Wave effects on submarine groundwater seepage measurement

Advances in Water Resources 32 (2009) 820–833

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Wave effects on submarine groundwater seepage measurement Anthony J. Smith a,*, David E. Herne b, Jeffrey V. Turner a a b

CSIRO Land and Water, Private Bag No. 5, Wembley, WA 6913, Australia Physical Systems, Woodvale, WA 6026, Australia

a r t i c l e

i n f o

Article history: Received 10 March 2008 Received in revised form 6 February 2009 Accepted 7 February 2009 Available online 20 February 2009 Keywords: Groundwater Submarine Wave Seepage Error AQUIFEM-P

a b s t r a c t Variation of sea-surface and water pressure above the sea bed induces temporal variation of submarine groundwater discharge over time scales ranging from seconds to years. Hydrodynamic theory and measurements suggest that wave-induced exchange between a permeable sediment bed and overlying water column is significant but conventional seepage-meter studies have focussed mainly on tidal dynamics and ignored waves. At Cockburn Sound in Western Australia we measured wave-induced flow reversals 1 through a seepage meter of amplitude 60 cm d ; however, it was unclear whether the measured flows were real seepage or, partly or wholly, an artifact of wave action on the seepage meter. A numerical model of seepage patterns beneath a vented benthic chamber demonstrated an observer effect introduced by the chamber and not previously identified. Placing a chamber on the sediment bed disturbed the pressure field and changed both the pattern and magnitude of the wave-induced flow. A separate analysis of benthic-chamber movements under the action of shallow surface waves established that micron-scale movements of the chamber at the wave frequency were sufficient to produce apparent seepage amplitudes of O(1–100) cm d1. We concluded that wave action is a key control on bed seepage and should not be neglected without justification in direct-measurement studies of marine bed discharge. A systematic error during each wave cycle can accumulate to a significant measurement error if the wave cycle error is large or if wave-induced flow is the dominant component of the seepage. In the latter case, the error could potentially be misinterpreted as a steady seepage component. Crown Copyright Ó 2009 Published by Elsevier Ltd. All rights reserved.

1. Introduction Sea-surface fluctuation induces water exchange between the sea and permeable bed sediments. Temporal variation of submarine groundwater discharge (SGD) in response to tides and longerperiod sea-surface dynamics have been previously reported [21,9,40,20] but variation of SGD at subtidal frequencies remains largely un-investigated [27] and un-measured. This may be an oversight in many studies because accepted hydrodynamic theory suggests that the wave-induced component of SGD can be the same order-of-magnitude as both the tide-induced component and the freshwater—terrestrial component—albeit at a much shorter time scale. Awareness about the biochemical and ecological significance of SGD has matured during the past 30 or so years. Groundwater has different quality to seawater and clear links between groundwater discharge and ecological zonation in marine environments have been found [15,32]. If groundwater fluxes across marine beds were uniform and steady then these relationships might be easy to study; however, groundwater seepage is notoriously difficult to

* Corresponding author. Fax: +61 8 93336211. E-mail address: [email protected] (A.J. Smith).

quantify because it is spatially and temporally variable across a broad range of space and time scales. As pointed out by Burnett et al. [2] ‘‘reliable methods to measure these fluxes need to be refined and the relative importance of the processes driving the flow needs clarification and quantification”. In this study, seepage-meter measurements of SGD at Cockburn Sound in Western Australia exhibited flow reversals at the wave period but we were uncertain whether the fluctuations were real seepage induced by waves or an artifact of wave action on the seepage meter. Wave-induced flows are short lived, fluctuate rapidly compared to tides and watertable dynamics in the inshore aquifer, and penetrate only small distances into the sediment bed. Nevertheless, seepage meters are placed directly at the water–sediment interface and are theoretically sensitive to all fluid exchange between the sediment bed and the water column. Wave-induced pore-water flow contributes a zero net groundwater flux to the sea because the sea is both the source and sink for the flow. The wave-induced component of seepage is normally overlooked in seepage-meter studies on the assumption that the device can accurately measure and sum to zero such fluctuations without significant error. Waves also may act mechanically on a benthic chamber to produce apparent seepage flows caused by subtle movements of the chamber relative to the sediment bed, or flexing of the chamber itself. The possibility that such fluxes oc-

0309-1708/$ - see front matter Crown Copyright Ó 2009 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.advwatres.2009.02.003

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but it ignores important properties of the sediment bed that can affect seepage—such as compressibility, inhomogeneity and anisotropy [19]—and may have limited utility for predicting bed seepage under field conditions. The other difficulty in applying Eq. (3) to seepage-meter measurements stems from the fact that the benthic chambers used in those studies disrupt the pressure field on which the formulation is based. These issues are explored further in the paper.

cur during seepage-meter studies has been noted by practitioners but not investigated theoretically or practically. This paper explores the potential for wave-induced flow and measurement error during seepage-meter studies, including both real and apparent flows. The theory of wave-induced bed seepage is reviewed to confirm that real seepage of the magnitude observed in Cockburn Sound is theoretically feasible and consistent with the wave climate and sediment properties. The possibility that the measured fluctuations were an artifact of wave action on the seepage meter is then considered. Specifically, a model of wave-induced pressure fluctuation in the sediment bed beneath a vented benthic chamber is presented, and the potential significance of movement or flexing of the benthic chamber in response to the mechanical action of waves is analysed.

1.2. A note on non-flat bed forms This paper is concerned specifically with cyclic bed seepage that occurs in response to unimpeded wave motion over a flat bed form. It also considers the potential influx and efflux of fluid from the benthic chamber in response to movement of the chamber caused by the mechanical action of waves. It does not contribute to the investigation of bed seepage induced by steady surface currents moving over non-flat bed forms (e.g., dunes and ripples) [43,41,7,13,31,22] or impermeable obstructions on the sediment bed (e.g., partially buried rocks and benthic chambers) [14,29,4].

