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.

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