate the applicability of our generalized Poiseuille’s law in porous media.