1.1. Description of wave-induced bed seepage Pore-fluid motions caused by surface waves propagating over permeable beds were first studied by marine researchers investigating the interaction between bed percolation and wave-energy damping [23,25,26,19,12]. They found that sea water flows into the sediment bed beneath the propagating wave crests and porewater efflux occurs beneath the wave troughs. For the case of free plane sine waves propagating over a flat, horizontal sediment bed (Fig. 1), the pressure-head field above the water–sediment interface, hðx; z; tÞ [L], is usually given as [16]

h ¼ ðD  zÞ þ

g coshðkðD  zÞÞ

2. Results from previous seepage-meter studies Seepage-meters studies of SGD have typically used measurement-integration periods that preclude the detection and analysis of wave-induced flows (Table 1). This does not mean that the measurements necessarily contain wave-induced errors, though it remains unclear whether wave action was significant in those studies and influenced the results. Only one of the 20 seepage-meter studies listed in Table 1 reported seepage measurements at a temporal resolution that could demonstrate wave effects. Although those seepage measurements were taken in a lake [27] the results contained variation in the 1 range ±300 centimetres per day ðcm d Þ in apparent response to wind-generated surface waves. The authors suspected that some of the flow variation was in response to flexing of the flat top of the benthic chamber. Riedl et al. [26], who did not use a seepage meter, also reported pore-water fluctuation at wave frequency of 1 amplitude O(164–389) ðcm d Þ based on the analysis of a ‘‘constant temperature hot-thermistor anemometer” placed within the upper 30 cm of a sandy sea bed. It is clear from Table 1 that seepage rates of less than 1 cm d1 1 and typically within the range 1—100 cm d Þ are reported in the hydrological literature. The implication is that seepage meters can resolve specific discharge across the sediment bed O(0.1–10) micrometres per second ðlm s1 Þ. Thus, 1 mm of seepage would take from 2 min to well over 2 h to emerge from the sediment. During that time it is further implied that the instrument can sum any bi-directional flow constituents without introducing an

ð1Þ

;

where D [L] is mean sea-surface elevation above the sediment bed, gðx; tÞ ¼ a sinðkx  xtÞ [L] is the sinusoidal fluctuation of the seasurface elevation, a [L] is the wave amplitude, k ¼ 2p=k ½L1  is the wave number, k [L] is the wavelength, x ¼ 2p=P ½T1  is the angular frequency, P [T] is the wave period, x is the horizontal coordinate in the direction of wave propagation, z is the vertical coordinate measured positive-upward from the water–sediment interface and t is time. For an incompressible, homogeneous, isotropic and infinitelythick sediment bed, and assuming Darcy flow, the pressure-head field below the water–sediment interface /(x,z,t) [L], and the resultant amplitude of wave-induced flow across the interface qa ½L T1 , are given by [23,25,26,11,12]

/ ¼ D þ exp kz

g

ð2Þ

cosh kD

and

qa ¼ K

ak ; cosh kD

ð3Þ

where K ½L T1  is the sediment hydraulic conductivity. Eq. (3) provides a succinct estimate of the wave-induced seepage amplitude

(,) a D

Sea

z Sediment

x Fig. 1. Instantaneous isopotentials (lines) and velocity field (arrow heads) generated by free plane sine waves of the form g ¼ a sinðkx  xtÞ propagating over a flat bed. In 1 this example D ¼ 1 m; a ¼ 0:1 m; k ¼ 45 m; P ¼ 15 s; K ¼ 5 m d and t ¼ 0 s.

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Table 1 Direct measurements of SGD from seepage-meter studies. Seepage device

Study

SGD measurement interval or integration interval

Specific discharge, 1 cm d

Lee-type [17]

Lake Michigan [5] Turkey Point [3] Cockburn Sound [34] Kahana Bay [9] Lake Biwa [36] Turkey Point [38] Waquoit bay [20] Indian River Lagoon [4]

4 days 1–125 min 60–120 min 3–12 h 20–30 min 20–30 min 60 min 60 min

0–0.01 0.5–20 3 to 32 0–10 0–20 0–50 2–40 0–13

Heat-pulse [36]

Lake Biwa [36] Shelter Island [28] Suruga Bay [40] Cockburn Sound

5 min 15 min 1 min 2 min

0–20 4–11 4–155 60 to 60

Continuous-heat [37]

Turkey Point [38] Cockburn Sound [39]

10 min 5 min

0–75 0–45

Ultrasonic [21]

Shelter Island [21] Shelter Island [28] Cockburn Sound [34] Turkey Point [38]

15 min 5 min 15 min 12 h

45 to 175 50 to 200 12 to 61 15–35

Dye-dilution [30] Electromagnetic [27]

Shelter Island [28] Ashumet Pond [27]

5 min 15 s

2–30 300 to 300

unacceptable error. This ability of seepage meters appears to be un-verified as is the ubiquity of bi-directional flow in seepage-meter studies. 3. Evidence of wave-induced bed seepage at Cockburn Sound Groundwater discharge into Cockburn Sound in southwest Western Australia (Fig. 2) was investigated during 2001 using a heat-pulse seepage meter (Fig. 3) that was deployed for brief periods in Northern Harbour and BP Boat Harbour. A description of the seepage meter and head-pulse method is contained in Appendix A. At both locations the seepage meter was positioned close to the shoreline and within a narrow 5-m-wide strip of sandy unconsolidated sediment where fresh submarine pore-water had been mapped [34] using a conductivity probe [35]. Seepage measurements were made automatically every two minutes. The results in Fig. 4 were initially surprising because they indicated that both the direction and magnitude of the seepage measurements varied considerably between readings. Closer examination of the sea-surface elevation data that were recorded during each measurement cycle indicated that wave action was the most-likely explanation for the apparent fluctuation of seepage. In particular, it became obvious that the typical range of sea-surface variation during a measurement period (66 s) was larger than the typical change in average sea-surface elevation between each measurement (120 s). Applying Eq. (3) to Northern Harbour (Table 2) we estimated a theoretical amplitude of wave1 induced bed seepage in the range Oð6—138Þ cm d , which was comparable in magnitude to the field results. 3.1. Heat-pulse temperature data during quasi-steady flow Fig. 5 depicts temperature data recorded for a known flow rate during laboratory calibration of the heat-pulse detection unit. Each temperature trace corresponds to the labelled thermistors T1–T6 in Fig. 3. In this example the heater was pulsed for 0.2 s and the heated water moved along the analyser tube in the direction of thermistors T4, T5 and T6. A temperature response also was recorded in the opposite direction at thermistor T3 due to heat

Fig. 2. Cockburn Sound is a shallow marine embayment located approximately 20 km south of Fremantle in southwest Western Australia. A shallow, unconfined, sand and limestone aquifer discharges groundwater along the coastline.

T1 T2 T3

Benthic chamber

+ve

T4 T5T6

amplification Water Sediment

-ve

Fig. 3. Heat-pulse seepage meter.

conduction (diffusion) against the direction of flow. The apparent flow velocity in the analyser tube was calculated by dividing the distance between two thermistors by the time taken for the heat-pulse to travel between them. Calibration curves (Appendix A) were established to relate these values to the true flow rates, which were determined by collecting the flow and logging the accumulated mass using an electronic balance and computer. It was evident from the fast ‘decay’ of the heat-pulse during the measurement period that diffusion due to heat conduction within the fluid phase and the analyser solid components (e.g., tube and thermistors) was significant.

