Viscous Flow past a Circular/Spherical Void in Porous Media - An Application to Measurement of the Velocity of Groundwater by the Single Boring Method

Abstract

The effect of the presence of a circular void or a spherical void of characteristic size R 0 in a porous media with permeability k on an otherwise undisturbed uniform flow U ∞ is investigated analytically. The entire flow field is determined by matching the solution of the Stokes equation inside the hole with that of the generalized Darcy's (Brinkman) equations in the porous region. The effect of the hole increases with the value of \(\zeta_{0} = R_{0}/\sqrt{k}\). When ζ 0 is much greater than unity, which corresponds to the case for typical groundwater flow, the velocity at the centre of the circular cylindrical hole U 0 becomes 3 U ∞ , while the volume flux Q 0 flowing into the hole from upstream-side becomes twice, in comparison with Q ∞ that flows into the same region in the absence of the hole. On the other hand, the velocity at the centre of the spherical void U 0 becomes 6 U ∞ , while Q 0 becomes 3 Q ∞ .