Multiscale tip asymptotics in hydraulic fracture with leak-off
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Bibliographic record
Abstract
This paper is concerned with an analysis of the near-tip region of a fluid-driven fracture propagating in a permeable saturated rock. The analysis is carried out by considering the stationary problem of a semi-infinite fracture moving at constant speed V . Two basic dissipative processes are taken into account: fracturing of the rock and viscous flow in the fracture, and two fluid balance mechanisms are considered – leak-off and storage of the fracturing fluid in the fracture. It is shown that the solution is characterized by a multiscale singular behaviour at the tip, and that the nature of the dominant singularity depends both on the relative importance of the dissipative processes and on the scale of reference. This solution provides a framework to understand the interaction of representative physical processes near the fracture tip, as well as to track the changing nature of the dominant tip process(es) with the tip velocity and its impact on the global fracture response. Furthermore, it gives a universal scaling of the near-tip processes on the scale of the entire fracture and sets the foundation for developing efficient numerical algorithms relying on accurate modelling of the tip region.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 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 it