A dipole solution for power-law gravity currents in porous formations

Abstract

A theoretical and experimental analysis of non-Newtonian gravity currents in porous media with variable properties is presented. A mound of a power-law fluid of flow behaviour index $n$ is released into a semi-infinite saturated porous medium above a horizontal bed, and can drain freely out of the formation at the origin. The porous medium permeability varies along the vertical as $z^{({\it\omega}-1)}$, porosity varies along the vertical as $z^{({\it\gamma}-1)}$, $z$ being the vertical coordinate and ${\it\omega}$ and ${\it\gamma}$ constant numerical coefficients. A self-similar solution describing the space–time evolution of the resulting gravity current is derived for shear-thinning fluids with $n<1$, generalizing earlier results for Newtonian fluids. The solution conserves a generalized dipole moment of the mound. The spreading of the current front is proportional to $t^{{\it\gamma}n/(2+{\it\omega}(n+1))}$. Expressions for the time evolution of the outgoing flux at the origin and of the current volume are derived in closed form. The Hele-Shaw analogue is derived for flow of a power-law fluid in a porous medium with vertically variable properties. Results from laboratory experiments conducted in two Hele-Shaw cells confirm the constancy of the dipole moment, and compare well with the theoretical formulation.