Multilayer Asymptotic Solution for Wetting Fronts in Porous Media with Exponential Moisture Diffusivity
Bibliographic record
Abstract
We study the asymptotic behavior of sharp front solutions arising from the nonlinear diffusion equation , where the diffusivity is an exponential function . This problem arises, for example, in the study of unsaturated flow in porous media where θ represents the liquid saturation. For physical parameters corresponding to actual porous media, the diffusivity at the residual saturation is so that the diffusion problem is nearly degenerate. Such problems are characterized by wetting fronts that sharply delineate regions of saturated and unsaturated flow, and that propagate with a well‐defined speed. Using matched asymptotic expansions in the limit of large β, we derive an analytical description of the solution that is uniformly valid throughout the wetting front. This is in contrast with most other related analyses that instead truncate the solution at some specific wetting front location, which is then calculated as part of the solution, and beyond that location, the solution is undefined. Our asymptotic analysis demonstrates that the solution has a four‐layer structure, and by matching through the adjacent layers, we obtain an estimate of the wetting front location in terms of the material parameters describing the porous medium. Using numerical simulations of the original nonlinear diffusion equation, we demonstrate that the first few terms in our series solution provide approximations of physical quantities such as wetting front location and speed of propagation that are more accurate (over a wide range of admissible β values) than other asymptotic approximations reported in the literature.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.002 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".