Analytical Front Tracking in Numerical Modelling of Two-phase Flow in Porous Media

uities in phase saturation are the obligatory attribute of any solution. Any numerical method should contain specific procedures capable to treat the discontinuities. We propose a specific two-scale numerical method which is based on replacing the saturation field by the field of discrete “elementary fronts”, whose movement is calculated on the basis of an algorithm similar to the dynamic invasion percolation. The pressure field is calculated within a macroscopic grid (scale l), while the movement of fronts is calculated inside each macroscopic cell, so that a step of the front motion h may be much lover than l. The equation of saturation transport becomes mono-dimensional within a cell and has the analytical solution. This solution gives the analytical relation for the front velocity. The time step for front motion is selected in such a way that the most rapid front would reach the limit of the corresponding macroscopic cell. Respectively the time step is variable and may be very small. When the elementary front reaches the inlet of the cell, the conditions of its penetration in the neighbouring cells are verified, including the connectivity of the displacing phase, the capillary counter-force, and so on. The connectivity of phase clusters is calculated on the basis of a special iterative algorithm developed by the group. The validity of such a method is proved theoretically: taking into account the very slow variation of the saturation far from the fronts, it is possible to replace the saturation field by a piece-wise constant approximation in the overall domain. Then the problem is reduced to the movement of the discontinuity surface. The advantage of the present method is its absolute physical and numerical stability, so that it can be applied to model the unstable displacement and analyse the fingering process. We illustrate the possibility of the method by simulating several examples of the unstable flow as (i) the gravity driven NAPL penetration in an aquifer (the Reyleigh-Taylor instability) and the (ii) displacement of heavy oil by gas (Saffman-Taylor instability), abd comparing them with experimental results.