  Imbibition, the displacement of a nonwetting fluid by a wetting fluid, plays
a central role in diverse energy, environmental, and industrial processes.
While this process is typically studied in homogeneous porous media with
uniform permeabilities, in many cases, the media have multiple parallel strata
of different permeabilities. How such stratification impacts the fluid dynamics
of imbibition, as well as the fluid saturation after the wetting fluid breaks
through to the end of a given medium, is poorly understood. We address this gap
in knowledge by developing an analytical model of imbibition in a porous medium
with two parallel strata, combined with a pore network model that explicitly
describes fluid crossflow between the strata. By numerically solving these
models, we examine the fluid dynamics and fluid saturation left after
breakthrough. We find that the breakthrough saturation of nonwetting fluid is
minimized when the imposed capillary number Ca is tuned to a value Ca$^*$ that
depends on both the structure of the medium and the viscosity ratio between the
two fluids. Our results thus provide quantitative guidelines for predicting and
controlling flow in stratified porous media, with implications for water
remediation, oil/gas recovery, and applications requiring moisture management
in diverse materials.
