On the Buckley-Leverett Equation With Constant-Pressure Boundary Conditions
Bibliographic record
Abstract
Summary The classical Buckley-Leverett theory applies to 1D linear flow of two immiscible phases under the fundamental assumption that the flow rate (or total volumetric flux) is constant as a function of time. One phase is injected into the medium at a constant rate, thereby displacing the other phase. If the displacing phase is instead injected at constant pressure and the outlet pressure is also constant, the problem is still well-defined; however, the classical Buckley-Leverett theory does not apply. This is because the injected phase and the in-situ phase have different properties such as viscosities. If the boundary pressures are kept constant, the flow rate will, therefore, vary over time. The main result of this paper is to show that the solution of the displacement problem can be obtained from the constant-rate solution through an analytical determination of the flow rate as a function of time, given constant-pressure boundaries. The theory developed in this paper also provides an analytical solution for the location of the displacement front at any given time, the time for frontal breakthrough at the outlet end, and the pressure distribution as a function of time inside the medium. It is demonstrated through computed examples that the constant-flow-rate solution, in general, cannot be used to approximate the corresponding solution for constant-pressure boundaries because the variation in the flow rate is very significant. A standard numerical method has also been applied and compared with the analytical solutions, demonstrating that fine numerical-simulation grids are required for acceptable comparison with the analytical solution.
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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 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.003 | 0.002 |
| Insufficient payload (model declined to judge) | 0.012 | 0.002 |
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".