Dunes, turbulent eddies, and interfacial exchange with permeable sediments



[1] Interfaces between a water column and underlying porous media are ubiquitous in nature. Turbulent flow over an irregular interface separating a water column and underlying porous media drives advective-fluid exchange between the two domains. We investigate the dynamics of this coupled system for unidirectional flow in the water column and a triangular interface modeled on dunes. Numerical simulations solve the Reynolds-averaged Navier-Stokes equations for the water column and then Darcy's Law and the continuity equation for the porous media. The two sets of equations are coupled via the pressure distribution along the interface. The pressure maximum and minimum along the interface, which are tied to the presence of an eddy, dominantly control the configuration of and flux through the interfacial exchange zone (IEZ) in the porous media. Since the length of eddies for fully developed turbulent flow is insensitive to the Reynolds numbers (Re), the configuration of the IEZ remains stable across a range of Re, although flux through the IEZ is strongly dependent on the Re. The flux is a power function of Re and a linear function of the current-topography induced pressure gradient along the interface. Mean residence times for fluids in the IEZ follow an inverse-power relationship with Re since the volume of the IEZ is insensitive to Re. The IEZ depths and fluxes can be predicted via equations fitted to the simulated IEZ depths and fluxes, scaled by the dune steepness, for different Re, granted that other systems maintain close dynamic similitude with those studied here.

1. Introduction

[2] Interfaces between a water column and underlying porous media are found in streams, rivers, lakes, estuaries and oceans. The advective exchange of water between the water column and the porous media controls a variety of physical, biogeochemical and thermal processes. Field and experimental observations, and modeling studies, indicate that the advection is mainly in response to pressure gradients along the interface, in a process sometimes referred to as “pumping.” The pressure gradients are typically generated by current flowing above a topographically-varied interface or by oscillatory flow due to waves and tidal fluctuations [Webb and Theodor, 1968; Thibodeaux and Boyle, 1987; Shum, 1992; Elliott and Brooks, 1997a; Huettel and Webster, 2001]. Aside from advection, the movement of bed forms also results in the trapping and release of solutes, a process called “turnover” [Elliott and Brooks, 1997a], while some benthic organisms also generate flow across the interface, referred to as bioirrigation and bioturbation. This paper focuses on advective “pumping” due to interaction of the water column current with interface topography to generate an irregular pressure distribution along the interface thus causing water circulation within the underlying porous media. The underlying porous media can be porous rock, coral, or even debris, but is usually permeable sediments. For permeable sediments the interface typically has spatially periodic triangular topography [Yalin, 1979; Cardenas and Wilson, 2007], with wavelengths ranging from centimeters up to meters in streams, and up to kilometers in marine settings. In this paper, the process, especially the porous flow portion, is referred to as current-topography driven fluid flow.

[3] Thibodeaux and Boyle [1987] demonstrated water column-sediment exchange for fluvial systems where current-topography interactions generated flow through a gravel bed. Similar processes occur in sandy sediments found in rivers [Elliott and Brooks, 1997a] and in lacustrine, estuarine and marine settings [Webb and Theodor, 1968; Huettel and Webster, 2001]. These processes affect the ecology and biogeochemistry of hyporheic and riparian zones in fluvial systems up to the watershed scale [Triska et al., 1989; Findlay, 1995; Rutherford et al., 1995; Harvey and Fuller, 1998; Worman et al., 2002]. Similar impacts are observed for processes in marine settings including early diagenesis [Huettel and Gust, 1992; Huettel et al.,1996; Huettel et al., 1998; Jahnke et al., 2000; Huettel and Webster, 2001; Jorgensen and Boudreau, 2001]. A complete understanding of most if not all thermal, chemical and biological processes occurring near the interface and within the porous media begins with the physical template– the fluid dynamics of the advective-exchange process.

[4] There have been several modeling advances describing current-topography driven fluid flow in porous media. Most previous modeling studies have decoupled the two systems. For example, instead of explicitly accounting for fluid flow in the water column, it has become customary to consider only the lower half of the domain (the underlying porous media). Many of these studies represent the impacts of current on the porous media via a sinusoidal pressure distribution imposed on an interface that is assumed to be flat [Elliott and Brooks, 1997b; Cardenas et al., 2004]. The few porous media modeling studies that explicitly consider topography of the interface use pressure distributions measured from flume experiments [Savant et al., 1987; Salehin et al., 2004]. Until recently no one has attempted to simultaneously model the conditions both above and below a topographically varied interface accounting for viscous flow in the water column. Using a multiphysics numerical modeling approach, the authors recently coupled viscous laminar flow in the water column and with porous flow in the underlying porous media [Cardenas and Wilson, 2006a, 2007]. However, most natural water column flows are turbulent.

