4.2. Rainfall on a Sloping Plane
An idealized system with rainfall on a sloping plane has been used by Sulis et al. [
45], Maxwell et al. [
46], Wu et al. [
47], and provides validation of the surface-subsurface coupling approach in Frehg. The system uses a simple sloping plane in the
x direction with uniform properties in the transverse
y direction, which can be modeled as a 2D
x-
z vertical section. The setup of the model is provided in
Table 1 and
Table 2. The physical system is characterized by the plane length (
L), slope (
), and the model parameters discussed in
Section 3, above. The vertical and horizontal saturated hydraulic conductivities are set to a uniform value (
) in each test, with three different values tested (
,
,
). The discretization uses uniform values for the horizontal grid (
, vertical grid (
), and time step (
). The initial conditions are a uniform water table depth (
) and zero water depth above the sloping plane. Boundary conditions include a rainfall rate (
), rainfall duration (
), recession duration (
) and a fixed surface elevation at the downstream boundary. Surface runoff at
m is used for model-model comparison, which is consistent with Sulis et al. [
45], Maxwell et al. [
46]. Note that to use the Dirichlet downstream boundary condition for the free surface, the total plane length
L is doubled compared to that in Sulis et al. [
45], but the subsurface domain beyond
m is deactivated. The fixed free surface elevation at outlet is much lower than
m, which guarantees the measured runoff at
m is unaffected by the downstream boundary conditions. The selection of the rainfall rate and the three
values implies that
will require the subsurface to saturate before runoff is generated, whereas
and
will generate runoff while unsaturated.
Frehg is run separately with the DW and SWE algorithms for each of the
values in
Table 2. The results are compared with those simulated by the CATHY and ParFlow models [
45], which use the kinematic wave (KW) approximation of the SWE (see
Section 3.2). The comparitive study by Maxwell et al. [
46] reveals negligible difference between KW and DW models for this test problem. For low slopes and low rainfall rates inertial terms can be expected to be small, so the results from the SWE should be close as well.
Figure 3 shows the surface runoff, at
m of the sloping plane, which corresponds to the downstream outlet in Sulis et al. [
45].
Figure 3a corresponds to a saturated hydraulic conductivity of
m/s, which is higher than the rainfall rate. Thus, the unsaturated subsurface domain will be filled first before runoff is generated. In
Figure 3b,c, the hydraulic conductivity is less than the rainfall rate, meaning that infiltration and runoff generation occur simultaneously. Since a deeper initial water table is used in
Figure 3b,c, less runoff is generated despite their small conductivity. In all scenarios, Frehg has good agreements with CATHY and ParFlow in terms of runoff. Compared with solving the SWE, applying the DW approximation leads to slightly higher surface runoff, but the difference between SWE and DW does not exceed the difference between CATHY and ParFlow, indicating both SWE and DW are reasonable methods for this test problem. For
Figure 3a,b, the DW method produces the highest flow rate on the rising limb, which is likely due to the asynchronous coupling scheme [
46].
The depth and velocity at the
m are shown in
Figure 4. Similar data are not reported in Sulis et al. [
45], Maxwell et al. [
46], but are recommended as providing insight by more recent studies [
18]. For all test scenarios, the DW approximation overestimates velocity while underestimating depth. A similar finding is reported in Cea and Blade [
16]. Since flow depth affects inundation area, which affects surface-subsurface exchange, these results suggest the DW approximation could influence the surface-subsurface exchange in the intertidal zone that experiences wetting/drying. This idea is investigated in
Section 4.3 and
Section 4.4.
Figure 5 shows the numerical dimensionless numbers (Equations (
18)–(
20)) as functions of time. It can be seen that
is close to 1 for all tested conductivity values. Furthermore,
and
are orders of magnitudes smaller than
. Because these non-dimensional numbers represent the relative importance of terms in the momentum equations (see
Section 2), these results indicate that the pressure gradient and bottom friction are well-balanced and the inertia terms are relatively unimportant for this rainfall-runoff case. Thus, the use of the DW approximation is justified and the similarity of the SWE and DW results is expected for this test case.
