Cross-hole fluid injection tests were performed on shallow (0.5 km depth) crustal fault zones to characterize their internal flow structures on scales of 3-30 m. The data were acquired at the Grimsel Rock Laboratory, Switzerland, after the zonal isolation of damage zones in boreholes equipped with multipacker systems. We show that 80% of the pressure responses detected evolved as a power function of time due to fracture-controlled flow and were best described by a fractional model with a flow dimension (n) in the range of [1.3,1.5]. The scaling of characteristic times (t c ) and intermediate transitions in the flow dimension is shown to be inconsistent with the trend expected for normal diffusion, indicating a diffusion slowdown process caused by the spatial integration of structurally different damage zones. An analysis of the entire data set points to a subdiffusive regime where the mean squared displacement of pressure fronts, ⟨r 2 (t)⟩, scales as ⟨r 2 (t)⟩ ∝ t 0.59 , similar to an anomalous diffusion process on synthetic fractal systems. We characterize this diffusion slowdown by calculating a network connectivity exponent ( ) of 1.4, from which we estimate a fractal dimension of d f = 2.23 for the tested fault zones. Such a value close to 2.0 supports the view of fault damage zones as hierarchical systems promoting flow channeling. Our results provide further support to theoretical models of fractional flow and their application to fractured media in nature.