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

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 and porosity vary along the vertical as $z^{omega-1}$ and $z^{gamma-1}$, respectively, being $z$ the vertical coordinate and $omega$ and $gamma$ 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 $t^{gamma n/(2+omega(n+1))}$. Expressions for the time evolution of 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 successfully with the theoretical formulation