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Flux, -1 cm.d

Water Depth, m

a

Time 11:30 0.8 0.6

12:00

12:30

Range of sea-level variation during measurement period

Water Depth, m

13:30

Mean water depth above chamber

0.4 0.2 0.0 11:30 80 60 40 20 0 -20 -40 -60 -80

12:00

12:30

13:00

13:30

Outer thermistor pair Inner thermistor pair

Porewater Efflux

Seawater Influx

b

Flux, -1 cm.d

13:00

Time 09:00 0.8 0.6

09:30

10:00

10:30

Range of sea-level variation during measurement period

0.4 0.2 0.0 09:00 80 60 40 20 0 -20 -40 -60 -80

Mean water depth above chamber 09:30

10:00

10:30 Outer thermistor pair Inner thermistor pair

Porewater Efflux

Seawater Influx

Fig. 4. Heat-pulse seepage measurements at (a) Northern Harbour on 8 May 2001 and (b) BP Boat Harbour on 12 May 2001. Measurements were taken every 2 min and required 66 s each to complete. Pore-water efflux was positive and seawater influx was negative. The plots of water depth show the mean depth during each measurement period (line) and the range of depths (vertical whiskers). Water depth was recorded at 5 Hz, providing 330 depth records for each measurement.

24.0

Property

Symbol

Value

23.5

Gravity acceleration Wave amplitude Mean water depth Wave speed Wave period Wavelength Wave number Sediment hydraulic conductivity Bottom current amplitude

g a D pffiffiffiffiffiffi c ¼ gD P k ¼ cP k ¼ 2p=k K ua

9:81 m s2 O(0.05–0.2) m O(1–3) m Oð3—5Þ m s1 15 s 45 m 0:14 m1 1 Oð5—50Þ m d Oð0:15—0:64Þ m s1

Temperature, deg. C.

Table 2 Wave conditions and sediment permeability in Northern Harbour.

19-02-2001, 12.50 pm T4

23.0 T5 22.5 T6 22.0 T1 T2

T3

21.5

21.0

Two estimates of the flow rate could be made using the inner pairs of thermistors, either T4–T5 or T3–T2, and the outer pairs T4–T6 or T3–T1. The heat-pulse travel time could be calculated as either the time difference between peak temperatures at the thermistors, or the time difference between the arrival of the temperature front at the thermistors. The latter method was used in this study because the detection of temperature peaks under submarine conditions ultimately proved to be problematic. Wave-induced oscillation of flow direction in the analyser tube during the measurement period resulted in multiple local maxima and minima.

0

10

20

30

40

50

60

Time, s Fig. 5. Heat-pulse temperature traces recorded at 5 Hz under steady flow conditions in the laboratory; in this example the flow rate was approximately 4 mm s1 in the analyser tube.

3.2. Heat-pulse temperature data during oscillatory flow Figs. 6 and 7 are examples of heat-pulse temperature traces obtained under shallow submarine conditions in Northern Harbour.

A.J. Smith et al. / Advances in Water Resources 32 (2009) 820–833

23.0

23.0

Water depth

22.5

0.3 21.5 T3 0.2

T2

21.0

T4

T1

T5

T6 0.1

20.5

0.0

20.0 0

10

20

30

40

50

60

Time, s Fig. 6. Heat-pulse temperature traces at Northern Harbour at 11.46 am on 8 May 1 2001; the estimated flow rate was Oð20—35Þ cm d into the sediment, which was equivalent to a flow rate in the analyser tube of approximately 8—15 mm s1 .

Water depth above the instrument—as plotted on the right-hand axes of the figures—was recorded simultaneously at a frequency of 5 Hz using a pressure transducer mounted on the seepage meter. The temperature data from Figs. 5–7 are combined in Fig. 8 to provide a more effective visualisation of heat transport within the analyser tube during the measurement period. Though it is atypical of most of the heat-pulse seepage measurements from Northern Harbour, the first example (Fig. 6) is familiar from the previous analysis of heat-pulse data under steady

Temperature, deg. C.

22.0

0.6

Northern Harbour: 08-05-2001, 11.52 am

0.5

22.5

0.4

Water Depth above Benthic Chamber, m

Temperature, deg. C.

0.5

Northern Harbour: 08-05-2001, 11.46 am

Water depth 22.0

0.4

T3

0.3

21.5 T4 T5 T6

21.0

T1

T2

0.2

Water Depth above Benthic Chamber, m

824

0.1

20.5

0.0

20.0 0

10

20

30

40

50

60

Time, s Fig. 7. Heat-pulse temperature traces at Northern Harbour at 11.52 am on 8 May 1 2001; the estimated flow rate was Oð9—20Þ cm d out of the sediment, which was equivalent to a flow rate in the analyser tube of approximately 4—8 mm s1 . Effects from wave action are pronounced in this example.

flow conditions. The temperature and pressure data confirm that the heater was pulsed at t ¼ 20 s after which time the sea-surface elevation above the seepage meter remained relatively constant for the next 30–40 s. Short-period, O(1–2) s fluctuation of water depth had negligible affect on the measured seepage rate. At the time of the measurement the apparent direction of seepage was into the 1 sediment at a rate of Oð20—35Þ cm d . Fig. 8 shows that the heat-pulse was mostly advected beyond thermistor T1 at t ¼ 30 s.

a

b

c

Fig. 8. Heat transport in the heat-pulse analyser tube during laboratory calibration and under submarine conditions in Northern Harbour; the three plots correspond to the temperature traces in (a) Fig. 5, (b) Fig. 6 and (c) Fig. 7. In the field examples, heat advection toward thermistor T3 indicated pore-water efflux from the sediment and heat advection toward thermistor T4 indicated sea water influx to the sediment.

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A.J. Smith et al. / Advances in Water Resources 32 (2009) 820–833

6000

Waves P

5000

PSD

4000 3000

Ripples P

2000 1000

0.1 0.2 0.3 0.4 0.5

Fig. 9. Power spectral density of sea-surface elevation in Fig. 7. The dominant frequencies are evident as peaks and correspond to shallow surface waves with period 15.2 s (0.0658 Hz) and wind surface ripples with period 2.4 s (0.417 Hz).

