Higher-order Differencing for Phase-front Propagation in Geothermal Systems

We are testing higher-order differencing total variation diminishing schemes implemented in the reservoir simulator TOUGH2 to reduce numerical dispersion of phase fronts in geothermal flow problems. The schemes are called total variation diminishing because they employ flux limiters to prevent spurious oscillations that sometimes occur with other higherorder differencing schemes near sharp fronts. Thus it appears that total variation diminishing schemes rely on an implicit assumption that the overall variability of advected quantities stays constant or diminishes with time. We use the Leonard total variation diminishing scheme in two special problems designed to test the applicability of the scheme for cases where this implicit assumption is violated. In the first problem, we investigate the isothermal propagation of a phase front in a composite porous medium where phase saturation increases as the front enters the second medium. In the second problem, we investigate the propagation of a phase front where boiling increases the saturation difference across the front as i t propagates. In the composite porous medium problem, we find that spurious phase saturations can arise if the weighting scheme is based on relative permeability; for weighting based on phase saturation, no such oscillation arises. In the boiling front propagation problem, the front position is highly sensitive to weighting scheme, and the Leonard total variation diminishing scheme is more accurate than upstream weighting because it decreases numerical dispersion in the thermal energy equation.