4.3. Inundation of an Intertidal Zone
Of particular interest in modeling a tidally-driven marshland is the behavior of the surface-subsurface exchange with successive wetting and drying of a sloping landscape. As an idealized test case, we examine SWE and DW behaviors in the test domain shown in
Figure 6, which has a uniform sloping face with a bed slope of 0.001. The subsurface domain extends 3 m below the lowermost point of the sloping surface plane. This domain is 2D and driven by sinusoidal oscillation of surface elevation, which represents idealized tidal forcing with period
and amplitude
.
Model parameters for a baseline simulation are presented in
Table 3. According to Equation (
1) and (
2), the DW approximation favors large time scale (
T) and bottom friction (
n). As previously discussed, due to strong variation of spatial scales in tidal-driven marshes,
and
have limited practical use as criteria to assess the validity of the DW approximation. However, the basic scaling relations in these equations should still hold, i.e., the DW model should converge to the SWE model as
T and
n increase. Thus, the following three test cases are examined (
Table 4): (1) baseline with 24 h tidal period (
) and Manning’s n of 0.03, (2) reduced tidal period, and (3) reduced Manning’s n. The initial water table elevation is same as the initial tidal elevation, which is 0.1 m above the zero-elevation plane in
Figure 6. Note that the initial water table is set to a constant elevation, in contrast to the constant depth below surface used in
Section 4.2. A model spin-up period (
) is applied, during which the horizontal conductivity is increased to
the baseline value of
Table 3 to rapidly create reasonable head and moisture fields in the subsurface domain. Data are collected and analyzed from the SWE and DW simulations over the
testing period at the 1/2 and 3/4 locations shown in
Figure 6. The first corresponds to the tide’s neutral level (the center of the intertidal zone), and the second is the upper edge of the continuously inundated zone.
Figure 7 shows evolution of the surface water over two days of simulation. In frames (a), (b) and (c), the normalized inundation area is defined as the inundation area divided by the maximum inundation area of the baseline SWE model.
Figure 7a indicates the DW inundation area is similar to the SWE model, but has a slight phase lag. In
Figure 7b the phase lag is increased when
is halved, and in
Figure 7c the phase lag is decreased (and almost eliminated) when Manning’s n is reduced by an order of magnitude. In these figures we can also see that increased phase lag is associated with a slight damping of the inundation area. However, as illustrated in
Figure 7d–f, the phase lag in different tests does not significantly affect the depth evolution at the 1/2 and 3/4 sampling locations.
Surface profiles on the rising and falling tides are shown in
Figure 8 to illustrate how the phase lag affects the water distribution. These effects can be interpreted as the DW introducing an artificial dissipation that scales directly with the bed roughness and inversely with the time scale.
For insight into the behaviors illustrated above, the discrete dimensionless numbers
,
and
of Equations (
18)–(
20) are plotted in
Figure 9 for the SWE simulations. All three cases clearly meet the DW requirement with
, which is similar to that proposed by [
13]. This result indicates advective inertia (the nonlinear term in SWE, neglected in DW) is unimportant. However, if we focus on cases with larger
n, i.e.,
Figure 9a–d, we find conditions where the DW requirements of
and
are violated. In particular, we see
during several episodes when
, indicating that the unsteady term dominates the friction term, hence the DW approximation is violated. Thus, the discrepancies between DW and SWE in
Figure 7a,b are largely explained by neglect of the unsteady term that becomes episodically important.
The results with the lower value of Manning’s
n,
Figure 9e,f are intriguing. These indicate the approximations of the DW method are never satisfied, i.e.,
and
, which indicates that the SWE momentum reduces to a balance between pressure gradient and unsteadiness throughout the simulation. Furthermore, yet, the phase lag and inundation area errors for this case in
Figure 7c are substantially smaller than those in
Figure 7a,b where the DW approximation is only violated episodically. The explanation for this behavior lies in the numerical discretization approach to using the DW approximation in an unsteady solver. The DW approximation, Equation (
6), is formally a steady-state relationship between
and friction (driven by
u) that is used to close the unsteady equation for mass conservation, Equation (
3). Thus, although the unsteady term is not included in momentum, unsteadiness will still exist in the coupled equation set. Furthermore, it can be shown (see
Appendix A) that a discrete balance of unsteady momentum and the pressure gradient is equivalent to the DW approximation for small values of Manning’s n when
u is slowly varying relative to the time step. Note that the collapse of these equations under limited discrete conditions should not be taken as a reason to use the DW approximation for flows where
and
. Such conditions (with
) should arguably be modeled with a linearized unsteady momentum equation (e.g., a wave equation).
