We study the behavior of time periodic three-dimensional incompressible flows modelled by three-dimensional volume preserving maps in the presence of a leakage. The distribution of residence times, and the chaotic saddle together with its stable and unstable invariant manifolds are described and characterized. They shed light on typical filamentation of chaotic flows whose local stable and unstable manifolds are always of different character (plane or line). A possible application to magma flows is discussed.