  In this article, we present new random walk methods to solve flow and
transport problems in unsaturated/saturated porous media, including coupled
flow and transport processes in soils, heterogeneous systems modeled through
random hydraulic conductivity and recharge fields, processes at the field and
regional scales. The numerical schemes are based on global random walk
algorithms (GRW) which approximate the solution by moving large numbers of
computational particles on regular lattices according to specific random walk
rules. To cope with the nonlinearity and the degeneracy of the Richards
equation and of the coupled system, we implemented the GRW algorithms by
employing linearization techniques similar to the $L$-scheme developed in
finite element/volume approaches. The resulting GRW $L$-schemes converge with
the number of iterations and provide numerical solutions that are first-order
accurate in time and second-order in space. A remarkable property of the flow
and transport GRW solutions is that they are practically free of numerical
diffusion. The GRW solutions are validated by comparisons with mixed finite
element and finite volume solutions in one- and two-dimensional benchmark
problems. They include Richards' equation fully coupled with the
advection-diffusion-reaction equation and capture the transition from
unsaturated to saturated flow regimes. For completeness, we also consider
decoupled flow and transport model problems for saturated aquifers.
