Volumetric material transports across distinct regions in the Chaos Screw (CS) system were described in terms of the volume-preserving lobe dynamics. Kinematic properties of a spatially periodic Poincare map were studied first with the volume- and orientation-preserving two-dimensional map, in order to provide mathematical frame works for analyses of manifold structures. The perturbed hyperbolic fixed point and the associated stable and unstable manifolds were obtained numerically. These manifolds form homoclinic tangles, and they divide the cross-sectional area into three distinct regions: left, right, and outer. Six volumetric flow rates between the three regions were described in terms of the associated lobe dynamics. As the perturbation strength increases, representative flow rates between these regions were found to increase linearly as long as the fraction of no-barrier zone is small.