2.3 Numerical Modeling
We numerically simulated all flume runs in HydroGeoSphere to visualize longer duration flow paths moving through the subsurface (Aquanty Inc. 2015; Brunner & Simmons, 2012). The model domain mirrors our flume dimensions of 1.2 m wide, 9.0 m long, and 0.1 m deep. We defined the base and sides of the model as no-flow boundaries to represent the bottom and sides of the flume environment. At the upstream boundary of the flume domain, a specified inflow flux was assigned to simulate the stepped spillway. At the downstream outlet, a critical depth boundary condition was given to match flume observations.
The HydroGeoSphere model solves wave approximations of the Saint Venant equation for surface flow, the Richards equation for subsurface flow, and the advection-dispersion-diffusion equation for groundwater age. The surface flow equation is solved on a 2-D finite-element mesh stacked upon the subsurface grid. The surface of the model domain was discretized with an unstructured, triangular mesh with maximum element length of 2.0 cm. For comparison, the grain diameter of flume sediments ranges up to 17 mm, therefore choosing a finer mesh would have been inconsistent with the concept of a porous continuum. The subsurface domain of our model is discretized into ten vertical layers. The three near-surface sediment layers represent the gravel and coarse sand top coat layer of the flume and are assigned element heights of 0.005 m and hydraulic conductivity of 2.5×10-3 m/s. The deeper seven mesh layers represent the underlying coarse sand layer and are assigned an element height of 0.028 m and hydraulic conductivity of 8.9×10-5 m/s. Sediment hydraulic properties are consistent with Wilhelmsen et al. (2021) (Table 3). Logjams were simulated as a porous medium with varying permeability (Table 3) to test the effect of this parameter on hyporheic flow and subsurface transient storage. The hydraulic conductivity of the more permeable logjam(s) is 1 m/s, while the value for less permeable logjam(s) is 0.5 m/s. It is important to note that because we treat the jams as a porous medium, we also chose to include flow paths through the jams as part of “subsurface” transient storage.
For each simulation in HydroGeoSphere, we visualized subsurface flow paths in Tecplot. We also visualized groundwater age throughout the subsurface, which was computed using the groundwater age mass-transport equation embedded in HydroGeosphere. Groundwater age is similar in concept to a residence time and includes the effects of both advection and dispersion (Therrien et al. 2006). We calculated this age only for visualization purposes, as it could not be used to construct residence time distributions because of the zero-age boundary condition at the bottom of the water column. To compute residence time distributions in the subsurface, we released particles at the water-sediment or water-jam interface and analyzed the flux-weighted residence times of the 9000-18000 particles that returned to the surface water. From the flux-weighted residence time distribution, we calculated the mean travel time in the subsurface and the skew. It is important to note that the model-derived mean travel time in the subsurface is not directly comparable with the flume-derived mean travel time, as the former only considers travel due to advection in the subsurface, while the latter considers salt transport due to advection and dispersion in the surface and subsurface. We report both values, as both represent different aspects of solute travel through the interconnected stream-groundwater system. Similarly, skew in the model-derived hyporheic residence time distribution is not directly comparable with skew in the salt breakthrough curve. We report both values to quantify the weighting of longer solute residence times (in one case, for the entire flume system, and in the other case, for the hyporheic zone specifically). We also calculated average hyporheic fluxes for each simulation. Model parameters are included in Table 3 and more on the governing equations is in the Supplemental Information.
Table 2. Flume and numerical model runs with discharge and logjam characteristics (number of jams and permeability). All runs were conducted in flume and numerically simulated unless otherwise noted. Replicates were run for all flume trials.