[5] This paper focuses on a turbulent water column flowing unidirectionally over a sand bed with immobile porous dunes. The two-dimensional results are directly transferable to other settings maintaining geometric and dynamic similitude such as dunes in streams and rivers, and unidirectional currents over estuarine and marine dunes. Numerical simulations of turbulent flow in the water column are sequentially coupled to Darcian porous media flow in the underlying sediments. The coupled simulations explicitly show how the two systems are interrelated. Simulations for different dune geometry (steepness and asymmetry) and current Reynolds Number allow us to investigate and compare advective exchange for different dynamical scenarios. We define an interfacial exchange zone (IEZ) as the zone within the sediments and near the water column-sediment interface that is influenced by current-topography driven advection. The spatial configuration of and fluxes through the IEZ are determined for various scenarios. These results are synthesized into simple predictive relationships that describe water column-porous media exchange.

2. Methodology and Validation

[6] This section presents the modeling scheme and governing equations, focusing first on turbulent flow in the water column and model validation using flume experiment results. The discussion of the coupling scheme and its validity follows. The sequentially-solved porous media flow formulation is presented last, along with another flume validation experiment that considers both turbulent flow in the water column and Darcy flow in the underlying sediments.

2.1. General Formulation for Turbulent Flow Over Dunes

[7] We simulate steady state two-dimensional turbulent water flow over triangular dunes (Figures 1a and 1b) by numerically solving a finite-volume formulation of the Reynolds-averaged Navier-Stokes (RANS) equations for an incompressible, homogeneous fluid:

equation image
equation image

where ρ and μ are fluid density and dynamic viscosity, t is time, Ui (i = 1, 2) and ui (i = 1, 2) are time-averaged and instantaneous velocity components in xi (i = 1, 2) directions, P is time-averaged pressure. Sij is the strain rate tensor defined as:

equation image

The Reynolds stresses are related to the mean strain rates by:

equation image

where νt (or μt/ρ) is the kinematic eddy viscosity, δij is the Kronecker delta, and k is turbulent kinetic energy. We adopt the k-ω turbulence closure scheme [Wilcox, 1991] since it has been demonstrated to work well for separated flows with adverse pressure gradients, including flow over dunes where there is a pronounced eddy [Yoon and Patel, 1996; Cardenas and Wilson, 2006b]. The eddy viscosity in this closure scheme is:

equation image

where the specific dissipation ω is the ratio of the turbulence dissipation rate ɛ to k:

equation image

and β* is a closure coefficient. The steady state transport equations for k and ω are:

equation image
equation image

We use standard closure coefficients for the k-ω scheme: α = 5/9, β = 3/40, β* = 9/100, and σk = σω = 0.5.

Figure 1.

Modeling formulation, numerical grids (a, c) and typical results (b, d) for turbulent flow in the water column (a, b) and porous media flow in the sediments (c, d). Mean flow is from left to right. Dune length L = 1.0 m. The color scale for the two outputs (b, d), representing pressure, are similar but not exactly the same; warmer colors: higher pressure, cooler colors: lower pressure. The top right figure (b) shows streamlines in the water column, while the bottom right figure (d) shows flow directions (but not magnitude) in the porous media. A few selected streamlines are shown for the porous media in (d); the bottom streamline defines the interfacial exchange zone (IEZ) which has area Az. This streamline separates all of the streamtubes originating from and returning to the interface, from the deeper part of the porous media that is dominated by the mean flow from left to right (i.e., underflow). The depth of the IEZ, dz, is the vertical distance between the trough and the deepest portion of the bottom streamline delineating the IEZ. The other streamlines in (d) delineate individual flow cells within the IEZ.