Figure 10 shows the surface-subsurface exchange flux in the intertidal zone and the fully inundated zone. In the intertidal zone, the exchange flux switches between infiltration (negative) and exfiltration (positive) in accordance to the tidal cycle. During falling tide, evaporation and topography-driven exfiltration leave the top layer of the subsurface unsaturated. During rising tide, a strong infiltration flux is observed that fills the unsaturated subsurface voids. In the fully inundated zone, the exchange flux is two orders of magnitude smaller and is monotonically downward during the entire simulation period. These results highlight the significant role of the intertidal zone in enhancing surface-subsurface exchange. Neglect of tidal wetting/drying and the variably-saturated subsurface flow is likely to significantly underestimate the exchange fluxes.
The value of the full SWE, as compared to the DW, can be seen in
Figure 10a,c,e. The surface water phase lag illustrated in
Figure 7 reappears here as a change in the timing of infiltration and exfiltration. The SWE model in
Figure 10 predicts higher maximum infiltration fluxes and longer exfiltration period because of its fast response to the tidal boundary condition. These results illustrate that model simplifications (i.e., using DW rather than SWE) affect both timing and magnitude of surface-subsurface exchange, especially in the intertidal zone.
4.4. Surface-Subsurface Exchange in the Trinity River Delta
The third test problem uses terrain of a real coastal wetland—the Trinity River Delta near Houston (Texas, USA). High-resolution lidar data was used to characterize the topography at the Trinity Delta and build the Trinity Delta Hydrodynamic Model (TDHM) reported in Li et al. [
48]. The TDHM used the FrehdC code [
40], which was a precursor of the Frehg code presented herein. Numerical studies with TDHM highlighted the important roles of boundary conditions, subgrid-scale topography and groundwater flow on surface hydrodynamics and salinity transport [
48]. However, TDHM only models fully saturated groundwater flow, which provides limited insights regarding the surface-subsurface exchange process at the intertidal zone. The Frehg model enables simulation of variably-saturated subsurface flow, which allows for a preliminary investigation of surface-subsurface exchange on a realistic wetland topography. Unfortunately, obtaining adequate field data for validation of both models remains an unresolved issue.
Figure 11 shows the bathymetry used in the present study at 150 m resolution, which is upscaled from available 1 m resolution lidar data. The coarse resolution was selected to minimize the computational costs while allowing simulation over complex topography. Admittedly, this coarse resolution would be insufficient for validating the model or conducting detailed analyses of flow, but it should suffice for the purposes of evaluating differences between the DW approximation and the SWE over a sufficiently complex system. Further details on the topographic data and the Trinity River Delta are available in Li et al. [
49]. All of the elevations used herein are given relative to the NAVD88 elevation datum.
Model parameters for the Trinity River Delta simulations are presented in
Table 5. In the surface domain, the river boundary condition on the north side of the domain is provided by a constant water level,
. The southern boundary is open water in Galveston Bay, to which we apply a time-varying tidal elevation as an open boundary condition. The tidal data is extracted from the NOAA record for tide & current station 8770613 (Morgans Point, Barbours Cut, Texas) for year 2018, which is available at 6 min intervals. The subsurface domain extends down to
m with a uniform vertical grid resolution. The bottom and side boundaries are all impermeable. The soil parameters are same as those in
Table 3. As in the simpler study in
Section 4.3, above, during
the model is run with increased horizontal conductivity to rapidly build up reasonable head and moisture profiles in the subsurface domain. Analyses (below) is based on
period that covers a 14-day spring-neap cycle with a 3-day buffer on either side.
A total of 4 test scenarios (
Table 6) are established to examine sensitivity to model simplifications and environmental processes. When evaporation is included, a constant evaporation flux of
m/s is applied to wet regions. Equation (
17) is used for dry regions. When wind stress is included, a constant wind speed of
2 m/s pointing towards the north direction is enforced. Note that all tests in
Table 6 are performed with the same
as listed in
Table 5 for consistency, but a much larger value of
can be used when surface flow is modeled with SWE.
