Fractal diffusion analyses of periodic pumping tests

We investigated the usefulness of the fractal diffusion equation, also known as generalized radial flow (GRF) equation, to characterize hydraulic properties and flow dimensions of the subsurface. Unlike other methods for deriving hydraulic properties that require selecting the flow dimension, analyses based on the GRF equation in principle constrain both, flow dimension and hydraulic properties. We utilized the GFR equation to analyze periodic pumping tests carried out in boreholes penetrating gneiss rocks in the research mine Reiche Zeche, Freiberg, Germany. These tests involved one injection borehole, where flow rate and injection pressure were recorded, and four monitoring boreholes, where pressure responses were monitored. Phase-shifts and amplitude ratios were derived through interference analysis, involving a comparison of the periodic signals of injection and monitoring pressure, as well as injectivity analysis, consisting of a comparison of the periodic flow rate and injection pressure. The pumping tests were conducted at three distinct intervals within the injection borehole, isolated by a double-packer probe and selected based on the characteristics of the fractures intersecting the borehole.  One interval contained a natural fracture zone characterized by a high fracture density with a high mean aperture. The others were previously hydraulically stimulated. While one of them had a single pre-existing fracture, the other was entirely intact before the stimulation that led to an induced fracture with feather geometry, as typical for a borehole that does not follow a principal stress axis. Several observations suggest that the gneiss volume is hydraulically heterogeneous: a) the hydraulic properties and flow dimensions vary with pumping period; b) estimated diffusivity values and flow dimensions differ for interference and injectivity analyses; c) discernible differences in diffusivity values and flow dimensions along diverse hydraulic paths, as determined by interference analysis. Furthermore, pressure dependence in hydraulic properties and flow dimensions are observed for all intervals. The hydraulic response of the fault-zone interval exhibits a greater sensitivity to variations in mean pumping pressure than the two stimulated intervals.