The second example (Fig. 7) is a more typical measurement and shows clearly the influence of waves on the direction and rate of measured flow. At t ¼ 20 s the sea-surface was falling and the apparent direction of seepage was out of the sediment at a rate 1 of Oð9—20Þ cm d . Crests and troughs in the temperature traces at later times indicate that there were subsequent oscillations of the flow direction past the thermistor array. The effect is illustrated most clearly in Fig. 8c, wherein the heat-pulse oscillates back and forth within the analyser tube.

T1

Deploying a benthic chamber to measure bed seepage introduces an observer effect because the chamber disrupts the pressure distribution that would drive the seepage if the chamber were not present. In the following modelling, the chamber was as-

T3

T2

3.5

3.5

3.5

3

3

3

2.5

2.5

2.5

2

PSD

4

2

2

1.5

1.5

1.5

1

1

1

0.5

0.5

0.5

0.1 0.2 0.3 0.4 0.5

4

0.1 0.2 0.3 0.4 0.5

0.1 0.2 0.3 0.4 0.5

T4

T5

T6 4

3.5

3.5

3.5

3

3

3

2.5

2.5

2.5

2

PSD

4

PSD

PSD

4. Simulation of pressure disturbance and wave-induced bed seepage beneath a vented benthic chamber

4

PSD

PSD

4

Power spectral densities (PSDs) of both the sea-surface elevation and the heat-pulse temperature data (Figs. 9 and 10) supported the above visual evidence of flow oscillations. Shallow surface waves with period 15.2 s, and wind ripples with period 2.4 s, were evident in the PSD plots of water depth. There were corresponding peaks in the PSD plots of temperature at the shallowwave period (15.2 s) but no evidence of flow variation caused by the surface wind ripples. With the benefit of hindsight, it is clear that the heat-pulse method was not suited to marine conditions when wave action resulted in flow oscillations at sub-measurement frequency. Only the flow order-of-magnitude and direction obtained by that method are considered to be reliable; however, the results provide valuable insight about the potential affects of wave action on seepage-meter measurements. Though the results suggested that wave action was a key control on the magnitude and direction of flow in the heat-pulse detection unit we were unsure whether the flows were representative of the natural wave-induced seepage rate, were perturbed by a wave-induced observer effect, or were a result of the mechanical action of waves on the seepage meter. Those possibilities are considered in the following analyses and discussion.

2

2

1.5

1.5

1.5

1

1

1

0.5

0.5

0.5

0.1 0.2 0.3 0.4 0.5

0.1 0.2 0.3 0.4 0.5

0.1 0.2 0.3 0.4 0.5

Fig. 10. Power spectral density of heat-pulse temperature traces in Fig. 7. The dominant frequency corresponds to shallow surface waves with period 15.2 s (0.0658 Hz). Effects from higher-frequency surface ripples are not evident.

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(x,0,0) -W/2 W/2

- /2

/2

a (x,0) D

Benthic chamber

W L

H

(0,0)

Sediment

B

Fig. 11. Model configuration; the upper graph depicts the instantaneous head potential along the sediment bed at t ¼ 0; pressure in the benthic chamber was assumed to be uniform and equal to pressure at the vent outlet.

sumed to be vented to the sea, and pressure change at the vent outlet was assumed to propagate uniformly and instantaneously to the sediment–water interface beneath the chamber. The model geometry and boundary conditions are depicted in Fig. 11. 4.1. Flow equations The porous sediment bed was modelled in vertical section and assuming two-dimensional Darcy flow [1]

S0 @ U @ 2 U @ 2 U ¼ 2 þ 2; K @t @x @z

ð4Þ

where Uðx; z; tÞ [L] is the hydraulic-head potential, S0 ½L1  is the specific storativity of the sediment and K ½L T1  is the sediment hydraulic conductivity. This approach ignores the finite length of the benthic chamber in the y-direction and provides only a twodimensional approximation of the three-dimensional flow pattern along the centerline of the chamber. If the water and porous medium are assumed incompressible (i.e., S0 ¼ 0) then Eq. (4) simplifies to the two-dimensional Laplace equation on which Eq. (3) is based. The following boundary conditions were applied along the top, sides and bottom of the model domain: Uðx; 0; tÞ ¼ /ðx; 0; tÞ for k=2 6 x < W=2; Uðx;0;tÞ ¼ hðL;H;tÞ for W=2 6 x 6 W=2; Uðx;z;tÞ ¼ /ðx;z;tÞ for x ¼ k=2 and k=2, and @ U=@z ¼ 0 for z ¼ B. The vertical boundary conditions on x ¼ k=2 and k=2 were adopted from Eq. (2) and were strictly correct only for an incompressible porous medium and fluid. Nevertheless, k was large compared to W and for realistic values of K in this study the far-field boundary fluctuations were rapidly attenuated and did not affect the chamber seepage. Along the base of the sediment bed, a no-flow condition was reasonable if the sediment was assumed to be resting on an impermeable basement or if vertical flow was considered to be insignificant at that depth. The latter condition was approximately satisfied by setting B ¼ k in all examples. From the exponential damping term in Eq. (2) it follows that wave-induced flow at z ¼ k is reduced to less than 0.2 percent of the maximum wave-induced flow at the water–sediment interface. For a compressible sediment bed the amount of damping with depth is greater.

4.2. Solution method The flow equation and boundary conditions were solved numerically using AQUIFEM-P [42], a finite-element modelling package for simulating linear, periodically-forced porous-media flow in two dimensions. The boundary conditions and solution from AQUIFEM-P are specified in the frequency domain. In application, the modelled system can be forced by head and flow boundaries conditions with arbitrary frequency, amplitude and phase. AQUIFEM-P simulates the response of the system at the specified frequencies and outputs nodal arrays of head amplitude and phase, and element arrays of velocity amplitude and phase. A more detailed description of AQUIFEM-P and its use can be found in the ‘‘User’s Manual and Description” [42]. Validation of the code has included benchmarking against algebraic solutions for both onedimensional and two-dimensional problems [33]. The finite-element mesh representing the sediment bed was designed with maximum nodal density directly beneath the benthic chamber (Fig. 12). It was impractical to properly incorporate the chamber rim as an impermeable intrusion within the sediment

n1 nodes : n2 ‘Effective’ depth

x ,t

x t L

H n1 n2

n2 n1

Fig. 12. Detail of the finite-element mesh and boundary nodes beneath the benthic chamber; the distance between nodes n1 and n2 on each side the chamber wall represents an effective intrusion depth of the chamber rim within the sediment.

