Observations of wave-induced pore-pressure gradients and bed level response on a surf zone sandbar

Dylan Anderson1, Dan Cox1, Ryan Mieras2, Jack Puleo2, Tom Hsu2

Oregon State University1 University of Delaware2

OS32A-06

Physical mechanisms driving sandbar migration

Large waves break on the sand bar Undertow currents directed offshore

Bar moves offshore

(adapted from Hoefel & Elgar 2003)

Offshore migrations

1

Physical mechanisms driving sandbar migration

Calmer wave conditions, weak currents Asymmetric wave shapes

Bar moves onshore

(adapted from Hoefel & Elgar 2003)

𝐴𝑠 =𝜕𝑢 𝜕𝑡 3

𝜕𝑢 𝜕𝑡 2 3/2Onshore migrations

2

𝜏𝑏

Shields parameter

𝜃 =𝜏𝑏

𝜌𝑠−𝜌𝑤 𝑔𝑑50

(adapted from Jack Puleo)

𝜃 = 0.8

Sheet flow

3

𝜏𝑏 𝜏𝑏

𝑝 +𝜕𝑝

𝜕𝑥𝑑𝑥 𝑝

Sleath parameter

𝑆 =𝜕𝑝

𝜕𝑥

𝜌𝑠−𝜌𝑤 𝑔

Shields parameter

𝜃 =𝜏𝑏

𝜌𝑠−𝜌𝑤 𝑔𝑑50

Considering pressures within the sediment bed: horizontal forcing

4

𝑝 +𝜕𝑝

𝜕𝑥𝑑𝑥 𝑝

Sleath parameter

𝑆 =𝜕𝑝

𝜕𝑥

𝜌𝑠−𝜌𝑤 𝑔

Shields parameter

𝜃 =𝜏𝑏

𝜌𝑠−𝜌𝑤 𝑔𝑑50

Considering pressures within the sediment bed: horizontal forcing

Bed failure ≈ “plug flow” Sf = 0.1 (Foster et al. 2006) (adapted from Sleath 1999) 5

𝜏𝑏 𝜏𝑏

𝑝 +𝜕𝑝

𝜕𝑥𝑑𝑥 𝑝

Sleath parameter

𝑆 =𝜕𝑝

𝜕𝑥

𝜌𝑠−𝜌𝑤 𝑔

Shields parameter

𝜃 =𝜏𝑏

𝜌𝑠−𝜌𝑤 𝑔𝑑50

−1

𝜌𝑓

𝜕𝑝

𝜕𝑥=

𝜕𝑢

𝜕𝑡+ 𝑢

𝜕𝑢

𝜕𝑥+ 𝑤

𝜕𝑢

𝜕𝑧𝐴𝑠 =

𝜕𝑢 𝜕𝑡 3

𝜕𝑢 𝜕𝑡 2 3/2

6

Can we develop a more holistic understanding of the forces within the bed?

Wave propagation

Oscillatory flow

7 spatial and temporal gradients in the flow

Piston-type wavemaker: -Regular, Irregular, Tsunami-Max wave: 1.7m @ 5 seconds

Long Wave Flume Length: 104m Width: 3.7m Depth: 4.6m

8

Hybrid profile construction- limits large-scale bathymetric-

hydrodynamic feedbacks

- isolates small-scale bed response tovarying wave forcing

d50 = 0.17 mm

d50 = 0.27 mm

9

Hybrid profile construction

- limits large-scale bathymetric-hydrodynamic feedbacks

- isolates small-scale bed response to varying wave forcing

OS23B-2039 From the sand bed to the free surface: an experimental study of wave induced sediment transport over a sandbar (Ryan Mieras)

Pore-Pressure Transducer Array

- Buried within sediment bed- Druck PDCR 81s, sampling at 100 Hz- Finite differencing

10

Pore-Pressure Transducer Array

11

𝑃𝑥𝑖 = −2𝑃𝑖−1 − 3𝑃𝑖 + 6𝑃𝑖+1−𝑃𝑖+2

6∆𝑥=

−2𝑥2 − 3𝑥3 + 6𝑥4−𝑥5

6∆𝑥

𝑃𝑧𝑖 = −3𝑃𝑖 + 4𝑃𝑖+1−𝑃𝑖+2

2∆𝑧=

−3𝑥3 + 4𝑧1−𝑧2

2∆𝑧

Horizontal 𝜕 𝑝

𝛾

𝜕𝑥=

Vertical 𝜕 𝑝

𝛾

𝜕𝑧=

Conductivity Concentration Profiler

- Buried within sediment bed - Instantaneous bed levels - 32 mm window, sampling at 100z

12

62 trials consisting of 26 different wave conditions

13

Ensemble averaging to create pressure gradient vector ∇𝑝

14

shore

15

Rotations of ∇𝑝 with each wave cycle

shore

Steep, short period waves can produce same onshore S as less steep long period waves with larger As values

