Beyond the cubic law: A finite volume method for convective and transient fracture flow
Bibliographic record
Abstract
Abstract The reduced dimension fracture flow model (referred to as the GG22 model) is a recently derived extension of the cubic law model for flow through fractures of variable aperture with fluid inertia effects. Novel numerical methods are required to solve the nonlinear partial differential equations governing the GG22 model, as it is more complex than the cubic law. The GG22 model is derived from Navier–Stokes, which allows the adoption of similar numerical methods to the Navier–Stokes equations, but the model contains its own idiosyncrasies which must be addressed. This article presents the first numerical methods to solve the GG22 model. An explicit multi‐step finite volume method is developed and verified. The method is based on deriving a Poisson equation for pressure with an additional continuity correction to overcome numerical instabilities. The critical timestep is derived and shown to be a function of the fundamental frequency of the fracture‐fluid system and the maximum fluid velocity. The results show excellent agreement with analytical solutions, and the method demonstrates a first‐order rate of fluid flux convergence in time and a second‐order rate of pressure convergence in space. The model is applied to a traveling aperture wave which shows that higher pressures are required to generate lower average fluxes than predicted by the cubic law.
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How this classification was reachedexpand
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 | 0.002 | 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 | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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 teacher head, 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".