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because the rim of a chamber is typically only a few millimetre thick while the mesh was required to be tens-of-metres in width and depth. The distance between the pair of boundary nodes directly inside and outside of the chamber wall (n1 and n2 in Fig. 12) was considered to represent an ‘effective intrusion depth’ because it defined the sediment thickness separating the external and internal chamber pressures. This approach was convenient but prevented the solution’s mesh-independence from being properly verified because the effective intrusion depth varied as a function of the mesh refinement. Six mesh designs with nodal spacings beneath the chamber of 20, 10, 5, 4, 3 and 2.5 cm were tested to confirmed that the results were qualitatively consistent. The 2.5-cm mesh, consisting of 83,382 nodes and 165,692 elements, was used

Table 3 Amplitude of specific discharge at the water–sediment interface with no benthic chamber; values of the input variables not shown in the table were: k ¼ B ¼ 45 m; W ¼ 1 m; H ¼ 0 m; P ¼ 15 s and a ¼ 0:1 m. 1

K; m d

1 5 10 100 1000

qa ; m d

1

Eq. (3) Incomp.

Model S0 ¼ 0

Model S0 ¼ 104 m1

0.0138 0.0692 0.138 1.38 13.8

0.0138 0.0691 0.138 1.38 13.8

0.186 0.428 0.610 2.05 14.0

a

QWa

-1

-1

-1

-1

for all subsequent simulations because O(2–3) cm is a realistic intrusion depth and the mesh provided the best nodal resolution beneath the chamber. A comparison of the results obtained using a coarser mesh and larger intrusion depth is presented further in the paper. 4.3. Bed seepage with no benthic chamber The numerical model was validated against Eq. (3) for the particular case of an incompressible fluid–sediment system with no benthic chamber present (Table 3). A similar analytical result was not available to validate the results for a compressible system. Madsen [19] presented a general solution to this problem but it relies on bulk porous-media properties, such as Poisson’s ratio, shear modulus and the coefficient of volume decrease obtained from oedometer tests, which are neither readily available for sea beds nor easily related to S0 . Specific storativity is a bulk property of the fluid and porous medium and depends on the compressibilities of the fluid, solid phases and sediment skeleton. Water and rock compressibilities are very small but the bulk compressibility of the sea bed can be significant due to intergranular re-packing and the presence of small amounts of highly-compressible gas (partial saturation). These effects are well known to marine engineers and are incorporated into their geotechnical analyses of wave-induced sea bed response [19,24] in much more detail than is considered here. The numerical results in this study indicated that wave-induced seepage is potentially much greater when the

f

QWa

QCa

g

QCa

L

c

QCa

-1

-1

-1

-1

L -1

h

-1

QCa

L

d

-1

L

L

b

-1

L

QCa

-1

-1

QCa

-1

-1

i

QCa

-1

-1

QCa

-1

-1

L

e

Incompressible (S0 )

L

j

Compressible -4 -1 (S0 )

Fig. 13. Simulated pore-water flow paths for five cases: (a and f) no benthic chamber and (b–e and g–j) benthic chambers with vent outlets at L ¼ 0, 0.5, 1 and 2 m; the elliptical flow paths are plotted on a regular grid at 5-cm spacing and exaggerated in scale by a factor 1000; values of the other input variables were: 1 K ¼ 5 m d ; k ¼ B ¼ 45 m; W ¼ 1 m; H ¼ 0 m; P ¼ 15 s and a ¼ 0:1 m.

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fluid–sediment system is compressible and can store and release water in response to the fluctuating pore-fluid pressure. The magnitude of this effect diminished as K increased (i.e., S0 =K ! 0) and the system behaviour approached that of potential flow. Fig. 13 illustrates cyclic pore-water flow paths in the sediment bed directly beneath the benthic chamber for five cases: (a and f) no benthic chamber and (b–e and g–j) benthic chambers with vent outlets at L ¼ 0; 0:5, 1 and 2 m. The flow paths are mathematically elliptical [33] and are scaled in size by a factor 1000. With no benthic chamber present, the amplitude of bed seepage at the water– 1 sediment interface was 6:91 cm d ð0:80 lm s1 ) for an incom1 pressible system and 42:8 cm d ð5:0 lm s1 ) for a compressible system with S0 ¼ 104 m1 (Table 3). Summed over the length of sea bed that would be occupied by the benthic chamber, the integrated seepage amplitudes, denoted Q Wa , were 72 litre per day per 1 1 metre ðL d m1 Þ and 435 L d m1 , respectively. 4.4. Affect of the benthic chamber Introducing a benthic chamber onto the sea bed resulted in non-uniform bed seepage patterns and different seepage amounts. Considering first the case of an incompressible fluid–sediment system, the seepage pattern in Fig. 13b was non-uniform but symmetric because the vent location was directly above the chamber. Maximum bed seepage occurred near the rim of the chamber where the pressure gradient between the inside and outside of the chamber was greatest. Defining Q Ca as the amplitude of integrated seepage beneath the chamber, the values for incompressible 1 flow increased from 75 to 505 L d m1 as the location of the vent outlet was moved from 0 to 2 m away from the chamber (Fig. 13b– e). The corresponding phase shift, denoted Q Cp , increased from 0:001P (0.015 s) to 0:229P (3.44 s). For a compressible system the affect of the chamber on seepage was more complicated due to phase shift in the pore-pressure response. Although the absolute values of Q Ca were larger, the chamber seepage was relatively less affected by the outlet location because it did not alter the system

compressibility. For realistic values of L between 0 and 2 m, the simulated values of Q Ca were slightly less than predicted with no benthic chamber present. For larger values of L=k, then Q Ca =Q Wa was greater than one. The result systematics are summarised in Fig. 14, which depicts the normalised seepage amplitude Q Ca =Q Wa , and normalised phase shift Q Cp =P, graphed against the normalised outlet location L=k. A value Q Ca =Q Wa ¼ 2 would indicate that the amplitude of seepage beneath the benthic chamber was twice the value obtained with no chamber. Similarly, a value Q Cp =P ¼ 0:5 would indicate that the seepage beneath the chamber was half a period out of phase relative to the seepage with no chamber. The maximum value of Q Ca =Q Wa occurred for L=k ¼ 0:5 because the chamber was directly beneath the wave crest when the vent outlet was directly beneath the wave trough, which produced the maximum potential pressure difference between the inside and outside of the chamber. Although that situation would be unrealistic in practice, Fig. 14 serves to illustrates the general principle that can lead to amplification and phase shifting of wave-induced seepage beneath the chamber.