16

−1

𝜌𝑓

𝜕𝑝

𝜕𝑥=

𝜕𝑢

𝜕𝑡+ 𝑢

𝜕𝑢

𝜕𝑥+ 𝑤

𝜕𝑢

𝜕𝑧

17

Advective terms account for 14% of the horizontal pore pressure gradient

Shields parameter

𝜃 =𝜏𝑏

𝜌𝑠−𝜌𝑤 𝑔𝑑50

Sleath parameter

𝑆 =𝜕𝑝

𝜕𝑥

𝜌𝑠−𝜌𝑤 𝑔

18

Shields parameter

𝜃 =𝜏𝑏

𝜌𝑠−𝜌𝑤 𝑔𝑑50

Sleath parameter

𝑆 =𝜕𝑝

𝜕𝑥

𝜌𝑠−𝜌𝑤 𝑔

Erosion depth is function of both shear and pressure gradient

4 mm ≈ 24 grain diameters

18

Bed failures coincident with spikes in onshore-directed S

19

Sf is not universal, and has no clear dependence on As

20

Initiation of erosion likely depends on a combination of shear and pressure gradients competing to secure or destabilize sediment skeleton

(Foster et al. 2006; Frank et al. 2015; Cheng et al. 2016)

Oscillatory flow

Wave propagation

• Obtained a pore pressure gradient, 𝛻𝑝, within the sediment bed beneath complicated hydrodynamics over a surf zone sandbar

• Magnitude of onshore-directed Sleath not directly related to As

• Rapid drops in bed level were coincident with spikes in pore pressure gradient, but not at a universal

critical failure point

21

𝑑𝑝

𝑑𝑥−

𝜏𝑠ℎ𝑏

> 𝐾𝐶∗ 𝜌𝑠 − 𝜌 𝑔 + 𝐾𝑑𝑝

𝑑𝑧

(Sleath [1999], Foster et. al. [2006])

Oregon State Univ. of Delaware Myongji University (Korea) Dan Cox (PI) Jack Puleo (PI) Hyun-Doug Yoon Dylan Anderson Tom Hsu (PI) Wei Cheng Ryan Mieras Yokohama University (Japan) Hyoungsu Park Yeulwoo (Yaroo) Kim Takayuki Suzuki Jose Pintado Patino O.H. Hinsdale Lab Zheyu (Nancy) Zhou Kyoto University (Japan) Pedro Lomonaco Patricia Chardon William Pringle Tim Maddux Doug Kraft Cooper Pierson Wang Yanhong (National Hydraulic Research Institute of China)

Natural Hazards Engineering Research Infrastructure (NHERI) program Daniel Cox (dan.cox@oregonstate.edu), NHERI Project PI Pedro Lomonaco (Pedro.Lomonaco@oregonstate.edu), HWRL Director

−1

𝜌𝑓

𝜕𝑝

𝜕𝑥=

𝜕𝑢

𝜕𝑡+ 𝑢

𝜕𝑢

𝜕𝑥+ 𝑤

𝜕𝑢

𝜕𝑧

Deriving gradients by finite difference approximations

𝑃𝑖−1 = 𝑃𝑖 − ∆𝑥𝑃𝑥𝑖 + 1

2∆𝑥2𝑃𝑥𝑥𝑖 −

1

6∆𝑥3𝑃𝑥𝑥𝑥𝑖 + 𝑂 ∆𝑥4

𝑃𝑖 = 𝑃𝑖

𝑃𝑖+1 = 𝑃𝑖 + ∆𝑥𝑃𝑥𝑖 + 1

2∆𝑥2𝑃𝑥𝑥𝑖 +

1

6∆𝑥3𝑃𝑥𝑥𝑥𝑖 + 𝑂 ∆𝑥4

𝑃𝑖+2 = 𝑃𝑖 + (2∆𝑥)𝑃𝑥𝑖 + 1

2(2∆𝑥)2𝑃𝑥𝑥𝑖 +

1

6(2∆𝑥)3𝑃𝑥𝑥𝑥𝑖 + 𝑂 ∆𝑥4

𝑃𝑥𝑖 ≈1

∆𝑥 𝛼𝑗𝑃𝑗𝑗 ∈

= 𝛼𝑖−1 + 𝛼𝑖 + 𝛼𝑖+1 + 𝛼𝑖 1

∆𝑥𝑃𝑖

= −𝛼𝑖−1 + 𝛼𝑖+1 + 2𝛼𝑖 𝑃𝑥𝑖

= 1

2𝛼𝑖−1 +

1

2𝛼𝑖+1 +

4

2𝛼𝑖 ∆𝑥𝑃𝑥𝑥𝑖

= −1

6𝛼𝑖−1 +

1

6𝛼𝑖+1 +

8

6𝛼𝑖 ∆𝑥2𝑃𝑥𝑥𝑥𝑖

1 1 1 1−1 0 1 21

2 0 12

42

−16 0 1

6 8

6

𝛼𝑖−1

𝛼𝑖

𝛼𝑖+1

𝛼𝑖+2

=

0100

𝛼𝑖−1

𝛼𝑖

𝛼𝑖+1

𝛼𝑖+2

=

−

13

−12

1

−16

𝑃𝑥𝑖 = −2𝑃𝑖−1 − 3𝑃𝑖 + 6𝑃𝑖+1−𝑃𝑖+2

6∆𝑥=

−2𝑥2 − 3𝑥3 + 6𝑥4−𝑥5

6∆𝑥

𝑃𝑧𝑖 = −3𝑃𝑖 + 4𝑃𝑖+1−𝑃𝑖+2

2∆𝑧=

−3𝑥3 + 4𝑧1−𝑧2

2∆𝑧

Horizontal 𝜕 𝑝

𝛾

𝜕𝑥=

Vertical 𝜕 𝑝

𝛾

𝜕𝑧=