Renard et al. (2026) Upscaling Darcy–Weisbach flow in serial karst conduits of variable diameter and karst network simplification

Identification

• Journal: Advances in Water Resources
• Year: 2026
• Date: 2026-10-30
• Authors: Philippe Renard, Iván Colecchio, B. Nœtinger, Marco Dentz
• DOI: 10.1016/j.advwatres.2026.105546

Research Groups

• Centre for Hydrogeology and Geothermics, University of Neuchâtel, Switzerland
• IFP Energies nouvelles, Rueil-Malmaison, France
• Spanish National Research Council (IDAEA-CSIC), Barcelona, Spain

Short Summary

This study develops an upscaling strategy for Darcy–Weisbach flow in serial karst conduits of variable diameter, deriving analytical formulas for equivalent hydraulic diameter and rugosity. This approach enables significant computational cost reduction in network simulations while preserving global flow response and statistical flow rate distributions.

Objective

• To develop an upscaling strategy for serial arrangements (branches) of karst conduits with variable diameters.
• To derive analytical formulas for an equivalent hydraulic diameter and an equivalent rugosity that describe the global hydraulic behavior of a variable-diameter branch.
• To simplify complex karst conduit networks by replacing branches of varying diameters with single equivalent conduits, thereby reducing computational cost.
• To provide an explanation for the early onset of turbulence observed in variable-diameter pipes and karst conduits.

Study Configuration

• Spatial Scale: Serial arrangements of conduits (branches) with varying diameters. Individual conduit segments have lengths of 15 meters. Diameters are modeled as lognormal random variables with a geometric mean of 1 meter, logarithmic variance (σ²logD) ranging from 0.1 to 1, and truncated between 0.3 meters and 30 meters. Synthetic networks are generated on a 2D lattice (256 × 128 nodes) using a bond percolation model.
• Temporal Scale: Steady-state flow conditions.

Methodology and Data

• Models used:
  • Darcy–Weisbach equation for head loss in conduits.
  • Churchill equation for the Darcy friction factor (fD) across laminar, transitional, and turbulent regimes.
  • von Kármán equation for the fully rough turbulent regime.
  • A piecewise linear power law model (linear interpolation in logarithmic space) to approximate the friction factor in the transitional regime for equivalent conduits.
  • Picard iteration and Newton–Raphson method for solving the nonlinear flow problem on discrete karst conduit networks (DKN).
  • Theta-subnetwork model based on bond percolation on a 2D lattice for generating synthetic network geometries.
• Data sources: Numerical experiments conducted on synthetic karst conduit networks.

Main Results

• Analytical formulas were derived for the equivalent hydraulic diameter (harmonic mean of D⁻⁴) in the laminar regime and the equivalent rugosity in the fully rough turbulent regime for serial conduits of variable diameter.
• Diameter variability causes the coexistence of laminar and turbulent flow regions within a single branch, leading to the onset of the transitional regime at global Reynolds numbers significantly lower than 2000 (e.g., Re_g ≈ 500 in numerical examples).
• A piecewise linear power law model, bounded by global Reynolds numbers derived from the minimum and maximum local diameters, effectively approximates the friction factor in the transitional regime for equivalent conduits.
• Network simplification by replacing branches with equivalent conduits achieved substantial computational speedup (up to ~100x) while accurately reproducing the global flow response (total discharge relative error typically below 15%).
• The simplified networks maintained the statistical distribution of edge flow rates and preserved hydraulic heads at common junctions (maximum relative errors below 3%).
• The largest discrepancies between original and simplified models occurred in the transitional flow regime, attributed to biases in the approximate friction factor model.

Contributions

• First to derive a unique analytical expression for the equivalent rugosity for the fully rough turbulent regime in serial conduits of variable diameter, complementing existing equivalent diameter formulas.
• Provides a robust and physically grounded upscaling framework for modeling nonlinear flow in heterogeneous karst conduit networks.
• Offers a method for significant computational cost reduction in karst flow simulations, making large-scale modeling more feasible.
• Explains the observed early onset of turbulence in variable-diameter pipes and karst systems.
• Proposes a practical heuristic formula to define the range of global Reynolds numbers for the laminar and fully rough turbulent regimes in variable-diameter systems.

Funding

• European Research Council (ERC, KARST, 101071836)

Citation

@article{Renard2026Upscaling,
  author = {Renard, Philippe and Colecchio, Iván and Nœtinger, B. and Dentz, Marco},
  title = {Upscaling Darcy–Weisbach flow in serial karst conduits of variable diameter and karst network simplification},
  journal = {Advances in Water Resources},
  year = {2026},
  doi = {10.1016/j.advwatres.2026.105546},
  url = {https://doi.org/10.1016/j.advwatres.2026.105546}
}