We report non-circular, stable liquid propagation patterns in a stable displacement process in a confined thin patterned porous layer. For constant fluid injection rates, the average front location of the interface r(t) exhibits a power-law behavior r ∝ t(1/2); however, when surface tension effects become important, the interface displays non-circular shapes, e.g. square, rectangular, or octagonal, and maintains the same shape during most of the injection process. The interface shape is controlled by the value of a dimensionless group representing the strength of surface tension stresses relative to stresses accompanying injection. Furthermore, we show that the propagation patterns of the interface can be controlled by the relative orientation of the different porous layers.