5. Wave-induced movement and flexing of benthic chambers Under the action of waves, benthic chambers are subject to surface forces that may cause subtle movement or flexing of the chamber. Seepage meters are designed to measure seepage rates 1 of Oð1—100Þ cm d ðOð0:1—10Þ lm s1 ) and are inherently sensitive to very small but rapid volumetric changes beneath the chamber. Wave-induced movement of a chamber could include vertical displacement relative to the sediment bed, rhythmic tilting or rocking due to uneven placement or variable sediment compressibility, or elastic deformation of the chamber itself. In operation, the walls of the chamber are usually embedded into the sediment and a horizontal flange may be provided to stabilise the chamber against the sediment surface. Notwithstanding these types of practical design features, micron-scale movements in response to wave action are still plausible, even though such motions cannot be observed or readily measured under field conditions. 5.1. Apparent seepage due to periodic chamber motions

Legend: S0 = 0 ) S0 = 0 ) -4

-1

S0 = 10 m )

QCa / QWa

50 40 30 20 10 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

L/λ 1.0

QCp / P

0.8

The cyclic motion of a benthic chamber required to induce an apparent-seepage amplitude qa can be estimated from simple geometric considerations—as presented in Appendix B. Considering the simple chamber motions illustrated in Fig. 15, the following relationship are apparent: a ¼ 2qa =x; b ¼ 4qa =x and c ¼ 4qa =x, where the chamber motions are sinusoidal and a=2; b=2 and c=2 are the displacement amplitudes. It can be seen that qa is proportional to the frequency because rapid chamber displacements cause rapid volumetric fluctuation and larger flow rates. The significance of these expressions is illustrated by considering the magnitude of movement that is required to produce apparent seepage rates that are detectable by seepage meters. For 1 qa ¼ Oð1—100Þ cm d and P ¼ 15 s, the approximate motions of

0.6 0.4 0.2 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

L/λ Fig. 14. Amplitude and phase of benthic-chamber seepage normalised against the base case with no benthic chamber; all results were produced using the same input values listed in Fig. 13.

Fig. 15. Benthic chamber movements.

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a benthic chamber required to produce the same magnitude flows are a ¼ Oð0:5—55Þ lm and b ¼ c ¼ Oð1:1—110Þ lm. Clearly, these are very small movements that would be impractical to detect under field conditions even though seepage meters are theoretically sensitive to the flows they produce.

for benthic chambers are undetermined. A sensitivity analysis was possible using order-of-magnitude estimates from the fluid mechanics literature and the following values of fluid properties and chamber geometry used in Cockburn Sound: q ¼ 1025 kg m3 ; ua ¼ Oð0:15—0:64Þ m s1 (Table 2), where ua is the amplitude of the wave-induced bottom current, As ¼ 0:141 m2 ; Ap ¼ 0:283 m2 and V ¼ 0:056 m3 . For example, Cokgor and Avci [6] measured force coefficients of order C D ¼ Oð0:4—0:6Þ; C I ¼ Oð1—1:5Þ and C L ¼ Oð2—3:5Þ for a half-buried cylinder subject to wave-induced oscillatory flow. Applied to Cockburn Sound the calculated wave forces were F D ¼ Oð0:6—18Þ Newton ðNÞ, C I ¼ Oð4—23Þ N and C L ¼ Oð6—208Þ N, which are significant. In comparison, the force exerted by gravity on 1 kg is 9.81 N. Gaylord et al. [10] reported similar coefficient values of order C D ¼ Oð0:67—0:72Þ and C I ¼ Oð2—2:6Þ for a solid sphere in oscillating flow.

5.2. Wave forces Based on theoretical considerations, a wave-induced bottom current will impart a vertical lift force F L on a benthic chamber, and a horizontal drag force F D and inertia force F I in the direction of the flow. The total in-line force is usually expressed by the Morison formula F M ¼ F D þ F I . In problems that involve shallow surface waves (D < k=2) acting on submerged objects that are small compared to the wavelength, the following empirical wave-force equations are generally used [18,8]: F D ¼ 12 C D qAs us jus j; F I ¼ C I qVðdus =dtÞ and F L ¼ 12 C L qAp u2s , where C D ; C I and C L are dimensionless force coefficients that need to be determined experimentally. The other symbols denote the fluid density q, the wave-induce bottom current us , the submerged object’s volume V, its projected crosssectional area As and its planform area Ap . For a benthic chamber, the wave forces are countered by resistive forces that are related to the mass of the chamber, its embedded depth in the sediment and surface friction. The potential magnitudes of wave forces acting on a benthic chamber are of immediate interest; however, force coefficients

5.3. Phase relations The phase of each wave force is evident from its mathematical form and the graphical representations in Fig. 16. The drag force varies at wave frequency but is not sinusoidal. The inertia force is sinusoidal but has different phase to the drag force. The lift force fluctuates at twice wave frequency because it always acts in the upward direction. Thus, it seems possible that the frequency and phase of a seepage flow might help to identify the primary driving

t

us q FD FI FL

x

0

x

0

x

P/2

x

0

x

P/4

x t Max.

Max.

*

q

q*

x

P/4

x

3P/4

x

P/2

x

0

t Max.

Max.

q* q* t Max.

q*

Max.

x

Fig. 16. Phase relations between sea-surface elevation and seepage.

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A.J. Smith et al. / Advances in Water Resources 32 (2009) 820–833

