## Darcian Means

Darcian means provide a new mathematical and modeling framework in which ground water modelers may judge the appropriateness of their approximations for intergrid hydraulic conductivity means for vertical and horizontal unsaturated flow. They use and require knowledge of both the porous medium pressure-conductivity relation and the model vertical space step size. The latest work demonstrates that many of the simple intergrid conductivity means produce numerous violations of a basic mathematical principle, the min-max principle for elliptic boundary value problems (steady-state flow in the vertical). Drafts and preprints of relevant papers are available in the links below.

 Figure 3a: Arithmetic Mean Model Initial pulse of y = -1 m between 0.35xl and 0.45xl, relaxing and translating with time, for np = 40 at t = 0, 10, 0.512(106), 1.024(106), 2.048(106) and 4.096(106) s, in the example fracture Figure 3b: Darcian Mean Model for np = 40 running in parallel with the same time steps in the same fracture and the same output times. Figures 3a and b from Dispersive Errors Induced by a Non-Darcian Mean in a Model of Unsaturated Flow, showing how the arithmetic mean produces extremely non-physical spikes and oscillations in a transient model of a vertically-moving wet pulse, when the arithmetic mean violates the min-max principle for similar conditions of steady-state flow

 This is a model of vertical infiltration into a typical fracture at Yucca Mountain using the arithmetic mean. With n segments in the wetting front, n = 8, 10, 12, 16, 20, 24 28 and 38, converging to the fine-grid solution with n = 470 on the left. The grid is adapted to begin each time step with a constant ratio of hydraulic between adjacent grid points. This is the same model with an approximate Darcian integral mean, that accounts for both the fracture pressure-conductivity relation and the model vertical space step size, with n = 8, 10, 12 and 470. Any questions? Read on.

