A Novel Analytical Fracture-Permeability Model Dependent on Both Fracture Width and Proppant-Pack Properties
Bibliographic record
Abstract
Summary For an empty fracture, the fracture permeability (kf) is mainly influenced by the effect of viscous shear from fracture walls and can be analytically estimated if the fracture width (wf) is known a priori (i.e., kf=β2wf2/12, where β2 is the unit-conversion factor). For an adequately propped fracture, the fracture permeability is mainly influenced by the proppant-pack properties and can be approximated with the proppant-pack permeability (kf=kp, where kp is proppant-pack permeability). It can be readily inferred that as the effect of viscous shear fades (or the proppant-pack effect becomes pronounced), there should be a regime within which both the viscous shear and the proppant-pack properties exert significant influences on the fracture permeability. However, the functional relationship between fracture permeability, viscous shear (or fracture width), and proppant-pack properties is still elusive. In this work, we propose a new fracture-permeability model to account for the influences of the proppant-pack permeability, proppant-pack porosity (ϕp), and fracture width on the fracture permeability. This new fracture-permeability model is derived from a modified Brinkman equation. The results calculated with the fracture-permeability model show that with different values of the Darcy parameter, the fluid flow can be divided into viscous-shear-dominated (VSD) regime, transition regime, and Darcy-flow-dominated (DFD) regime. If the Darcy parameter is sufficiently large, the effect of proppant-pack permeability on fracture permeability can be neglected and the fracture permeability can be calculated with the VSD fracture-permeability (FP) (VSD-FP) equation (i.e., kf=β2ϕpwf2/12). If the Darcy parameter is sufficiently small, the effect of viscous shear on fracture permeability can be neglected and the fracture permeability can be calculated with the DFD-FP equation (i.e., kf=kp). Both the VSD-FP and DFD-FP equations are special forms of the proposed fracture-permeability model. For the existing empirical/analytical fracture-conductivity models that neglect the effect of viscous shear, one can multiply these models by the coefficient of viscous shear to make these models capable of estimating the fracture conductivity with large values of Darcy parameter.
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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 | 0.000 | 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".