(1a)L.
Li and D.
A. Barry
School of Civil and Environmental Engineering, The University
of Edinburgh, Edinburgh, EH9 3JN, U.K.
Abstract: Field and laboratory observations have shown that a relatively low beach groundwater table enhances beach accretion while a high water table promotes beach erosion. These observations have led to the beach dewatering technique (artificially lowering the beach water table) for combating beach erosion. The aim of this study is to quantify the interactions between the ocean and coastal aquifer. Such interactions affect swash sediment transport and beach profile changes. A process-based numerical model is developed to simulate the interacting wave motion on the beach, coastal groundwater flow, swash sediment transport and beach profile changes. The non-linear shallow water equation is modified to simulate swash/backwash motion interacting with the beach groundwater. Saturated flow in the coastal aquifer is governed by the Laplace equation. An additional term is added into the free surface boundary conditions for the water table to incorporate capillary effects. The instantaneous cross-shore sediment transport rate is calculated according to Bagnold’s sediment transport model. The net sediment transport rate is obtained for every swash/backwash cycle and is used to calculate the resulting beach profile changes. Results of model testing demonstrate that the model replicates (1) bar/berm formation at beaches under different wave conditions, (2) the equilibrium state of a beach exposed to constant wave conditions, and (3) accretionary effects of a low beach water table on beach profile changes.
Sediment transport processes in the swash zone are of fundamental importance to beach morphology and shoreline stability. Ultimately, it is the direction of the net sediment transport in the swash zone that determines the beach status, i.e., whether it is eroded, accreted or in an equilibrium state.
The swash/backwash motion, i.e., wave run-up and run-down in the swash zone, provides the driving force for swash sediment transport. The upwash moves sand on-shore while the backwash transports it offshore. The hydrodynamics of these processes are very complicated, involving highly non-linear transformations of broken and unbroken waves on a sloping beach. Moreover, the wave motion interacts with the beach groundwater flow. Seawater may infiltrate into the sand at the upper part of the beach (around the shoreline) during swash wave motion if the beach groundwater table is low. In contrast, groundwater exfiltration may occur across the beach with a high water table. Such interactions have been demonstrated to have a considerable impact on the swash sediment transport in field studies [Grant, 1948; Duncan, 1964]. Seawater infiltration under a low water table was found to enhance on-shore sediment transport, whereas groundwater exfiltration under a high water table promote offshore sediment transport. These field observations have led to the beach dewatering technique (artificially lowering the beach water table) for combating beach erosion [Turner and Leatherman, 1999]. Although some success has been gained in the practice of this technique, the understanding of the processes involved and underlying mechanisms is incomplete.
Quick [1991] analyzed the effects of infiltration and exfiltration on the beach slope through the force balance on a control volume (defined within the swash zone) averaged over a wave cycle. He concluded that beach gradient increases with sediment size due to the effects of increased permeability. Groundwater effects were shown to enhance the on-shore shear stress on the sediment particles and lead to beach accretion. In the analysis, infiltration was assumed to occur at the upper part of the beach, accompanied by exfiltration at the lower part. Such a flow exchange pattern is typical of a low beach water table situation. In addition to modifying the shear stress on the sediment particles, the infiltration and exfiltration processes affect the normal stress on sediment particles [Nielsen, 1992] and hence sediment movement. Turner and Masselink [1999] found that the infiltration/exfiltration in the swash zone can increase the sediment transport rate by up to 40% of the peak transport rate during upwash and decrease it by 10% during the backwash.
There have been many investigations into swash zone hydrodynamics [e.g., Guza and Thornton, 1982; Packwood, 1983]. Of these studies, only Packwood [1983] included the effects of beach groundwater flow. The infiltration, approximated as a purely vertical flow, was found to reduce the maximum wave run-up and the strength of the backwash. Although much research has been conducted on sediment transport in general, swash sediment transport processes have received little attention. Only a handful of field studies has been reported [Horn and Mason, 1994]. Moreover, quantitative inter-relation between the hydrodynamics and sediment transport processes were seldom examined in these studies except for Hardisty et al. [1984] and Masselink and Hughes [1999]. In these studies, the analysis was based on Bagnold's sediment transport model [Bagnold, 1966]. None of these studies, however, considered the effects of beach groundwater flow explicitly.
