Boundary conditions

The modeling of periodic groundwater flow in a regional scale model domain requires a high computational effort. To reduce the computational effort, a step- wise procedure was used to compute the overheight:

• Constant mean sea level (CMSL) model. First, the development of a fresh- water lens in an initially saline domain (with concentration C = 1) was simulated using constant boundary conditions (Figure 5.3a). The hydraulic head in the offshore region S is equal to mean sea level, whereby the water flowing into the model domain has a concentration C of 1. In the hinterland region P , the water level is controlled with level hP, again having an inflow

concentration C = 1. In the region L, a recharge flux was applied with con- centration C = 0. The simulations were ended when the total salt mass in the model reached a constant value.

• Periodic sea level (PSL) model. The concentrations and hydraulic heads of the CMSL model were used as initial conditions for the periodic sea level model. The differences compared to the CMSL model are the boundary con- ditions in the intertidal area (B) and the offshore region (S), where a tidal boundary condition was applied (Figure 5.3b and c). The simulations were ended when the time-averaged hydraulic heads (measured over one tidal cycle) and salt concentration in the intertidal area reached a constant value. • Time-averaged sea level model (TASL) model. The results of the peri- odic sea level model were used as the initial condition for the time-averaged sea level model. In this model, constant boundary conditions were applied, similar as in the constant mean sea level model. In the intertidal area and offshore regions a constant head boundary condition was applied based on the time-averaged hydraulic heads (measured over one tidal cycle) of the periodic sea level model with concentration C = 1. The simulations were ended when the time-averaged hydraulic head at the groundwater divide and the total mass in the model reached a constant value.

The regional scale overheight was computed from the difference in hydraulic head between the CMSL and TASL models, whereby the hydraulic head was analyzed at x = 0.5L. In order to validate this step-wise approach, the TASL model was compared with a model wherein only periodic sea level was simulated. This model is referred to as ’step-wise validation model’. The boundary conditions of the step- wise validation model are equal to the ones used in the PSL model, but the initial condition for this model is a saline domain with C = 1. The step-wise validation model was ended when the time-averaged hydraulic heads in the model reached a constant value.

L P S

a)

high tide mark

low tide mark no flow

no

flow no flow

all figures not at scale

sea level t = t + Δt

B

sea level t = t

inactive model cells t = t

actived model cells t = t + Δt active model cells

t = t msl intertidal area

b)

c)

Figure 5.3: a) Boundary conditions of the constant mean sea level (CMSL) model. b) Boundary conditions of the periodic sea level (PSL) model. α is the slope of the intertidal area, B is the width of the intertidal area. c) The GHB-DRN tidal boundary condition (Mulligan et al., 2011), with the method for a correct timing of the tidal signal in the intertidal area.

The tidal boundary condition in the intertidal area in the periodic sea level models consists of head-dependent flux boundary cells, which were implemented using the GHB-DRN (General Head Boundary - Drain) approach described by Mul- ligan et al. (2011). When, for the cells representing the surface of the intertidal area, the hydraulic head in the intertidal area is lower than the sea level during rising tide, inflow from a GHB cell occurs with an inflow concentration of 1. When the hydraulic head during falling tide is higher than the sea level and higher than the surface elevation of the intertidal area, there is outflow to a DRN cell. For further details, the reader is referred to Mulligan et al. (2011).

In the intertidal area, the maximum horizontal length of the cells was 5 m and the maximum thickness of the layers was 0.2 m. This is comparable with the number of cells Mulligan et al. (2011) used for discretizing the intertidal area. The cell size gradually increases away from the intertidal area to maximum values of 50 m for the horizontal and 5 m for the vertical dimension. Using stress periods of 15 minutes and 1-minute long flow and solute transport time steps, the following general form of the tidal oscillation was discretized:

h(t) = h0+ A sin(ωt − φ) (5.4)

where h(t) is the sea level at time = t, h0is the reference hydraulic head (mean

sea level), A is the tidal amplitude [L], ω is the frequency [T−1_{] ( 2 π T}−1_{, where T}

is the period [T]) and φ is the phase shift []. The effect of waves, spring and neap tides and wind effects on the water level were not considered. The terms ’tidal forcing’ and ’tides’ (a result of tidal forcing) therefore refer to diurnal variations.

The original GHB-DRN tidal boundary approach of Mulligan et al. (2011), only prescribes the boundary condition for the uppermost cells that represent the sur- face of the intertidal area. If the slope of the intertidal area is small and the hydraulic conductivity of the intertidal sediment is large (Table 5.1), a relatively large number of cells fall dry during falling tide (i.e., they become inactive in the simulation). Subsequently, upon inundation of the intertidal area during rising tide, a delay of the tidal signal occurs. This can be attributed to the method of activating the inactive cells again; cells are allowed to be wetted only from below, as wetting from the horizontally adjacent cells leads to non-convergence of the flow solution (Mulligan et al., 2011). Therefore, the GHB boundary condition was prescribed for one stress period at the semi-saturated cell and across all cells that fall dry during ebb, as soon as inundation of that cell occured due to the rising water level (Figure 5.3c). This approach effectively assumes that the saturation of the unsaturated zone is instantaneous, and that the vertical downward flow of infiltrating seawater is not associated with a large hydraulic head gradient. For implementation, this approach means that one has to know the number of de- saturated cells beforehand. Since this is not possible to determine a priori, this was done by trial and error for each model simulation. The forced activation of cells allows for the use of a relatively high (-0.1, or 10 % of the cell thickness and only wetting from below) WETDRY parameter of the BCF package. As a result, no problems with the convergence of the flow solution were encountered.

To justify this approach, a comparison was made with a simulation using the recently developed MODFLOW-NWT code (Niswonger et al., 2011). MODFLOW- NWT allows for a robust simulation of intertidal groundwater flow. Density vari- ations were ignored and salt transport was not simulated. The results indicate a correct timing of the tidal signal, and negligible differences in hydraulic heads and fluxes between the models. For brevity, these results are omitted in this chapter.