  We present a mathematical approach to optimize the material distribution for
fluid transport in unsaturated porous media. Our benchmark problem is a
simplified diaper model as an exemplary liquid absorber. Our model has up to
three materials with vastly different properties, which in the reference
configuration are arranged in parallel layers. Here, swelling is neglected and
the geometry of a swollen diaper is assumed and treated as a porous medium of
high porosity. The imbibition process is then simulated by solving an
unsaturated flow problem based on Richards' equation. Our aim is to maximize
the amount of absorbed liquid by redistributing the materials. To this end, a
density based multi-material topology optimization (based on the well known
SIMP model) is performed. A sensitivity analysis for the nonlinear transient
problem is provided, which enables the application of first order optimization
solvers. We perform two- and three-material optimization and discuss several
variants of the problem setting. We present designs with up to 45% more
absorbed liquid compared to the reference configuration.