[8] In all simulations, spatially periodic pressure boundaries are prescribed for the left and right boundaries of the RANS domain (Figure 1a), with a prescribed pressure drop dP, so that our domain approximates an infinite domain in the horizontal direction with flow from left to right. The interface at the bottom of the RANS domain is treated as a no-slip wall boundary. A fully-coupled slip boundary at the interface would be more appropriate for some other porous media conditions, but is not necessary for the sand-like porosity and permeability ranges considered in this paper. The no-slip boundary conditions are:

equation image

where y is the normal distance from the wall (interface), and η is the normal distance between the wall and center of the computational cell adjacent to the wall. This formulation for wall boundaries is convenient since it does not necessitate prescribing wall functions, which are not well described for cases of turbulent separated flow over dunes. A symmetry boundary is used at the top of the water column, effectively making this an “enclosed flow” problem (one that is surrounded by fixed boundaries, as in these simulations, or that has a flow field of such spatial extent as to be considered infinite, as in application of the model results to, e.g., the bottom of a river). Total pressure is the sum of a hydrostatic pressure and the variable dynamic pressure, P; there is no need to represent gravity explicitly in (2). In shallow water applications the top of the RANS domain is actually a free surface; but in our simulations the water depth is large enough to replace the free surface with the symmetry condition (water column Froude numbers, using the water column depth as the characteristic length, are ≪1 in all simulations). Consideration of a no-slip wall boundary at the bottom of the water column, periodic boundaries for the lateral boundaries, and a symmetry boundary for the top of the domain allows us to solve the complete and well-posed problem for turbulent flow while ignoring the presence of a porous media below the water column.

[9] RANS modeling, from pre-processing (including structured grid generation) through the solution to post-processing, is conducted using CFD-ACE+, a commercially available multiphysics modeling software, and its associated programs. CFD-ACE+ solves the finite-volume formulation of the coupled equations (2), (7) and (8). In the various simulations, the number of structured grid nodes varies from ∼16000 to more than 80000 (Figure 1a), with more densely spaced nodes near the bottom wall boundary. The maximum dimensionless wall y+ for all our simulations is less than 1, suggesting adequate resolution of the laminar sublayer at the wall.

2.2. Validation for Turbulent Flow

[10] We validate our RANS formulation by simulating and comparing with the flume experiments of van Mierlo and de Ruiter [1988], where they investigated detached or recirculating turbulent flow over a fixed dune with an impermeable surface. The observed eddy detached at or near the crest and reattached on the stoss side of the downstream dune. Modeled properties of water correspond to reported values (Table 1). In our simulations, the periodic pressure drop between the “upstream” and “downstream” boundaries is varied until the average velocity at the upstream inlet matched the reported value. Yoon and Patel [1996] previously used the same experiments to validate the k-ω scheme solved by their research code and found remarkable agreement between modeled and experimental values for flow and turbulence quantities. The reader is referred to Yoon and Patel [1996] for a detailed description of the flume experiments and of the nondimensionalization of velocity profiles. Our simulated dimensionless velocity profiles also agree very well with observed values (Figure 2). Note that we are able to simulate recirculating flow in the eddy from flow detachment to reattachment (profiles 2–6 in Figure 2). Our solutions for turbulence quantities k and νt are comparable to those in Yoon and Patel [1996]. Details on the turbulent flow solution can be found in Cardenas [2006]. The reported experimental parameters and the corresponding simulation input parameters in all validation runs can be found in Table 1.

Figure 2.

Comparison of experimental dimensionless water-column velocity profiles from van Mierlo and de Ruiter [1988] and simulated velocity profiles. Flow is from left to right. Profile numbers in the top locate the velocity profiles shown below. Note that both axes of the velocity profiles are nondimensional. The inlet average velocity is Uo = 0.63 m/s and the inlet depth is do = 0.29 m; these values are used for normalization.

Table 1. Parameters in Experiments Used for Validations (Top Entries) and Corresponding Simulation Input (Bottom Entries)
Parametervan Mierlo and de Ruiter [1988]Fehlman [1985] Run 4 ∣ Run 10Elliott and Brooks [1997a]
  • a

    average velocity above crest.

  • b

    depth above crest.

  • c

    flat crest.

