Simulation of propagating fronts in geothermal reservoirs with the implicit Leonard total variation diminishing scheme

Geothermal reservoir engineering requires accurate numerical solution of the advective± di€usive transport equations for strong advective ¯ows of multiphase nonisothermal ¯uids. Conventional interface weighting schemes such as upstream weighting cause numerical dispersion. Numerical dispersion can be reduced by grid re®nement, but this increases execution times and computer memory requirements. As an alternative, higher-order di€erencing schemes can be used to reduce numerical dispersion, but they often lead to spurious oscillations. These limitations have led to the development of higher-order schemes called total variation diminishing (TVD) schemes. For geothermal reservoir engineering, these schemes must be capable of handling ¯ows that may not be physically total variation diminishing. We have implemented TVD schemes into the implicit geothermal reservoir simulator TOUGH2. We verify the Leonard TVD (LTVD) scheme by comparison to an analytical solution for two-dimensional ¯ow and transport. The LTVD scheme reduces numerical dispersion for tracer transport in a two-phase geothermal reinjection problem. One-dimensional simulations show that the LTVD scheme works well even if the saturation variation increases with time. Because the location of the phase front is strongly coupled to temperature, phase front propagation is sensitive to grid resolution insofar as it a€ects the temperature ®eld. Phase front propagation in a composite porous medium Buckley± Leverett ¯ow problem, where phase saturations increase upon encountering a second medium, are slightly more accurate for the LTVD scheme as compared to upstream weighting. We ®nd that the LTVD scheme only performs well if the weighting and limiter are applied to saturation rather than to relative permeability. While there is some increasedcomputational cost with the LTVD scheme due to increased linear equation solution time and smaller time-step size, the LTVD scheme is a practical and robust method for reducing numerical dispersion in complex ¯ow problems relevant to geothermal reservoir engineering. # 2000 CNR. Published by Elsevier Science Ltd. All rights reserved.