Turner [1995] reported a modeling study that included the effects of groundwater on swash sediment transport. In the model, the net sediment transport rate is related to the up-wash sediment transport rate by a parameter, M, which is determined by the local slope and the equilibrium beach slope. To simulate groundwater effects, different values of equilibrium slope are used for the saturated (below the effluent line of the water table) and unsaturated (above the effluent line) beach face. Turner’s [1995] model produced some interesting results, for example, the slope break typically found within the intertidal zone of macro-tidal beaches. However, the model is primarily inductive and its representations of the swash sediment transport and groundwater effects are rather crude. In particular, questions regarding the physical interpretation of M and its dependence on other physical parameters are yet to be addressed.
The objective of the present study is to develop a process-based numerical model for studying the effects of infiltration and exfiltration on swash sediment transport and beach profile changes. The interacting swash/backwash and beach groundwater flow are to be simulated. The resulting hydrodynamics are then used to predict the sediment transport rate and beach profile changes. Various difficulties encountered in measurements have hindered the study of swash sediment transport in field and laboratory conditions. The numerical approach is expected to compliment experimental studies and provide some new insight into the phenomena.
There are three major processes involved in this problem: (1) wave motion on the beach; (2) beach groundwater flow; and (3) cross-shore sediment transport in the swash zone (Figure 1). All of these three processes are assumed to be two-dimensional in the cross-shore plane. In the following, we shall describe mainly the modelling aspects of these three processes and how they interact with one another.
Wave motion on the beach has been modeled using the depth-averaged nonlinear shallow water equation (SWE [Peregrine, 1972]). The original SWE is modified here to incorporate the infiltration/exfiltration on the beach face and bottom friction, i.e.,
(1a)
(1b)
where h is water depth, u is water flow velocity, t is time, x is the horizontal coordinate (cross-shore direction), I is the infiltration (negative) or exfiltration (positive), b is the beach angle, g is the magnitude of gravitational acceleration, and f is the friction factor. The coefficient f1 equals 0 if I > 0 (exfiltration), and 1 if I < 0 (infiltration). The term 1/2f|u|u is the quadratic approximation of the bottom friction [Kobayashi et al., 1987].
Hibberd and Peregrine [1979] developed a dissipative finite difference scheme based on the Lax-Wendroff conservation method [Lax and Wendroff, 1960] for solving the SWE to simulate a uniform bore's run-up on a beach. This scheme was found to be applicable for simulating the swash and backwash motion of wave trains [Kobayashi et al., 1987]. A prescribed incoming wave is combined with the reflective wave (calculated from the numerical solution based on the linear wave theory) to determine the seaward boundary conditions [Kobayashi et al., 1987]. At the landward side is the shoreline, a moving boundary. A special procedure developed by Hibberd and Peregrine [979] is applied to determine the moving shoreline for every time step.
The shallow water
equations and the numerical scheme were originally used by Hibberd and Peregrine [979] to simulate the run-up of an incident
uniform bore. The water depth inside the breakpoint was used as the input at the
seaward boundary. The simulation involved only wave evolution after breaking.
Although the shallow water equations do not include any direct mechanism of
energy dissipation effects, dissipation is implicit in the bore representation [Hibberd
and Peregrine, 1979]. Such a boundary condition based on a bore is however
restrictive and not always applicable in reality. Since existing wave theories,
such as cnoidal and Stokes wave theories, cannot describe the asymmetry of the
wave profile inside the breakpoint [Flick
et al., 1981], the seaward boundary needs to be located outside the
breakpoint, where an incident wave train can be specified using existing wave
theories [Kobayashi et al., 1987]. The
simulation may then involve wave evolution through breaking. In other words,
surf-zone dynamics are also included in the simulation. The one dimensional
shallow water equations no doubt are not capable of predicting the details of
the breaking process.
Despite its lack of an energy dissipation mechanism, the shallow water approach has been found to provide quantitatively correct run-up results even in the parameter range of breaking waves. Such a capability is well documented but unexplained [Titov and Synolakis, 1995]. Numerical dissipation is believed to play an important role in this model performance. However, the numerical dissipation is not controllable and its quantification is ad-hoc.
Beach groundwater responds to sea level oscillations, including low-frequency tide and high-frequency wave motion. The beach groundwater table has been demonstrated to fluctuate with the tide and waves. As the low frequency tidal fluctuation propagates inland in the aquifer, the phase changes. In contrast, the high frequency water table fluctuations appear to respond to the shoreline movement simultaneously. Two different mechanisms have been identified for these responses: horizontal mass movement is responsible for the aquifer's responses to low frequency tidal fluctuations, while capillary effects control the aquifer's response to high-frequency fluctuations [Li et al., 1997].
