Sensitivity of the pressure distribution to the fractional order <i>α</i> in the fractional diffusion equation
Bibliographic record
Abstract
In reservoir engineering, an oil reservoir is commonly modeled using Darcy’s diffusion equation for a porous medium. In this work we propose a fractional diffusion equation to model the pressure distribution, p(x, t), of fluid in a horizontal one-dimensional homogeneous porous reservoir of finite length, L, and uniform thickness. A chief concern in this work is to examine the sensitivity of the pressure distribution, p(x, t), to different forms of pseudo-diffusivity, K, including cases when it depends upon the order of the fractional derivative (α), 0 ≤ α < 1 (e.g., K ∝ (1 – α)), which may be more realistic for some types of rock formations. In all cases the systems show a near-linear increase in the pressure difference P(x, t) = (p(x, t) – p i )/p i in the reservoir for large times, where p i = p(x, t = 0). For x/L < 0.4, the rate of increase of P with time increases with α, but there is a crossover at x/L = 0.4 and this trend reverses for x/L > 0.4. When K = 10k (k is the conventional permeability when α = 0), the solutions are almost independent of α, and when K = 0.1k the rate of increase in P depends upon α. This effect is enhanced when K = (1 – α)k; furthermore, in this case towards the closed end of the reservoir the pressure distribution remains practically undisturbed as α → 1. These results show that the pressure distribution in a porous reservoir is very sensitive to the dependence of the pseudo-diffusivity on the order of the fractional derivative, α.
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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 | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| 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".