Mehdi Eliassi

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

[1] In the mid-1980s the discovery of finger persistence, both for constant infiltration as well as in subsequent infiltration events [Glass et al., 1988], suggested that standard unsaturated flow theory applied to gravity-driven fingering (GDF) was incomplete. Subsequent discovery of the nonmonotonic behavior in saturation (and thus pressure) at the finger tip [Glass et al., 1989] clenched this fate for the traditional conceptualization. While a standard hysteretic mechanism can fully explain finger persistence following from the nonmonotonic behavior at the tip, the cause (i.e., the initiation) of nonmonotonicity within the finger could not be explained by standard theory. This critical point lied dormant for many years. In the meantime, many assumed that standard porous-continuum theory, in combination with hysteretic equations of state, contained all the relevant physics and was sufficient to explain GDF. Therefore when Nieber [1996] presented his results, they were embraced as ‘‘the first to model experimentally observed unstable fingered flow successfully’’ [Deinert et al., this issue, second paragraph] (hereinafter referred to as DPCSS). Nieber’s [1996] simulations were purported to be solutions of the traditional unsaturated flow governing equation, the Richards equation (RE). However, we demonstrated that such assumed physics; that is, the RE in combination with standard monotonic properties (SMP) (defined by standard constitutive relations and hysteretic equations of state [Mualem, 1976; van Genuchten, 1980]) was not sufficient to model GDF [Eliassi and Glass, 2001a]. [2] 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. In fact, within all this work a statement is made that the governing equation being solved is the RE and that van Genuchten [1980] equations are used to describe (water) saturation-capillary pressure and saturation-hydraulic conductivity relations. 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. In light of this comment we endeavor to explain, once again, our results and their significance in section 2 below. [3] More recently, we have gone beyond the RE to consider its extension to include the experimentally observed hold-back-pile-up (HBPU) effect [Eliassi and Glass, 2001b, 2002] critical for the porous-continuum modeling of GDF. By postulating the HBPU effect as physically tied to wetting front sharpness, the HBPU can be mathematically formulated in a variety of ways to include hypodiffusive, hyperbolic, and mixed spatial-temporal forms. For each an extended flux relation comprised of the Darcy-Buckingham flux plus an additional component due to the HBPU effect can be inferred. 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 This paper is not subject to U.S. copyright. Published in 2003 by the American Geophysical Union.

Multiphysics Model of Palladium Hydride Isotope Exchange Accounting for Higher Dimensionality

This report summarizes computational model development and simulations results for a series of isotope exchange dynamics experiments including long and thin isothermal beds similar to the Foltz and Melius beds and a larger non-isothermal experiment on the NENG7 test bed. The multiphysics 2D axi-symmetric model simulates the temperature and pressure dependent exchange reaction kinetics, pressure and isotope dependent stoichiometry, heat generation from the reaction, reacting gas flow through porous media, and non-uniformities in the bed permeability. The new model is now able to replicate the curved reaction front and asymmetry of the exit gas mass fractions over time. The improved understanding of the exchange process and its dependence on the non-uniform bed properties and temperatures in these larger systems is critical to the future design of such systems.