• 03/11/2000: Two fitting functions for pressure-conductivity relations with links to Darcian means (571kb); An exponential functional form fits many pressure-conductivity relations for coarse soils and has a related analytic equation that can be iterated to produce a Darcian mean for vertical unsaturated flow. A power functional form works better for fine soils and has simplifying properties that may make it a useful for producing Darcian means. Simulated annealing is used to fit five different combinations of these two functions to 58 soils from the U.S. Salinity Laboratory UNSODA database. The best fits run from 0.0058 to 0.416 (for a very noisy data set) in mean absolute relative error. Supporting software is under development. (03/19/2000-Note some math errors found on 03/17/2000 have been corrected; this link leads to the corrected version) Removed from site Sept 2004
• Dr. Glenn Brown has a great new site on Darcy's original work.
• 01/06/2000: Some note on the summing properties of Darcian means; If Darcian means are to be applied in general, especially to tabulated, measured data sets, then it will be convenient to have a summing property that allows a set of basis functions and associated Darcian means to create composite functions for any data set. This paper shows that such a summing property exists for horizontal flow, but is in error under the influence of gravity. A numerical experiment demonstrates the size and shape of that error for a composite pressure-conductivity relation reminiscent of flow through fractured rock. Removed from site Sept 2004
• Correction, 12/18/1999: Equations in error. I believe that equations [30] and [65] in the DOE Report DOE/ER/82329-2 are in error. Equation [6] in "A Darcian integral approximation..." is correct.
• draft of 11/24/1999: Min-Max Violations Produced by Intergrid Conductivity Means; It is surprisingly common for historical, empirical intergrid hydraulic conductivity means, such as the arithmetic, harmonic and geometric to produce violations of the min-max principle for elliptic boundary value (steady-state flow) problems, even in the simplest three-point vertical grid test. Even the means that apparently do not produce min-max violations, such as the upstream and up-gravity, are generally inaccurate in estimating intergrid flow. This paper checks the validity and accuracy of thirteen means, both historical non-Darcian and Darcian approximations, against three sets of soil conductivity parameters associated with the Yucca Mountain nuclear waste site, over vertical space step sizes from 1 cm to 10 m, and relative conductivities from 0.996 down to about 1e-8 or 1e-10. The printed version (not available here) comes to 70 pages and about 200 2-D and 3-D graphs. Removed from site Sept 2004
• draft of 9/4/1999: Dispersive Errors Induced by a Non-Darcian Mean in a Model of Unsaturated Flow; Compared to a Darcian mean, the arithmetic interblock hydraulic conductivity mean can overestimate conductivity for dry-over-wet conditions and underestimate conductivity for wet-over-dry conditions, giving rise to non-physical separation and clumping of the mass flow. A simple three-point test is developed to show that the arithmetic mean can produce astable behavior in a model of a long, vertical fracture. A numerical experiment shows that in a vertical unsaturated flow model this produces dispersive oscillations in the leading edge of a wet pulse and a dry spike of excessively increased matric suction at the trailing edge. The dry spike tends to increase as the log of time in the model, out to approx. 2.66 years of model time, and can be delayed in onset by transient conditions. This does not bode well for such models attempting to predict flow on the scale of thousands of years. By contrast, a parallel model using an analytic Darcian mean demonstrates smooth, well-behaved response at all the model times and space step sizes.
• draft of 7/17/1999: Darcian Means Between Saturated and Unsaturated Modeling Points; Darcian means have previously been defined only for models of homogeneous media where adjacent grid points are unsaturated. This paper extends the concept to situations where one of the points (in the vertical) is saturated or has a positive pressure head. In such a case, the assumptions used predict that part of the porous medium between the grid points will be fully saturated. This can be accommodated by using a simple Newton-Raphson iteration. Removed from site Sept 2004
• draft of 4/30/1999: Some Methods to Examine the Validity of Darcian Means; Many current models of vertical unsaturated flow, some of which are used in models of transport of nuclear and hazardous waste, may be deficient or suspect on a very basic level. This paper demonstrates, for typical fracture flow in a Yucca Mountain, Nevada, medium, that the arithmetic mean is demonstrably nonphysical, even at small grid spacings, both by mathematical and model development. Where the arithmetic mean over predicts the advance of a wetting front by up to 18.75%, a new approximation to Darcian means is shown to hold the advance to within 0.36% of the fine grid solution over a range of 8 to 470 adaptive grid segments in the wetting front. Thus criticisms that Darcian means are nonphysical and invalid are refuted. Removed from site Sept 2004
• draft of 06/30/99: A Darcian integral approximation to interblock hydraulic conductivity means in vertical infiltration; This paper introduces a generally more accurate approximation to Darcian means (than Some analytic ... means, below), using as examples the matrix and fracture flow conductivity relations for Topopah Spring welded volcanic tuff at the Yucca Mountain Project in Nevada. It compares the new approximation to the geometric and arithmetic means for matrix and fracture flow, respectively. It offers to other investigators a reasonable tool for examining the properties of Darcian means in running models. Includes data sets of Darcian means for matrix and fracture flow. (This draft supersedes the draft of 12/22/98, which had transcription errors in equation [6].) Full version accepted for publication in Computers and Geosciences, 1999.
• draft: Some analytic and approximate Darcian means; If a "perfect" mass-conservative method reduces a 2.5% error in total mass infiltrated, how much better is the method that reduces the 50% error that the "perfect" method missed? This paper demonstrates that even in mass-conservative infiltration models with mass balance good to one part in 108, non-Darcian flow error is still alive and well. Darcian means explain why the arithmetic mean "loses" up to 50% of the infiltrated mass with very large vertical model space steps, and why physical measurements of unsaturated parameters at such places as the Yucca Mountain Project may be more critical than has been thought. Final version in GROUND WATER 37(4):532-538.
• draft: A piecewise Brooks-Corey approximation to Darcian interblock conductivity means for vertical unsaturated flow; a very long 260kb page, the precursor of Some ... Darcian means and Some ... new time steps ... with much more information, including an explanation of conductivity state space visualization concepts for Darcian means. (click here for pdf version of model code only) Removed from site Sept 2004
• Excerpts: Report DOE/ER/82329-2, Developing Darcian means in application to Topopah Spring welded tuff; A detailed tutorial on the concept of Darcian interblock hydraulic conductivity means for vertical unsaturated flow, with applications to media at the Yucca Mountain site. Demonstrates just how differently fracture and matrix flow must be handled, and that composite functions are not very useful.

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