μ [Pa-s]0.001 0.001none reported 0.001none reported 0.001
ρ [kg/m3]998 1000none reported 1000none reported 1000
Uoa [m/s]0.633 0.6360.43 ∣ 0.85 0.44 ∣ 0.850.132 0.130
L [m]1.60.9150.178
H/L [−]
Lc/L [−]0.95–0.975c0.750.74
kp [m2]naNa1.12 × 10−10
 1.12 × 10−10
porosity [−]naNa0.325

2.3. Linking the Two Domains: Pressure Along the Interface

[11] The RANS solution provides the averaged bottom pressures at the sediment-water interface (SWI), which is a no-slip wall in the turbulent flow model. This pressure solution is then imposed as a Dirichlet boundary at the top boundary of the underlying domain representing the permeable sediments (Figure 1c). This results in sequential coupling with no feedback and emphasizes the need for accurate RANS pressure solutions along the SWI.

[12] We validate the RANS bottom pressures by simulating the flume experiments of Fehlman [1985], in which they installed pressure taps on the surface of an artificial dune with an impermeable surface. Fehlman [1985] presented the spatial distribution of the differences between pressures (ΔP) measured at the pressure tap locations with respect to pressure at the crest of the dune. There were no pressure taps at the crest and this reference pressure was linearly interpolated between the measured values at pressure taps immediately surrounding the crest. This biases the pressure differences since the actual pressure at the crest does not necessarily fall between the measured values from the adjacent taps used for the interpolation. An additional source of error is our consideration of a symmetry boundary for the top of the RANS domain when it is actually a free surface. Despite these factors, our simulated pressure profiles agree reasonably well with the experiments (Figure 3). Numerical solution difficulties also contribute to the less than perfect agreement. The crest of the dune is a singularity, and most flows detach along this area. We expect an adverse pressure gradient at this point in order for separation to occur and from Bernoulli's Law. Although it appears that the simulated dip in pressure may be exaggerated (Figure 3), the pressure at a separation or detachment point, or at an abrupt corner, is extremely difficult to capture either numerically or experimentally. A direct comparison is impossible. To partially get around this, pressure profiles referenced to a pressure value for a node just upstream of the crest are also presented (Figure 3). Both pressure profiles along the dune, one referenced to the simulated pressure at the crest (black curve) and one referenced to a simulated pressure just upstream of the crest (gray curve), reasonably reproduce the measured values. Although the simulated pressure dip near the crest appears exaggerated, it is nevertheless consistent with observations from other flume experiments showing an adverse pressure gradient, where the eddy detachment occurs near the pressure dip (e.g., Huettel and Gust [1992] and Raudkivi [1963]).

Figure 3.

Comparison of measured pressure profiles along an impermeable dune surface (hollow squares) with simulated pressure profiles (gray and black curves). Flow in the water column is from left to right. The Run numbers correspond to original numbering scheme in Fehlman [1985]. Two simulated curves are presented. Fehlman defines the pressure change (ΔP) as the difference between the interpolated pressure at the crest and the pressure at other measurement points. The black curve corresponds to ΔP's calculated based on the original definition. The gray curve corresponds to the simulated pressure difference referenced to a point just upstream of and not exactly at the crest. The geometry of the experimental dune is illustrated (H = 0.137 m, L = 0.915 m, Lc/L = 0.75). Calculated average velocities taken at the crest are 0.43 m/s and 0.85 m/s, for Runs 4 and 10 respectively. Flow depth at the crest is 0.15 m for both runs.

2.4. Sequentially Coupled Simulation for Porous Media Flow

[13] The porous media domain is governed by the combination of Darcy's Law and the continuity equation for incompressible flow in a non-deformable media, i.e., the groundwater flow equations:

equation image
equation image

where qi (i = 1, 2) is the specific discharge (i.e. Darcy flux) and kp is intrinsic permeability.

[14] The top boundary of the porous domain, the SWI, is a Dirichlet boundary which takes on the pressure from the RANS simulations (Figure 1c). The lateral boundaries are periodic boundaries with an imposed pressure drop, dP, consistent with that for the RANS domain with which the porous domain is coupled. This formulation results in a continuous pressure distribution across the two domains (Figures 1b and 1d). It also results in what is commonly referred to as “underflow”, and is present in all or most simulations. Underflow is the ambient horizontal flow in the deeper parts of the porous domain which is unaffected by the irregular pressure boundary along the SWI (Figure 1d). Underflow is a direct result of the prescribed pressure drop between the lateral boundaries. The bottom boundary of the porous domain is a no-flow boundary. It may impact the flow field when it is too close to the top boundary. In order to approximate a semi-infinite vertical domain solution, where the bottom boundary no longer has an effect on the flow field near the SWI, we follow criteria in Cardenas and Wilson [2007]. Intrinsic permeability is 1 × 10−10 m2 for all simulations except for the validation experiment discussed below (§2.5), where we use the reported value for kp (Table 1). Model generation, solution and post-processing for the porous media domain is implemented through the generic finite element analysis code Comsol Multiphysics. The number of second-order triangular elements range from ∼15000 to ∼30000 (Figure 1c).

