A Generalized Hyperbolic Decline Equation with Rate-Time Dependent Function
Bibliographic record
Abstract
Abstract Nearly all the decline curve equations used today are based on the Arps hyperbolic equation1, given as: Using this equation, the production rate ‘Q’ at anytime ‘t’ can be calculated from the hyperbolic exponent ‘b’, the initial production rate ‘Q0’ and its corresponding decline rate ‘D0’ at time zero. Although Equation (1) is easy to use, the variation of the decline rate with time (except b = 0) limits the applicability of the equation. For a hyperbolic decline curve (b>0), if a different production rate ‘Qi’ on the curve is used an initial rate, a different corresponding decline rate ‘Di’ needs to be identified for the equation to represent the same decline curve. Moreover, if there is a rate or reference time change in the production forecast period, the identified hyperbolic equation from the production history is no longer applicable. (1) Q(t)=Q 0 × (1+b × D 0 × t) − 1 / b In this paper, a generalized hyperbolic equation is derived to overcome the above limitations. Once a set of ‘Q0’, ‘D0’ and ‘b’ is identified from the production history, the equation can be used to predict the future rate regardless of the initial rate or time change.
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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 | 0.002 | 0.006 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.003 | 0.002 |
| Research integrity | 0.003 | 0.003 |
| Insufficient payload (model declined to judge) | 0.007 | 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".