It is important to include these two mechanisms in modeling beach groundwater flow, especially its interaction with the ocean. Li et al. [1997] have developed a model which incorporates these two mechanisms to simulate beach groundwater flow. This model solves the Laplace equation for saturated flow in the aquifer,
(2)
where f
is the velocity potential,
(P is the pressure, r
is the water density and z is the
vertical coordinate). Note that the aquifer has been assumed homogeneous. Such
an assumption may not apply strictly at natural beaches. However, it simplifies
the complicated modelling to a workable extent in the first instance. Also such
an assumption should not reduce the significance of any findings concerning the
groundwater effects on sediment transport. The capillary effects are included in
a modified free surface boundary condition for the water table,
(3)
where K and ne are the hydraulic conductivity and effective porosity of beach sand, respectively. n is the local coordinate on the boundary (i.e., the water table) in the normal direction outward from the flow domain, Bc is the thickness of the capillary fringe and g is the angle between the water table and the horizontal axis. The first and second terms on the right hand side of equation (2) represent the first and second mechanisms for coastal groundwater responses to the oceanic oscillations, respectively.
In addition to the water table, three other boundaries exist (Figure 1): the landward boundary, the seaward boundary (i.e., the beach face) and the impermeable boundary at the base. The landward boundary condition is prescribed by a constant head (i.e., constant potential). At the base, the flux is zero and hence, ¶f/¶n = 0. The boundary conditions at the beach face are more complicated. Depending on whether the shoreline and the exit point of the water table couple with each other, a seepage face may exist. Once the boundary conditions are determined, equation (2) can be solved using a variety of numerical techniques. The boundary element method (BEM) is used in this study.
Sediment transport in the swash zone behaves like a sheet flow and its transport rate can be described by Bagnold's sediment transport formula for bed load [Hughes et al., 1997], i.e.,
(4)
|
|
In the sediment transport model, the effects of sediment size are included through the critical flow velocity (ucri). The relation between the critical velocity and sediment size is based on Shield's criterion. Apart from the effects on the critical flow velocity, sediment size influences the transport process of sand indirectly through the groundwater. A large sediment size results in a large permeability and hydraulic conductivity of the porous medium. This will increase the infiltration/exfiltration rate and hence affect wave motion that provides the driving force for sediment transport. Note that the model is yet to include the modification of the normal stress on the sediment due to infiltration and exfiltration.
Once the flow velocity is determined from the wave motion model, we can calculate the instantaneous sediment transport rate along the beach face. The net sediment transport rate (qnett) is obtained for each wave cycle and is then used to deform the beach. The beach profile change is governed by the continuity equation for sand, i.e.,
(5)
where zb is the elevation of the beach face and rs is the density of the sand.
As mentioned previously, the seaward boundary has to be located in a relatively deep region (in the surf zone) where the incoming wave conditions can be specified. This may raise questions regarding the applicability of equation (4) over the entire simulation domain, especially in the surf zone.
The three processes described above interact with one another. In the numerical model, the interactions are simulated through the boundary conditions of each process. For example, the boundary conditions at the beach face for the groundwater flow are determined by the shoreline position and the local sea surface elevation simulated by the wave model. The groundwater flow model is not fully coupled with the wave motion model. Rather, the results from the wave simulation of the previous time-step are used to determine the boundary conditions for the groundwater flow model at the current time step. The infiltration/exfiltration calculated from the groundwater flow model is then used in the wave motion model for the next time-step computation. An iterative procedure would be required if full coupling of the wave motion model and the groundwater flow model was attempted. For the purpose of simplicity, we adopt the partial coupling instead of a full coupling at this stage. The difference is expected to be of secondary importance, especially when a small time step is used.
The instantaneous sediment transport rate (q(t)) is calculated at each cross-shore grid point using the velocity from the wave motion model. The sum of the instantaneous rates over a wave cycle gives the net sediment transport rate, which is used to predict the beach profile changes. The location of the nodes at the beach face in the groundwater flow model, and the local beach slope in the wave motion model and the sediment transport model are then adjusted according to the beach deformation.