[15] Figure 1d illustrates a typical solution for the porous domain. The pressure distribution along the SWI, which by definition is continuous across the two domains, sets-up two flow cells in the porous domain. The streamlines delineating and dividing these flow cells are determined by the location of the maximum and minimum pressures along the SWI. Also illustrated in Figure 1d is the interfacial exchange zone (IEZ), separated by a dividing streamline from the area below which is dominated by underflow.

2.5. Sequentially Coupled Simulation Validation

[16] There are few laboratory experiments that allow for validation of coupled simulations of current-topography driven flow. The experiments by Elliott and Brooks [1997a] are a rare example, and we use their “Run 9” (see Elliott [1990] for a more detailed description of the flume experiments) for validation. Rather than a dune, they examined smaller bed forms– ripples. Fluid and sediment properties (e.g., porosity and permeability) in the simulation are the reported values (Table 1). The simulated SWI bed form topography, water column depth, and depth of the impermeable bottom boundary for the porous media domain (the floor of the flume) also follow those published. However, there are two major differences between our simulations and the experiments. First, we do not consider the top of the water column as a free surface. This does not have a substantial effect on the flow field since the Froude Numbers are below unity. Second, Elliott and Brooks [1997a] did not impose underflow in their Run 9 experiment; underflow is present in our simulation. They imposed underflow in subsequent experiments using an ad hoc method of having a separate fluid recirculating system for the sediments (see discussion in Elliott [1990] and Figure 1 of Elliott and Brooks [1997a]). As we shall discuss later, underflow is a natural condition in flumes with repeating bed forms and there may be no need to introduce it artificially. In our validation exercise, no parameters are calibrated. The only parameter that varies is the prescribed pressure drop for periodic boundaries, dP, which has been adjusted to match the reported volumetric flux of water recirculated through the flume.

[17] The only measured water column experimental flow parameters were the free-surface profile, interface topography, and amount of water being recirculated through the flume. By focusing on the circulation through the porous media, this experiment is a rigorous test for our sequential coupled formulation. Can we reproduce the flow field in the sediments by matching only to the volumetric flux circulating through the flume? To observe this flow field, Elliott and Brooks [1997a] released dye from discrete ports within the sediments. Since flow is steady, the visually mapped dye streaks effectively mapped streamlines in the sediments. Figure 4a superposes our simulated streamlines (gray dashed lines) with the observed dye streaks (black lines). Our formulation is able to reproduce the directions of flow accurately.

Figure 4.

(a) Comparison of dye streaks (black curves) from Elliott's [1990] Run 9 [24, 37] with simulated streamlines (dashed gray curves); (b) comparison of dye penetration fronts (black lines) mapped at 75, 150, 320, and 650 minutes to simulated advective solute fronts (dashed gray curves) for the same run. Flow in the overlying water column (not shown) is from left to right. H = 0.028 m, L = 0.181 m, Lc/L = 0.75.

[18] Elliott and Brooks [1997a] also introduced dye into the water column. The penetration front of the dye through the sediments was mapped through time. In order to further test our solutions to the porous flow problem, we coupled it with a solute transport simulation. Solute transport is governed by the classical advection-dispersion equation. The solute with a concentration Co is released at t = 0 from the SWI (top boundary of the porous domain) and its initial concentration is zero everywhere. The dispersion coefficient is set close to zero so that the mid-concentration line (i.e., C/Co = 0.5) represents the advective front. The simulated advective fronts are compared to the mapped dye fronts. Note that it was not reported how the experimental dye fronts were mapped. There is some subjectivity as to how a diffuse front is mapped by the human eye and it is difficult to tell if the dye fronts were mapped at the very edge of the plume where there is no discernible color change (e.g., C/Co < 0.1), at the plume interior zones where the color corresponds to pure dyed liquid (e.g., C/Co > 0.9), or at somewhere in between (e.g., C/Co ∼ 0.5). Another immeasurable complicating factor is the potential effect of flow channeling and enhanced porosity along the walls of the flumes; of course this is where the fronts are mapped. The above factors introduce some uncertainty in our comparisons. Despite these concerns, the simulated and mapped fronts agree well (Figure 4b).

