Exact solutions for flow through porous media with the Klinkenberg effect
Bibliographic record
Abstract
The effect of gas compressibility during flow through a porous medium modifies the governing equation for pressure, and for low pressures, this leads to the so-called Klinkenberg effect. Darcy’s law remains unchanged, and accounting for the equation of state leads to a parabolic governing equation for pressure. The increased flow rate at low pressure modifies the nonlinearity of this governing equation. We present two broad approaches for constructing exact solutions to the steady state problem. First, geometric reduction is employed to construct exact solutions in Cartesian and polar coordinates. Next, the governing equation is interpreted so that a stream function exists for the flow, and this is used to demonstrate that solutions can only be found when the flow is irrotational. A second, rather broad, class of exact solutions is thus constructed from potential flows and their generalization for variable permeability cases. The latter leads to a non-constant coefficient problem, and we provide both an algorithm illustrating how to use an existing linear numerical solver to solve the nonlinear problem and an explicit exact solution for an annular domain.
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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 | 0.000 | 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 | 0.000 |
| 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 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".