The cross-shore sediment transport behaves differently under different wave conditions at a beach. Offshore sediment transport dominates if the beach is exposed to waves with high wave steepness (high-energy waves, large amplitude and high frequency). On the other hand, on-shore sediment transport prevails if the waves have small wave steepness (low energy waves, small amplitude and low frequency). The transition between the two cases occurs for waves with intermediate steepness in which sand moves both on- and off-shore [Mizumura et al., 1991]. Correspondingly, three types of beach profiles exist: (I) the berm type, sediment moved on-shore; (II) the bar/berm type, sediment moved on- and off-shore; and (III) the bar type, sediment moved off-shore. The beach type can be related to a non-dimensional number, defined as
(6)
where the numerator is simply the deep water wave steepness and denominator combines the effects of sediment size and the mean beach slope. The following criteria for classifying the three types of beach profile have been suggested for a laboratory model beach,
(7a)
(7b)
(7c)
while for a prototype beach, critical values of 4 and 8 are replaced by 9 and 18, respectively [Mizumura et al., 1991].
Seven simulations were conducted for different values of C. The simulations started with a uniform beach slope (tanb = 0.1) and were run for 2800 waves. After that, the beach reached an equilibrium state (more details in the next section). The values of the model parameters for each simulation are listed in Table 1. In Figure 2, we show typical results of the three types of beach profiles from simulations 1, 4 and 7. As the wave steepness increases, the beach type changes from I to II and then III. These beach profiles are the results of cross-shore sediment transport and Figure 3 shows the net sediment transport rate for each simulation after 1000 waves. In simulation 1, sediment was transported predominantly on-shore, forming the berm. In simulation 4, sand moved both on-shore forming the berm and off-shore forming the bar. In simulation 7, opposite to simulation 1, sediment transport was predominantly off-shore.
To compare the model predictions of the beach types with the empirical criteria, we plotted the beach types using different symbols against C (Figure 4). The beach types were determined by the formation of berm, bar or both in the final beach profile. The dependence of the beach type on C with the three divisions is evident.
An important concept in the study of beach morphology is the equilibrium profile that a beach will attain if exposed to constant wave conditions for a sufficiently long time. The realization of an equilibrium beach profile is difficult under natural conditions due to the variability of wave conditions. However, numerous laboratory experiments have demonstrated that a beach approaches an equilibrium state under constant wave conditions and that the rate of beach change diminishes with time [Larson, 1988]. In terms of sediment movement, the magnitude of the net transport rate decreases with time. The maximum net sediment transport rate was found to be inversely related to time [Larson, 1988], i.e.,
(8)
where qm0 and qm are the initial and the time-varying peak net transport rate, respectively; and a is the rate coefficient for the decay of the peak net transport rate.
The time-varying net sediment transport rates for each simulation were recorded. A time series of these net transport rates from simulation 4 is shown in Figure 5a. The magnitude of cross-shore sediment transport clearly decreased with time. The peak transport rates (i.e., the minimum value of each time series) were collected at all time steps and then fitted to equation (8) with a good agreement obtained (Figure 5b). The model seemed to describe the temporal behaviour of the peak sediment transport.
The fitted values of qm0 and a are 1.3 ´ 10-4 kgs-1m-1 and 0.89 h-1, respectively. The value of qm0 may be used to estimate the extent of the initial profile changes at a beach that is subject to changing wave conditions. For example, the early stage beach erosion due to a storm may be predicted by qm0 estimated for high energy storm waves. On the other hand, the time scale for approaching the beach equilibrium may be given by 1/a.
Fig. 5
Simulated
beach equilibrium. b, circles are simulated data and solid line from fitting.
Two simulations (simulations 8 and 9 in Table 1) were conducted with lowered beach water tables achieved by setting low landward groundwater levels and with two different sand particle sizes. For the purpose of comparison, a reference simulation was conducted without groundwater effects. Notice that the beach was exposed to relatively energetic waves in the simulations.
Results from the simulations show that significant infiltration occurred in the swash zone (the upper part of the beach), accompanied by less intensive exfiltration at the lower part of the beach (Figure 6a). The infiltration/exfiltration rate has been averaged over a wave cycle and a negative value of the flux signifies infiltration. Such a flow exchange pattern implies that there exists a local circulation below the beach face, a phenomenon that has been demonstrated by Longuet-Higgins [1983] and Li and Barry [1999]. The magnitude of infiltration is larger than that of exfiltration due to the lower water table. Also, as the hydraulic conductivity in simulation 9 is larger than in simulation 8, both infiltration and exfiltration in the former are greater than those in the latter.
To examine the infiltration effects on the sediment transport process, we calculated the net cross-shore sediment transport rate at t = 1.42 h (after 800 waves), i.e., the sum of the on-shore sediment transport during upwash and the off-shore sediment transport during backwash. The results are compared with those from the reference simulation. Without the infiltration, the sediment was transported off-shore solely (Figure 6b). The effects of infiltration led to the enhancement of the on-shore sediment transport and the reduction of the off-shore sediment transport. Consequently, the net sediment transport rate changed. In both simulations, the net sediment movement at the upper part of the beach altered its direction from off-shore to on-shore (Figures 6b). The extent of the increase in the net on-shore sediment transport depends on the magnitude of the infiltration; it is larger in simulation 9 than in simulation 8. In the deep water region (x < 7 m), the groundwater flow was small and had little effect on sediment transport.
