Spreading of axisymmetric non-Newtonian power-law gravity currents in porous media

A relatively heavy, non-Newtonian power-law fluid of flow behavior index n is released from a point source into a saturated porous medium above an horizontal bed; the intruding volume increases with time as tα. Spreading of the resulting axisymmetric gravity current is governed by a non-linear equation amenable to a similarity solution, yielding an asymptotic rate of spreading proportional to t(α+n)/(3+n). The current shape factor is derived in closed-form for an instantaneous release (α = 0), and numerically for time-dependent injection (α ≠ 0). For the general case α ≠ 0, the differential problem shows a singularity near the tip of the current and in the origin; the shape factor has an asymptote in the origin for n ⩾ 1 and α ≠ 0. Different kinds of analytical approximations to the general problem are developed near the origin and for the entire domain (a Frobenius series and one based on a recursive integration procedure). The behavior of the solutions is discussed for different values of n and α. The shape of the current is mostly sensitive to α and moderately to n; the case α = 3 acts as a transition between decelerating and accelerating currents.

► We model spreading of non-Newtonian power-law gravity currents in porous media. ► A self-similar solution is derived for radial flow and different injection rates. ► Profiles are obtained analytically/numerically with Newtonian results as special case. ► Three different analytical approximations are derived for continuous injection rate. ► Results are functions of injection rate and flow behavior index.