Development of Explicit Formulas for Estimating Seepage Characteristics Underneath Aprons with Equal and Unequal Double Vertical End Piles
Publication: International Journal of Geomechanics
Volume 22, Issue 12
Abstract
Piles are walls that are installed underneath hydraulic structures to control seepage flow, uplift pressure, and exit hydraulic gradient. In this study, the effect of the ratio of width of the apron to the foundation depth (B/D), the ratio of the width of the apron to the downstream pile depth (B/d2), and the ratio of the upstream and downstream pile depths (d1/d2) on seepage flow characteristics is numerically investigated using the finite-element method. From the numerical simulations, design charts, and explicit formulas for estimating seepage flow characteristics, such as the percentage residual uplift pressure at key points, the discharge flow factor, and the exit hydraulic gradient factor underneath aprons with end double-piles for both states of equal and unequal depths are developed. The simulations are validated with existing analytical solutions for simple cases. To assess the accuracy of the presented formulas, statistical metrics; that is, the determination coefficient, root mean square error, and percent relative error (RE%) were used. The statistical metrics are satisfactory, and their values quantified the reliability of the proposed formulas. The proposed explicit formulas, with RE% less than 5%, could be used by design engineers instead of implicit analytical equations that involve elliptic integrals.
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Data Availability Statement
Some or all data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.
Notation
The following symbols are used in this paper:
- B
- bottom width of the dam (m);
- D
- foundation depth of the dam (m);
- d1
- depth of the upstream pile (m);
- d2
- depth of the downstream pile (m);
- GE
- exit hydraulic gradient (−);
- H
- dam reservoir water head (m);
- h
- total head (m);
- Hc
- water head at key point C (m);
- HE
- water head at key point E (m);
- k
- hydraulic conductivities (m/s);
- ksat
- saturated hydraulic conductivities (m/s);
- kx
- hydraulic conductivities in the x-direction (m/s);
- ky
- hydraulic conductivities in the y-direction (m/s);
- mw
- slope of the water storage curve (1/kN);
- Pc
- percent residual uplift pressure at key point C (%);
- PE
- percent residual uplift pressure at key point E (%);
- Q
- boundary flux (m3/m3s);
- q
- seepage discharge (m2/s);
- t
- time (s);
- uw
- pore water pressure (kN/m2);
- γw
- unit weight of water (kN/m3); and
- θ
- volumetric water content (m3/m3).
References
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Information & Authors
Information
Published In
International Journal of Geomechanics
Volume 22 • Issue 12 • December 2022
Copyright
© 2022 American Society of Civil Engineers.
History
Received: Mar 3, 2022
Accepted: Jul 1, 2022
Published online: Sep 28, 2022
Published in print: Dec 1, 2022
Discussion open until: Feb 28, 2023
ASCE Technical Topics:
- Analysis (by type)
- Engineering fundamentals
- Foundations
- Geomechanics
- Geotechnical engineering
- Hydraulic engineering
- Hydraulic gradients
- Hydraulic pressure
- Hydraulic properties
- Hydraulic structures
- Numerical analysis
- Pile foundations
- Piles
- Seepage
- Soil dynamics
- Soil mechanics
- Soil properties
- Uplifting behavior
- Water and water resources
Authors
Metrics & Citations
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