force; however, real-world complexity may mean that such situations are rare. A brief discussion is presented below to illustrate the potential uncertainties of inferring driving forces from seepage response. If the fluid and sea bed are incompressible then wave-induced seepage with no benthic chamber is predicted to have opposite phase to the sea-surface elevation. A measured seepage fluctuation with different phase may indicate apparent seepage caused by wave-induced motion or flexing of the benthic chamber. On the other hand, the phase shift might be caused by fluid–sediment compressibility or the influence of the benthic chamber and outlet position, or both. The drag force is bi-directional and has the same phase as sea-surface elevation. If a benthic chamber was tilted a maximum distance into the sediment (expelling chamber water) when the drag force had maximum strength in either the positive or negative flow direction, and was tilted a maximum distance out of the sediment (drawing water into the chamber) when the drag force had maximum strength in the opposite direction, then the apparent seepage would lag sea-surface elevation by either P=4 or 3P=4. The inertia force also is bi-directional but has maxima and minima when the drag force is zero. If the benthic chamber was tilted back and forth by the inertia force, in the same manner as described above for the drag force, then the magnitude of apparent seepage would be either the same phase as sea-surface elevation or would be P=2 out of phase. The lift force is uni-directional and has two maxima per wave period; one when the bottom current is strongest in the positive flow direction and one when it is strongest in the negative flow direction. Apparent seepage caused by the lift force should be obvious because it would fluctuate at twice wave frequency. Based on the Fourier analyses in Fig. 10, the wave-induced seepage rate was found to lag sea-surface elevation by approximately 0:4P. In comparison, the theory for an incompressible fluid–sediment system predicts that wave-induced seepage with no benthic chamber should lag sea-surface elevation by 0:5P. The discrepancy of around 0:1P could be explained by the outlet location, which was 0.5–1 m from the chamber, or fluid–sediment compressibility effects. Alternatively, the inertial force may have been significant. A hemispherical stainless-steel dome was used and therefore effects due to flexing were less likely. More generally, it is evident that water movement through a seepage meter in response to wave action is likely to be complex while the above theoretical approach is relatively simplified. It is reasonable to expect that phase shifts will be present due to fluid–sediment compressibility and benthic-chamber observer effects, and it is reasonable to expect that true micron-scale motion and flexing of a benthic chamber deployed in submarine conditions would be highly complex in response to interacting wave forces that are superimposed with other water-body dynamics.

wave-pumping through the seepage meter. All internal flows are subject to friction losses associated with the roughness and geometry of the flow conduit (e.g., material type, bends, inlets, outlets, restrictions, etc.). A small directional bias in flow resistance through a seepage meter is likely to exist unless friction losses are identical in both directions. A systematic error also could arise if the flow measurement contained a directional bias resulting from the instrument design, imperfect construction, or drift in the instrument calibration. This error could occur even if the flow was perfectly periodic and would reflect an inability of the instrument to detect, measure and integrate flow identically in both directions. In practice, such errors are expected to be small if care is taken to minimise them; however, it is of interest to determine the conditions that might lead to a significant error over many wave cycles. A seepage meter deployed for 1 h, 1 day or 1 week would be exposed to 240, 5760 or 40,320 wave cycles with 15-s period. A systematic volumetric error would grow incrementally larger with each wave cycle but the flow-rate error would remain constant because the total measurement time also would accumulate. Consider an arbitrary sinusoidal bed seepage as follows:

QðtÞ ¼ Q s þ Q a sinð2pt=PÞ;

Both field and theoretical results in this paper indicate that wave-induced flow through a vented benthic chamber can be significant under realistic marine conditions. Whether such periodicities contribute a negligible net flow through the chamber over multiple wave cycles is un-verified. If wave action is significant enough to cause flow reversals within a seepage meter then it follows that a uni-directional seepage meter would accumulate a systematic measurement error and should not be used. Two additional systematic errors are conceivable for bi-directional seepage meters: (1) directional bias in the flow itself and (2) directional bias in the flow measurement. A directional bias in the flow conducting components of a benthic chamber or the flow detection unit could lead to slow

1

where Q s is the average flow rate ½L T ; Q a is the amplitude and P is the period. If Q a > Q s then the seepage is bi-directional and a systematic measurement error is possible. The time intervals for inflow and out flow during each period are determined from the roots of Eq. (5), which are t 0 , ðP=2Þ  t 0 and P þ t 0 , where t 0 ¼ ðP=2pÞ arcsinðQ a =Q s Þ. The corresponding inflow and out flow volumes are as follows:

  Q s ðP  4t 0 Þ Q a P 2pt 0 ; cos þ 2 p P t0   Z Pþt0 Q ðP þ 4t 0 Þ Q a P 2pt 0 : V in ¼ Q ðtÞdt ¼ s cos  P t 2 p P 2 0 V out ¼

Z

P t 2 0

Q ðtÞdt ¼

ð6Þ ð7Þ

Thus, as Q s ! Q a the potential for a significant measurement error reduces. The maximum potential for measurement error exists when Q s ¼ 0 or Q a =Q s is large. 6.1. Case: Q s ¼ 0 For the above condition the flow volumes are V out ¼ Q a P=p and V in ¼ Q a P=p, which sum to zero each wave period. A simple error analysis can be made by introducing the relative measurement errors in and out as follows:

QP ¼ 6. Wave-induced measurement error

ð5Þ 3

V out out þ V in in Q a ð   Þ; ¼ P p out in

ð8Þ

where Q P ½L3 T1  represents the net flow each period that results from either a directional flow bias through the instrumentation or a net measurement error that results from an inability of the instrument to measure and integrate inflow and outflow identically. For a relatively large systematic error, e.g., out  in ¼ 1, the rate of error accumulation is around 30% of Q a . This also corresponds to the rate of error accumulation for a uni-directional seepage meter with in ¼ 0 and out ¼ 1. For a 10% error between seepage inflow and outflow, i.e., out  in ¼ 0:1, the rate of error accumulation would be around 3% of Q a . Overall, a systematic error is likely to be significant only if both out  in and Q a =Q s are large. In such cases the potential exists to misinterpret the error as a real seepage.

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7. Conclusions

Acknowledgements

Wave action is an important control on seepage across marine beds and should not be neglected without justification in directmeasurement studies of SGD. In particular, it is clear that wave climate should be recorded in seepage-meter studies, including wave height, amplitude and wavelength. Continuous measurement of bottom currents at seepage meter locations also would be an advantage. Combined with a knowledge of the sediment permeability, this information would enable a theoretical assessment of the potential for wave-induced seepage into and out of the sediment bed, as well as the potential for wave-induced apparent seepage. Continuous or sub-wave frequency measurements of seepage are required to make the above assessment. Ultrasonic-type [21] and electromagnetic-type [27] seepage meters offer the best potential because the flow measurements are quasi-instantaneous. Lee-type and tracer-type (heat and dye) seepage meters are not suited for use in marine environments with prominent wave action because they cannot resolve seepage measurements at sub-wave frequency. It is possible that these instruments can accurately integrate wave-induce seepage but this utility has not been proved and contributes significant uncertainty to those measurement studies. When deploying seepage meters in wave-affected environments, every effort should be made to ensure that the benthic chambers are stationary and firmly seated against the sediment, particularly if high wave energy is present. It is advisable that benthic chambers have rigid constructions that do not flex or distort, even at micron-scale, under the action of wave- and current-induced forces. In the absence of experimental data, it is suggested that rigid hemispherical domes should be used in preference to other geometries. In particular, the traditional ‘drum top’ benthic chamber appears to be a poor design for marine conditions because the flexible top surface of the drum is susceptible to movement cause by wave-induced lift forces. Systematic measurement errors associated with wave-induced seepage flows could be significant if the error is large or wave-induced flow is the dominant component of the seepage. Measurement errors are less likely to occur using ultrasonic-type and electromagnetic-type seepage meters in which the analyser unit has symmetric geometry and the internal flow is unimpeded by protruding components (e.g., heater elements, thermistors and injection ports).

