Gravity Currents in Horizontal Porous Layers:
Transition from Early to Late Self-Similarity
with Hamdi Tchelepi, Brian Cantwell and Franklin Orr, Jr.
We investigate the evolution of a finite release of fluid into an
infinite, two-dimensional, horizontal, porous slab saturated with a
fluid of different density and viscosity. The vertical boundaries of
the slab are impermeable and the released fluid spreads as a gravity
current along a horizontal boundary. At early times the released
fluid fills the entire height of the layer, and the governing
equation admits a self-similar solution that is a function of the
viscosity ratio between the two fluids. This early similarity
solution describes a tilting interface with tips propagating as
x ~ t1/2. At late times the released fluid has spread
along the boundary and the height of the current is much smaller
than the thickness of the layer. The governing equation simplifies
and admits a different similarity solution that is independent of
the viscosity ratio. This late similarity solution describes a point
release of fluid in a semi-infinite porous half-space, where the tip
of the interface propagates as x ~ t1/3. The same
simplification of the governing equation occurs if the viscosity of
the released fluid is much higher than the viscosity of the ambient
fluid. We have obtained an expression for the time when the solution
transitions from the early to the late self-similar regime. The
transition time increases monotonically with increasing viscosity
ratio. The transition period during which the solution is not
self-similar also increases monotonically with increasing viscosity
ratio. Numerical computations describing the full evolution of the
governing equation show good agreement with the theoretical results.
Estimates of the spreading of injected fluids over long times are
important for geological storage of CO2, and for the migration of
pollutants in aquifers. In all cases it is important to be able to
anticipate when the spreading regime transitions from x ~ t1/2 to x ~ t1/3.
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