[19] There are notable differences between the experimental observations and simulation results. The main or common point of origin of the streamlines and dye streaks are slightly different. This point nominally corresponds to a pressure high along the SWI. The solute penetration fronts are also shallower than the observed dye fronts at earlier times. Several factors may result in these subtle discrepancies including effects of turbulent momentum and mass transfer across the SWI. These are discussed later on. Note that we also assume that the flow is spatially periodic. It may not be perfectly periodic for both the water column and the sediments. Nonetheless, our numerical model formulation reproduces aspects of both the turbulent flow in the water column (Figures 2 and 3) and the underlying Darcy flow in the sediments (Figure 4).

[20] The results, where a simulation with underflow agrees with experiments that have no artificially induced underflow, suggest that underflow is occurring in the experiments. It is a natural consequence of a flume configuration where the lateral ends are impermeable boundaries, but in which there are many repeating bed forms. We surmise that underflow results because of the configuration of the first bed form in the series of repeating bed forms. The very first bed form cannot have two flow cells, as in Figure 1d, since it is bounded by the impermeable flume end. The fluid that enters the sediments in the first bed form cannot immediately return to the SWI via the “upstream” circulating flow cell shown in Figure 1d. It has to move downstream via underflow and eventually returns to the SWI at the downstream end of the flume.

3. Dynamical Relationships Between Turbulent Flow and Porous Media Flow, and its Effects on Interfacial Exchange

[21] The relationship between features of the turbulent and porous media flow fields are discussed in this section. First, salient aspects of how the two domains are linked are discussed with particular emphasis on the dynamics of the turbulent eddy and the pressure distribution along the SWI, and how these influence flow in the porous media. Second, results of sensitivity analysis are presented, focusing on the sensitivity of the exchange process to the dune geometry (steepness and asymmetry) and to the flow dynamics in the water column as indicated by the Reynolds Number.

[22] In the ensuing discussions, the length of the dune is L = 1.0 m and the depth of water above the trough is dw = 0.5 m. The dune height H and the horizontal distance between the crest and the upstream trough Lc are varied to effectively change the dune steepness and asymmetry (Figure 1). These dune geometries, particularly the steepness, are typical of subaqueous dunes [Yalin, 1979]. As stated earlier, kp = 10−10 m2. The Reynolds Number Re is defined as [Cardenas and Wilson, 2006a, 2007]:

equation image

where Uave is the depth averaged horizontal velocity U taken directly above the crest, and ν is kinematic viscosity of the fluid. Refer to Figure 1 for schematic definition of geometric parameters.

3.1. Eddy and Dune Geometry, Bottom Pressure Distribution, and the Interfacial Exchange Zone

[23] The pressure distribution along the SWI drives the flow in the underlying porous media and determines the configuration of the resulting porous flow field. The SWI pressure distribution is represented by the values and locations of a pressure minimum and a pressure maximum, which are somewhat analogous to a dipole field. The minimum pressure, which represents a “sink” from the view of the porous media, is invariably located at or near the crest of dunes and almost always coincides with the eddy detachment point (Figure 1b). The pressure maximum, a “source”, is typically found near where the eddy reattaches (Figures 1b and 1d). As is true with most dipole potential fields (note that this is only an analogy; the field is not actually a dipole field), its zone of influence is related to the distance between the two poles. (Huettel et al. [1996] used this concept to model current-mound induced flow in sediments with a sink-source potential flow model.) We previously presented this aspect of the current-topography driven fluid flow system for laminar flow conditions (see Figures 2–4 of Cardenas and Wilson, 2007), where we showed that the laminar eddy grows with increases in Re, controlling the growth of the interfacial exchange zone (IEZ). We define the IEZ as the area adjacent to the SWI receiving water from above the SWI. This zone is delineated by the streamline which bounds all streamlines originating and ending at the SWI (see illustration in Figure 1d). Stream ecologists commonly refer to the IEZ as the “hyporheic zone.”