Fig. 6 Simulated beach groundwater flow (a) and net sediment transport rates (b). Dashed line shows data from sim 9, solid line from sim 8 and dot-dashed line from the reference simulation
The responses of the beach profile displayed in Figure 7 show that beach accretion occurred as a result of the enhanced on-shore sediment transport. A berm has been formed in both simulations. The predicted profiles with and without infiltration effects are much similar to those from laboratory experiments. Without groundwater effects, the beach, exposed to high energy waves, is eroded and an off-shore bar is formed (upper panel). Under a low water table, infiltration leads to beach accretion and, hence, the formation of a berm (lower panel). Again, the accretion is greater in simulation 9 due to the larger hydraulic conductivity, resulting in a steeper beach slope than that in simulation 8. This indicated that the sediment size affects the beach slope (particularly in the swash zone) as it determines the permeability of the beach sand.
Fig.7 Simulated beach profiles. Dashed line shows data from sim 9, solid line from sim 8 and dot-dashed line from the reference simulation.
The ratio of the total infiltration and exfiltration over a wave cycle to the total swash volume in these four simulations is estimated to range from 0.54% to 3.5% [Li et al. 1999], a rather small proportion. However, the ratio of the net sediment transport rate to either the on-shore or off-shore sediment transport rate is also small and ranges from 1% to 5% at maximum; in other words, it is in the same order of magnitude as the ratio of infiltration/exfiltration to the swash volume. Note that the net sediment movement over a wave cycle is due to the difference between the on-shore sediment transport during the upwash and the off-shore sediment transport during the backwash. Therefore, even though there is only a small water exchange across the beach face relative to the total swash volume, this amount of water seems significant enough to modify the net sediment transport and so affect the resulting beach profile.
Table 1 Values of model parameters used in the simulation.
|
Simulation No. |
Model parameters |
|||
|
H0 (m) |
T (s) |
K (m/s) |
HLW (m) |
|
|
1 |
0.08 |
3.19 |
|
|
|
2 |
0.12 |
3.19 |
|
|
|
3 |
0.16 |
3.19 |
|
|
|
4 |
0.20 |
3.19 |
|
|
|
5 |
0.24 |
3.19 |
|
|
|
6 |
0.28 |
3.19 |
|
|
|
7 |
0.32 |
3.19 |
|
|
|
8 |
0.32 |
3.19 |
0.0004 |
0.6 |
|
9 |
0.32 |
3.19 |
0.0008 |
0.6 |
d
= 1 m, f = 0.1, initial uniform beach
slope = 0.1, ne = 0.45 and L
= 45 m.
A process-based model has been developed to study swash sediment transport and beach profile changes. In particular, the effects of infiltration/exfiltration on beach stability are included in the model. Three sub-models are formulated for the three physical processes: wave motion on the beach, beach groundwater flow and swash sediment transport, leading to the prediction of the beach profile changes. The interactions among the processes are incorporated in the model. The results of model testing presented in this paper demonstrate that the model is capable of replicating the beach system's behaviors, in particular, bar/berm formation, beach equilibrium and accretionary effects of low beach water tables on sediment transport and beach profile changes.
The model is yet to be validated against experiments. There are also theoretical concerns about the model formulation: for example, (1) application of the SWE in the deep water region is likely to over-predict the wave steepening (the nonlinear effects) due to the lack of wave dispersion; (2) application of the lumped bed-load sediment transport model to surf zone is questionable since suspended load is significant there; and (3) how the numerical dissipation is compared with the energy dissipation due to wave breaking. Improvement on the numerical solution should also be carried out. The numerical scheme used here is second-order accurate but oscillatory at the steep wave front [Richtmyer and Morton, 1967]. These numerical oscillations become obvious and cause the solutions to become unstable when a very energetic wave condition is applied at the seaward boundary. Recently, Liska and Wendroff [1996] developed a composite scheme that combines Lax-Wendroff and Lax-Friedrichs schemes. The Lax-Friedrichs step serves as a consistent filter removing the unwanted numerical oscillations. Such a composite scheme should be incorporated in the future development of the model to enable the simulation of a wide range of wave conditions.
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