A Semi‐Analytical Solution for Heat Transport in Rock With Parallel Fractures and a Heat Source in Both Fracture and Matrix
Bibliographic record
Abstract
Abstract In recent decades, numerous analytical solutions have been developed to quantify temperature perturbations in fractured rock having mobile and immobile fluid phases. The assumption of one‐dimensional heat conduction in the matrix, or neglecting heat dispersion and storage in fractures, however, are typical simplifications adopted to overcome the difficulties in mathematically representing the problem. In this study, we propose a two‐dimensional semi‐analytical solution framework based on Green's function approach for a flexible heat source definition, including source dimensions, energy delivery strength and duration, and the presence of a heat source in the matrix and/or fracture. The solution fully accounts for heat conduction, advection, dispersion, and transient heat exchange between the fracture fluid and rock matrix in a system of parallel fractures. The solution having a strip heat source extending from a fracture into the matrix indicates that one‐dimensional heat conduction in the matrix underestimates and overestimates temperature responses at early and later times, respectively. Additionally, the temperature peak arrival time is also substantially delayed by simplification. The fracture temperature grows slower near the heat source area as the fracture aperture increases. The fracture temperature growth is enhanced via the overlapped heating areas between the parallel fractures. The transient temperature analyses imply that the spatial temperature variation is strongly associated with heat delivery strength. The early time temperature variances are closely related to the heat source configurations, and the later time temperatures in the domain are mainly determined by the total energy being delivered into the 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 | 0.001 | 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 | 0.001 |
| 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".