[24] Before focusing on the dynamics of the IEZ, we discuss the dynamics of the water column eddy since it ultimately controls the pressure distribution along the SWI and therefore processes in the IEZ. In laminar cases, where IEZ development tracks eddy growth [Cardenas and Wilson, 2007], the eddy length (as projected on the horizontal axis), Le, is also a measure of the distance between the pressure minimum and maximum since the eddy detaches and reattaches near these points [Cardenas and Wilson, 2007]. However, eddies more or less stay the same length for fully developed steady state turbulent flows across a broad range of Res. Similar behavior was observed by Armaly et al. [1983] for turbulent flow past a backward-facing step. Armaly et al.'s [1983] flume experiments suggested that the length of the separation zone grows abruptly under laminar flow conditions, decreases through the transitional flow regime and eventually increases again to an asymptotic length when fully developed turbulence is attained (Figure 5a). The fully turbulent eddy length for a backward-facing step is smaller than the maximum length achieved under laminar conditions but larger than the minimum length attained during transitional flows. These aspects of eddy behavior under evolving flow conditions (increasing Re) were reproduced in our earlier simulations of laminar [Cardenas and Wilson, 2006a, 2007] and turbulent flow over dunes [Cardenas and Wilson, 2006b]. Figure 5b plots results of both laminar and turbulent flow simulations. The results of laminar flow simulations were obtained following Cardenas and Wilson [2007]. The simulated dynamics of eddies over dunes from laminar through turbulent flow, shown in Figure 5b but without the transitional regime, is similar to those observed by Armaly et al. [1983] for flow over a step. Note that the RANS simulations are incorrect at low Res while the laminar flow simulations based on the classical steady Navier-Stokes equations are incorrect at high Res. Both fail to properly simulate the transitional regime, although from these results we can't be sure exactly where that is.

Figure 5.

Dimensionless eddy length (Le/S or Le/H, S = back-step height) as a function of Reynolds Number (Re). The data in (a) are from experiments for flow past a backward-facing step by Armaly et al. [1983], while those in (b) and (c) are from our numerical simulations for dunes. Note that dimensionless eddy lengths and Re's for the backward-facing step are computed differently in Armaly et al. [1983]. Based on the behavior of the eddy length, Armaly et al. [1983] suggest that their experiments covered the three flow domains labeled in (a) and indicated by different shades. Data shown in (b) include results from laminar flow simulations (hollow squares), from Cardenas and Wilson [2006a, 2007], and turbulent flow simulations (solid circles), both for H/L = 0.05. Results in (c) are from turbulent flow simulations representing different dune steepness (H/L). L = 1.0 m and Lc/L = 0.9 in all cases.

[25] The invariance of eddy length for fully-turbulent flow over dunes at high Re has been observed in flume experiments, but there are no experiments for laminar to transitional flow. Experiments by Engel [1981], and the references therein, show that for fully developed turbulent flow the ratio of eddy length to dune height (Le/H) varies within a narrow range of 4–6. We recently verified this behavior by simulating turbulent flow for a few Re [Cardenas and Wilson, 2006b], and we extend those results here.

[26] In this study, a suite of simulations representing different dune shapes and flows across a broad range of Res are presented. Two measures are used for dune geometry: the dune steepness or height-to-length ratio (H/L) and dune asymmetry (Lc/L), which is the location of the crest relative to the length of the dune (see Figure 1 for illustration of geometric parameters). The eddy length Le is the horizontal distance between the detachment and reattachment points of the eddy. These points are determined by picking the points at which the horizontal-velocity changes directions along a profile of the computational cells that are located adjacent to the wall boundary. Dimensionless eddy lengths (Le/H) for five different dune steepnesses, covering the typical range for subaqueous dunes, are presented in Figure 5c. At low Res, the results of these simulations are incorrect since flow is transitional or laminar. These low Re results are presented since a critical Re where flow becomes transitional or turbulent for flow over a variety of dune geometries has not been documented. At higher Res, where we expect fully turbulent flow, all the values for Le/H fall within the narrow experimental range reported by Engel [1981].

[27] If eddy length is a surrogate for the relative locations of the pressure minimum and maximum that drive flow through the IEZ, then the spatial extent of IEZs for turbulent open channel flow conditions should be more or less constant and also confined to a narrow range. Figure 1 suggests, however, that the eddy reattachment point does not exactly correspond to the maximum pressure point and is slightly upstream of it. This is also apparent in the simulations presented by Yoon and Patel [1996]. In turbulent flow the eddy length is an indirect measure of the locations of the critical pressure points relative to each other. In laminar flow the co-location of the reattachment point with the maximum pressure is more pronounced although still not perfect [Cardenas and Wilson, 2007].

