  In this paper, the interaction between two immiscible fluids with a finite
mobility ratio is investigated numerically within a Hele-Shaw cell. Fingering
instabilities initiated at the interface between a low viscosity fluid and a
high viscosity fluid are analysed at varying capillary numbers and mobility
ratios using a finite mobility ratio model. The present work is motivated by
the possible development of interfacial instabilities that can occur in porous
media during the process of CO$_2$ sequestration, but does not pretend to
analyse this complex problem. Instead, we present a detailed study of the
analogous problem occurring in a Hele-Shaw cell, giving indications of possible
plume patterns that can develop during the CO$_2$ injection. The numerical
scheme utilises a boundary element method in which the normal velocity at the
interface of the two fluids is directly computed through the evaluation of a
hypersingular integral. The boundary integral equation is solved using a
Neumann convergent series with cubic B-Spline boundary discretisation,
exhibiting 6th order spatial convergence. The convergent series allows the long
term non-linear dynamics of growing viscous fingers to be explored accurately
and efficiently. Simulations in low mobility ratio regimes reveal large
differences in fingering patterns compared to those predicted by previous high
mobility ratio models. Most significantly, classical finger shielding between
competing fingers is inhibited. Secondary fingers can possess significant
velocity, allowing greater interaction with primary fingers compared to high
mobility ratio flows. Eventually, this interaction can lead to base thinning
and the breaking of fingers into separate bubbles.
