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- Initial reviews of this work have been mixed, degenerating in at least one case to verbal abuse, including a charge of fraud, based more on partisan methodologies than mathematics. But for those willing to make the effort to follow the math, significant new insights are available.
- The definition of Darcian means for vertical unsaturated flow require only three deceptively simple equations: Darcy's law, the assumption of constant flow in space between grid points for a single time step, and the unsaturated relative conductivity relation, kr(y).
- The Darcian mean for total flow, Kv, can be decomposed into the Darcian mean for gravity flow, Kx, and the Darcian mean for horizontal or diffusive or capillary flow, Kh. Kh can be calculated from the integral of kr(y)·dy, but Kx must be determined from the actual steady-state distribution of kr(x) on the microscale between grid points. When kr(y) is more complicated than an exponential relation, say a Brooks-Corey or van-Genuchten-style relation, kr(x) must usually be determined implicitly from a numerical solution to an elliptic boundary-condition problem, or its equivalent.
- Darcian means, Kx, Kv and Kh, were generated for a supposed
composite relative conductivity relation for Topopah Spring welded
tuff, over a range of 10
^{-10}to 1 in adjacent vertical grid point conductivities, {(x1,kr1) & (x2,kr2), with x1 below x2} and over a range of vertical space step size, Dx, from 0.01m to 1000m, assuming both the flow and fractures to be vertical. As Dx increases, Kx transforms from a prototypical form to the upper grid point conductivity, kr2. Kv also transforms from Kh to Kx, even getting to kr2 faster than Kx. The Darcian total relative flow, rq, also transforms with Dx, exhibiting some odd regions of flow that remain nearly constant regardless of changes in the adjacent grid point conductivities. None of the standard means chosen for demonstration, the arithmetic, geometric, upstream and an integral form, show this full range of characteristics. None of them are useful for replacing Kv to calculate rq except in limited conditions. All of the standard means predicted flows with errors, some ranging in excess of six orders of magnitude. - Darcian means of an exponential conductivity relation have easily-derived explicitly analytic solutions. Manipulation of these solutions provides two new logistic weighting functions for estimating Kv and Kx from kr1, kr2 and Kh, as well as a scaling parameter, u, that explains part of the variation of Darcian means with vertical space step size and conductivity relation parameters. These approaches prove very useful in developing approximate Darcian means for Brooks-Corey conductivity relations.
- Examining Darcian means for Brooks-Corey conductivity relations
shows that for Dx à
0, the total flow and gravity flow means, Kv and Kx, can be described
by a single equations (called here the
*manta function*) with a single change in parameter value, m. This function encompasses several standard interblock conductivity means (the arithmetic, geometric and harmonic) and shows them to be special cases, valid and Darcian only under limited conditions, such as when Dx à 0. The manta function is the prototype function for Kx and Kv mentioned above. Kx and Kv transform from it, with different values of m, to kr2 with increasing Dx. Plots, with Brooks-Corey parameters chosen to represent Topopah Spring matrix and fracture flow, demonstrate how differently matrix and fracture flow must be handled via interblock conductivity means at similar space steps. - The scaling parameter, u, derived from the exponential relation case, depends on parameters from both the medium and the model. It allows a reduction in dimension, collapsing 3-D surface plots of Darcian means on the (kr1,kr2)-plane into 2-D curves against a (kr2/kr1)-axis. This allows easier visual comparisons of Darcian and standard means, and a new way to plot the errors of approximations. Plots in this manner show that, compared to standard means, new Darcian approximations develop less error over a wider range of model conditions.
- A common approach to separate calculation of gravity and capillary (or diffusive) flows is investigated using the new Darcian visualization tools, where the capillary flow mean is Kh and the gravity flow mean is a standard mean. These cases demonstrate a consistent violation of the Darcian balance between total, gravity and capillary flows, regardless of mass balance error, that results in large positive and negative singularity-type errors in dry-over-wet conditions.
- Darcian approximations for Brooks-Corey relations are subject to this kind of error, but offer some hope of compensation by applying different approaches to different modeling regimes. It is suggested that perhaps calculating separate gravity and capillary flows is wrong, that it is better to use a total-head formulation that calculates only total flow. This is buttressed in the subsequent section with additional examples using the chosen composite relation for Topopah Spring welded tuff.
- Darcian approximations for the exponential and Brooks-Corey relations are applied piecewise to the composite relation for Topopah Spring tuff, along with a modified published approximation using the arithmetic mean for Kx, and Kh for capillary flow. Both the modified published approach and a similar piecewise Darcian Brooks-Corey approach demonstrate significant areas of backwards flow and serious effective displacement in the regimes for hydrostatic conditions. The Darcian piecewise exponential approximation and the Brooks-Corey piecewise approximation that directly estimate Kv and calculate Kx from Darcian balance work best, but still cannot adequately handle the regime where flow is dominated by near-saturation in the matrix. All of the approaches studied as substitutes for true Darcian means in running models had this difficulty. This suggests that the composite relation itself causes the problem to be poorly posed in those regimes, and that the correct approach to vertical fractures and flow may well be separate, linked models of fracture and matrix flow with simpler conductivity relations.
- Since this investigator does not know the details of the models used by Los Alamos National Lab, Lawrence Berkeley National Lab and others to describe the Yucca Mountain site, he cannot comment on the applicability of these findings to those studies. But it would seem wise for those who are cognizant to take another look. It would be unfortunate to have done all that work only to have opposing counsel in an environmental court case point out that some of the calculated flows are backwards or otherwise in error.
- If this investigator is right about the significance of these results, then this is a small but critical part of the problem. This approach will not answer all questions or solve all problems, but it cannot be neglected or ignored. Much work still remains to be done to extend these concepts to such regimes as capillary barriers, heterogeneous media, ponded heads, and multiphase flow.