Figure 12 shows the numerical dimensionless numbers at locations L1 and L2 defined in
Figure 11. L1 is located in a permanently-wet lagoon distant from the open boundary with surface fluxes in both
x and
y directions. In contrast, L2 is located in the intertidal zone close to the open boundary and has surface fluxes only in the
x direction. It can be seen that at L1,
is around 1 in both directions, indicating unsteadiness is important at this location, which is consistent with results for the simpler study of inundation in the intertidal zone, in
Section 4.3. Similarly,
is typically less than 1. We see flow direction becoming important with
, which is less than 1 in the x direction, but close to 1 in y direction. At L2,
is close to 1, while
and
are mostly much less than 1. However, a few higher values of
are also observed. Detailed examination (not shown) reveals that these high
values at L2 occur during high tide.
Overall, the above observations imply that due to unsteadiness, the DW approximation is generally invalid in the Trinity Delta unless the depth is small. This result seems contradictory to the simple scaling argument of Equation (
2) that states increasing importance of unsteadiness at small depth. The reason might be the thin-layer drag model used in Frehg [
10], which increases bottom friction at small depth and stabilizes the flow field (this drag model is deactivated in
Section 4.2 for a consistent comparison to the benchmark results). It should be noted that the coarse grids used in this test problem prevents investigation on the effects of small-scale topographical features. At finer grid resolution with more complex topographical features, advective inertia could become important, which brings further question to the validity of the simplified DW equations [
50].
Figure 13 shows the modeled depth, exchange flux and water content at L1 and L2.
Figure 13a indicates that, although L1 is relatively far from the tidal boundary, the depth (hence water surface elevation) still oscillates at tidal frequency. The reason is that we use a relatively coarse grid resolution (
m), which neglects small-scale water-blocking structures and enhances surface connectivity [
10]. At L1, the 3 scenarios with SWE produce negligible differences, but the DW model overestimates amplitude of depth oscillations.
Figure 13b shows that during spring tide, the intertidal zone experiences wetting/drying transitions. During neap tide, the intertidal zone experiences continuous dry period. The effect of evaporation in removing thin layers of water during falling tide is evident. Without evaporation, L2 never becomes completely dry during spring tide. Compared to SWE+evap, additional inundation events are found on day 1 and day 3 when wind effect is modeled. Additional inundation events are found on day 1, 3, 12, 17 and 20 when SWE is replaced by DW.
These differences in modeled inundation patterns have influence on surface-subsurface exchange.
Figure 13c–f display the exchange flux and the water content of the top-most subsurface cell at L1 and L2. As long as evaporation is modeled, the peak infiltration flux at L2 is two orders of magnitudes larger than that at L1, which highlights the critical role of the intertidal zone in promoting surface-subsurface exchange. By comparing
Figure 13b,d,f, the occurrence of these extreme infiltration fluxes is caused by evaporation and wetting/drying: during low tide or neap tide, evaporation leads to reduced water content in the subsurface domain (hence large negative pressure head). When the intertidal zone is flooded again, since increased head gradient is formed and plenty of void space becomes available, large infiltration flux is observed. Factors affecting the wetting/drying status of the intertidal zone, such as wind and use of the DW approximation, alters timing and magnitude of the exchange flux because they have influence on the evolution of the inundation area, which affects subsurface water content (
Figure 13f). These findings and mechanisms are similar to those revealed in
Figure 10, but they further illustrate the role of spring-neap cycle in depleting the subsurface domain. In permanently-wet regions such as L1, the subsurface domain is unaffected by evaporation and is always fully saturated (
Figure 13e), so switching between SWE and DW has minimal influence on the exchange flux (
Figure 13c). Thus, the key point for simulating surface-subsurface exchange is to accurately capture the wetting/drying status in the intertidal zone. This relies on the use of the SWE (rather than the DW), the inclusion of the variably-saturated subsurface flow and important environmental processes such as wind and evaporation. Arguably, accurately predicting the evolution of inundation area also depends on topography and grid resolution, which will be investigated in future studies.
To quantitatively illustrate the importance of the intertidal zone, we can compare the area of the intertidal zone (defined as the area that experiences wetting/drying transition at least once during the simulation period) and the exchange flux through this zone to the total area and total exchange flux. Our analysis indicates that for SWE+evap, the intertidal zone is only 3.34% of the total wet area in the Trinity Delta but provides 11.2% of the surface-subsurface exchange flux; hence the intertidal zone is of outsized importance in the exchange.