The research described in this paper was supported by CSIRO Land and Water, Australia. The authors wish to thank Dr Ian Webster and Dr Mike Trefry from CSIRO Land and Water, and the four anonymous reviewers for AWR.

Appendix A. Heat-pulse method for seepage measurement The seepage meter deployed in Cockburn Sound used heat as a tracer for bi-directional fluid motion [36]. Water exchange across the sediment–water interface within a 600-mm diameter benthic chamber was amplified by a factor 3600 by routing the flow through a vented, 10-mm diameter tube containing an in-line heater and thermistor array. That configuration enabled measurements of specific discharge to a minimum value of around 1 0:5 lm s1 ð4 cm d Þ which corresponded to a flow rate in the analyser tube of approximately 2 mm s1 . In operation, the heater was pulsed for 0.2 s and the direction and velocity of flow in the analyser tube was estimated based on the direction and speed of travel of the heated water. The heater element and thermistors were equally spaced at 10-mm intervals and temperature at each thermistor was recorded discretely at frequency 5 Hz. The water around the heater was raised to a temperature that was detectable by the thermistors but not to the point that natural convection perturbed the flow or there was excessive residual heat by the time of the next measurement. The analyser tube had horizontal orientation in all deployments. Seepage rates were determined from calibration curves (Fig. A.1) relating the apparent flow velocity in the analyser tube to the true flow rate measured by electronic mass balance. The apparent flow rates differ from the true flow rates because the transport of heat in the analyser tube was by advection and conduction. In general, the calibration models had larger error bars at high flow rates because the heat-pulse travel times were smaller and had larger relative errors. The minimum measurable flow rate was approximately 1 mm s1 . At lower flows, the Péclet number was sufficiently small that most of the heat transport was achieved by conduction. In that situation, the heat-pulse travelled away from the heater at approximately the same speed in both directions and a reliable flow rate could not be determined.

(b) Outer Thermistors

(a) Inner Thermistors 12 -1

Seepage Meter Flux, mm.s

Seepage Meter Flux, mm.s

-1

12 10 13

8

16

15 17

6

15

48 28 14

4

23

20

Linear fit of the means Slope = 0.965 R = 0.995 P-value < 0.0001

2

10

16 13

15

8 17

6

48 28

4

15

23 14 Linear fit of the means Slope = 1.265 R = 0.998 P-value < 0.0001

20

2 0

0 0

2

4

6

8

10 -1

Mass Balance Flux, mm.s

12

0

2

4

6

8

10

12

-1

Mass Balance Flux, mm.s

Fig. A.1. Heat-pulse calibration curves for (a) inner thermistor pair and (b) outer thermistor pair; data labels indicate the number of measurements at each flow rate, a whisker indicates the range of measured values and a symbols indicate the mean value.

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Appendix B. Derivation of apparent seepage rates B.1. Generic-shape chamber with vertical sides and vertical displacement

b Tilt motion: aðtÞ ¼ 2L sin xt. Seepage length: lðtÞ ¼ cosL a. Seepage area: AðtÞ ¼ lW. a  cosb aÞ. Enclosed water volume: VðtÞ ¼ LWðH  L tan 2 b sin xt xt 0 bxðLþ2b sin 2L Þ secb sin ðtÞ 2L Specific discharge: q ðtÞ ¼ VAðtÞ  cos xt. 4L

Amplitude of specific discharge: qa ¼ b4x. (circle)

B.4. Hemispherical chamber with tilting in two dimensions

H

h

R

Water Sediment

b

(circle)

Vertical motion: mv ðtÞ ¼ a2 sin xt. Embedded depth: dðtÞ ¼ b  mv . Cylinder height: hðtÞ ¼ H  d. Seepage area: A ¼ pR2 . Enclosed water volume: VðtÞ ¼ pR2 h. 0 cos xt. Specific discharge: q ðtÞ ¼ V AðtÞ ¼ ax 2 . Amplitude of specific discharge: qa ¼ ax 2

pivot point

a

b Tilt motion: aðtÞ ¼ 2L sin xt. Dome height: hðtÞ ¼ R  b  R sin a. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dome base (seepage) radius: rðtÞ ¼ R2  ðb þ R sin aÞ2 . Seepage area: AðtÞ ¼ pR2 . 2 3 Enclosed water volume: VðtÞ ¼ pRh  13 ph . 0 ðtÞ ¼ Specific discharge: q ðtÞ ¼ VAðtÞ b sin xt b sin xt b sin xt bx cos 4R ðbRR sin 4R ÞðbþRþR sin 4R Þ cos xt. xt 2 4ðR2 þðbþR sinb sin Þ 4R Þ  2 2 bx Amplitude of specific discharge: qa ¼ RR2 b . 4 þb2

(circle)

h

r

Water Sediment

r

b

B.2. Hemispherical chamber with vertical displacement

R

h

R

Water

b Sediment

B.5. Cylindrical chamber with flexing top

Vertical motion: mv ðtÞ ¼ a2 sin xt. Embedded depth: dðtÞ ¼ b  mv . Dome height: hðtÞ ¼ R  d. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Dome base (seepage) radius: rðtÞ ¼ R2  d . Seepage area: AðtÞ ¼ pr 2 . 2 3 Enclosed water volume: VðtÞ ¼ pRh  13 ph . 0 ðtÞ ¼ ax cos x t. Specific discharge: q ðtÞ ¼ VAðtÞ 2 . Amplitude of specific discharge: qa ¼ ax 2

volume

(circle)

h

R

H

Water Sediment

Flexing motion (dome height): hðtÞ ¼ 2c sin xt. 2 þR2 . Dome radius of curvature: rc ðtÞ ¼ h 2h 2 Seepage area: A ¼ pR . 2 3 Enclosed water volume: VðtÞ ¼ pR2 H þ pr c h  13 ph .

B.3. Box chamber with tilting in two dimensions

0

2

Specific discharge: q ðtÞ ¼ V AðtÞ ¼ cxð4R

(rectangle)

þc2 sin2 xtÞ 16R2

cos xt.

W

. Amplitude of specific discharge: qa ¼ cx 4

References

H b piv o axi t s

l L

a

Water Sediment

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