6 Critical perspective and challenges

[page 2335, §1]   
[2335.1.1] Theoretical analysis and the numerical results suggest, that propagation of travelling and non-travelling saturation overshoots must be expected in the traditional setting for hysteretic relative permeability functions of jump-type. [2335.1.2] Although hysteresis in the relative permeabilities may be small the hysteresis in the flux functions can be significant and the saturation overshoot can be pronounced depending on the parameters. [2335.1.3] Note, that hysteresis in Pc is not needed to obtain propagation of the overshoot.

[2335.2.1] At first glance this situation seems to be at variance with mathematical proofs of the unconditional stability of the traditional theory [57, 58, 11, 61, 68] where it is argued that saturation overshoot is impossible with or without capillary hysteresis. [2335.2.2] Note however, that the mathematical proofs are based on the elliptic-parabolic Richards equation (19). [2335.2.3] Analytical arguments and numerical solutions seem to show that overshoot saturations (travelling or non-travelling) are possible in the more general case of eq.  (17) with hysteresis of jump type in the relative permeabilities. [2335.2.4] Thus, failure to reproduce saturation overshoot cannot be considered a valid criticism of the established traditional theory. [2335.2.5] More theoretical and analytical work is needed to clarify the situation and the crossover between these regimes.

[2335.3.1] Extensive numerical solutions of eq.  (29) reveal a strong dependence on the initial conditions. [2335.3.2] This could provide partial answers to the question how saturation overshoot can be initiated. [2335.3.3] It suggests also that the experimental observations of saturation overshoot could be largely explained as a consequence of hysteresis in the relative permeability and fractional flow functions. [2335.3.4] There are, however, numerous other experimental phenomena (related to the residual, irreducible and trapped [page 2336, §0]    phase fractions) that cannot be adequately predicted by the traditional model with hysteresis [24, 41, 46, 47, 49, 50].

[2336.1.1] The current situation with respect to saturation overshoot is still unsatisfactory in several respects and poses a number of challenges for future research. [2336.1.2] Experimentally, the assumption of travelling waves [55, 62], i.e. the assumption cim=cdr, has to the best of our knowledge, not been clearly demonstrated [54]. [2336.1.3] The uncertainties in saturation measurement and the relatively short columns make it difficult to establish that the two fronts move at the same speed. [2336.1.4] Theoretically, the existence of overshoot solutions needs more rigorous analysis, especially in higher dimensions. [2336.1.5] Also the relation with front stability and fingering poses challenges for theory and experiment.