The Sunset Solution: Closure on Leaking Proppant
Bibliographic record
Abstract
ABSTRACT: Recent analytic work (Peirce and Detournay, 2022a) has established the tip asymptotics for receding hydraulic fractures that close due to leak-off to the permeable rock. This asymptotic result enabled the development of rigorous numerical schemes to explore the recession dynamics of hydraulic fractures that develop in a state of plane strain (Peirce and Detournay, 2022b) and radially symmetrically (Peirce, 2022). These detailed studies enabled the identification (Peirce and Detournay, 2022c) of the so-called Sunset Solution, which is a similarity solution that emerges close to the ultimate collapse of the fracture. The asymptotic analysis performed in (Peirce and Detournay, 2022c) establishes that the existence of the sunset solution is due to a fundamental decoupling of the kinematics from the dynamics in the governing equations, which leads to a robust way to measure the Carter leak-off coefficient from the rate of change in the fracture aperture at the wellbore. These results were established assuming that no fluid is leaking from the closed section of the fracture. In this paper, we explore the viability of the procedure to estimate the Carter leak-off coefficient using the sunset solution when the fracture is assumed to close on proppant that is filled with fluid upon closure and which continues to leak off and exchange fluid with the open portions of the fracture. 1 INTRODUCTION The diagnostic fracture injection test (DFIT) is used in petroleum engineering to assess the minimum in situ stress, Carter's leak-off coefficient, and other reservoir parameters. The test consists of monitoring the well pressure during injection of fluid over a time long enough to create a hydraulic fracture, as well as after shut-in of the well. Interpretation of the pressure decline that follows the end of pumping was pioneered by Nolte (1979, 1986). His original analysis relies on two main assumptions: (i) the fracture stiffness remains constant after shut-in until the fracture completely closes and (ii) Carter's leak-off coefficient CL is a constant. In Nolte's analysis, the coefficient CL is interpreted from the pressure decline following the sudden drop of pressure observed when the well is shut-in, by expressing the declining pressure as a linear function of G(Δt), which is a function of Δt - the time elapsed since shut-in divided by the shut-in time ts (Nolte, 1979, 1986).
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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) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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 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".