Modelling viscous fingering during reinjection in geothermal reservoirs

Viscous fingering is a hydrodynamic instability that can occur when two fluids of differing viscosity come into contact, and leads to complex intrusions (or fingers) of one fluid into the other. We are interested here in this instability within the context of reinjection into a geothermal reservoir, where the viscosity of the injected fluid differs from that in the reservoir due to its higher temperature. We first perform a classical linear stability analysis, which considers the early time growth of temperature perturbations. Here the governing equations can be simplified through linearisation, and this linear stability analysis shows which wave number, and hence fingering displacement shape, will initially be most unstable. Results illustrate that there are two key parameters that define the stability: the Peclet number (which is proportional to injection rate), and the log-mobility ratio (which is related to injection enthalpy). These critical parameters provide limits on injection temperature and flow rate that will be stable against the fingering instability for a reservoir of given temperature, and we see that the values of these suggest that viscous fingering will always likely occur in practice. We also use the TOUGH2 numerical simulator (Pruess et al., 1999) to solve the full (i.e. non-linearised) governing equations, and confirm that viscous fingering is indeed occurring at such low injection rates. Moreover, it enables us to examine cases where reservoir permeability is heterogeneous. Under these circumstances, for heterogeneity indices beyond some critical value, fingering displacements are dominated by the permeability structure (the precise value of which should factor in inherent numerical dissipation present in the TOUGH2 simulator).