Comment on “On the continuum‐scale modeling of gravity‐driven fingers in unsaturated porous media: The inadequacy of the Richards equation with standard monotonic constitutive relations and hysteretic equations of state” by Mehdi Eliassi and Robert J. Glass

Abstract

on ''On the continuum-scale modeling of gravity-driven fingers in unsaturated porous media: The inadequacy of the Richards equation with standard monotonic constitutive relations and hysteretic equations of state'' by Mehdi Eliassi and Robert

Reply to comment on “On the continuum‐scale modeling of gravity‐driven fingers in unsaturated porous media: The inadequacy of the Richards equation with standard monotonic constitutive relations and hysteretic equations of state” by Mehdi Eliassi and Robert J. Glass

“…Critical to the mimic is the use of downwind averaging of the hydraulic conductivity in the numerical solution approach. However, as we showed [ Eliassi and Glass , 2001a], the downwind averaging does more than simply “adjust the permeability” at the wetting front as was suggested by DPCSS (first paragraph). Actually, downwind averaging induces a numerical error that can be large enough to modify the underlying governing equation such that RE is no longer being solved.…”

“…Of course, with grid refinement this numerical error reduces and the GDF response vanishes. DPCSS states that we have “exaggerated” (final paragraph) our critique of Nieber [1996]. Of course, we do not believe that we have exaggerated our critique.…”

“…Our conclusion, as stated in our previous work, was [ Eliassi and Glass , 2001a, p. 2019]

Thus the RE along with standard monotonic hydraulic properties does not contain the critical physics required to model gravity‐driven fingers and must be considered inadequate for unsaturated flow in initially dry, highly nonlinear, and hysteretic media where these fingers occur.

Possible alternatives to the standard approach were listed to [ Eliassi and Glass , 2001a, p. 2032]

…include but are not limited to a dynamic capillary pressure resulting in nonstandard equations of state; modified nonmonotonic constitutive relations for the relative permeability; and/or entirely different formulations of the flux law itself.

We have gone on to consider additional physics and invite DPCSS and others to consider our recent work [ Eliassi and Glass , 2002], where we identify the experimentally observed HBPU effect and mathematically model the HBPU in a variety of forms. Support for each of these forms can be found within extended theory for single‐phase and multiphase flow and, interestingly, from very different underlying conceptualizations of the possible physics.…”

“…We are happy to see that our work has caused others to recognize the insufficiency of traditional theory. Of course, recognizing that traditional theory (i.e., the RE) cannot fully represent GDF requires the recognition that Nieber 's [1996] solution of the RE can only “mimic” GDF (DPCSS, first paragraph). We note, however, that such recognition cannot be found in six other works [ Ritsema et al , 1998a, 1998b; Nguyen et al , 1999a, 1999b; Nieber et al , 2000; Ritsema and Dekker , 2000] that faithfully used the basic numerical method suggested by Nieber [1996] after it was developed.…”

“…While parallels for each extended flux relation can also be found in the multiphase literature, it remains to be seen whether such porous‐continuum‐scale models can be applied such as to increase our basic understanding of the GDF process. DPCSS have suggested that the recent work of Deinert et al [2002] and Dautov et al [2002] shed light on extended theory in the context of modeling GDF. While such discussion may prove premature, we address extended theory and these preliminary works in section 3 below.…”