[28] Simulated IEZ depths, dz, for laminar and turbulent flow conditions are plotted together in Figure 6a. dz is defined as the vertical distance between the trough and the deepest portion of the dividing streamline that envelopes all porous flow originating from the SWI (Figure 1d). The simulated dz behavior tracks the behavior of the eddy in the water column. When the eddy length is still increasing under laminar water column conditions, so is the IEZ depth. When the eddy stabilizes to a shorter length under fully-developed turbulent conditions, the IEZ similarly stabilizes to a shallower depth. At higher Res where fully turbulent flow is expected, the dz remains essentially constant.

Figure 6.

Part a) shows dimensionless interfacial exchange zone depth (dz/L) as a function of Reynolds Number (Re) for laminar and turbulent flow simulations; H/L = 0.05. Part b) shows simulated dz/L for turbulent flow over different dune steepnesses (H/L). Also presented in (b) is the dimensionless area of the IEZ (Az/L2) for the case where H/L = 0.05, indicated by gray-filled triangles. Shown in (c) are the dimensionless interfacial fluxes (qint*) corresponding to the same cases in (b). Gray-filled triangles in (c) correspond to the dimensionless residence times (tr*) for H/L = 0.05. Curves in (c) are all fitted power models with R2 > 0.99 in all cases. L = 1.0 m and Lc/L = 0.9 in all cases.

[29] Figure 6b shows that steeper dunes (higher H/L) have shallower IEZs. This behavior is visualized in Figure 7a and explained by Figure 8a, which shows the normalized pressure distribution for simulations with different dune steepness. It illustrates that, with increasing dune steepness, the pressure maximum on the stoss (upstream) side of the dune moves downstream, closer to the pressure minimum near the downstream crest. The locations of these critical pressure points along the SWI are indicated by the streamlines dividing the IEZ into distinct flow cells (Figure 7a). Figure 8b plots the locations of the critical pressure points for different dune steepness. We measure the “distance” between critical pressure points as the horizontal separation between the pressure maximum along the stoss slope and the pressure minimum at the downstream, and not upstream, crest (that is, across the porous media flow cell which fills the center of each panel in Figure 7a). The results are consistent with a “dipole” analogy, the closer the “poles” are, the shallower the zone of influence (i.e., the IEZ and dz are smaller). This interpretation is also limited only to cases where there is a pronounced eddy that detaches at or near the crest. Our interpretation suggests that a bed form with a vanishingly small but finite H/L would have the largest depth compared to bed forms with larger steepness. We ran simulations with a flat bed but using the pressure solution from the triangular bed form case and found the same pattern of increasing IEZ depth with decreasing H/L suggesting that the pressure profile, i.e., distance between the maximum and minimum pressures, controls the IEZ depth. This is a counter-intuitive but interesting result. However, this result is apparent when interpreted within a context of a dipole in a potential field, at least when the dipole strength is constant. That strength is equivalent to the flux throughout the IEZ, which we examine below.

Figure 7.

Flow and pressure fields for simulations with different H/L's and Lc/L = 0.9 (a) and with different Lc/L's and H/L = 0.05 (b). L = 1.0 m in all cases. Warmer colors: higher pressure, cooler colors: lower pressure. Flow directions, but not magnitude, are shown for the porous media. Gray lines in the porous media delineate the interfacial exchange zone and two flow cells. Pressure is normalized as p* = (ppmin)/(pmaxpmin), where p is the pressure, and pmin and pmax are, respectively, the minimum and maximum for each simulation. pmin and pmax are located near where the streamlines intersect the top boundary. Illustrated in the water column is a streamline that delineates the eddy.

Figure 8.

(a) and (c) are dimensionless pressure profiles taken along the sediment-water interface for the different-dune-shape simulations presented in Figure 7a and 7b, respectively. Pressure is normalized as p** = (ppmid)/(pmaxpmin), where pmid is the midpoint between the maximum and minimum pressures, pmax and pmin. The ambient pressure gradient has been removed in these profiles. (b) and (d) show the x-locations of the critical pressure points and eddy detachment and reattachment points for the same simulations in (a) and (c). The troughs are located at x = 0 and 1 m for all figures. In (b) the crest is at x = 0.9 m for all 5 cases; in (d) the crest location increases from x = 0.5 to 0.9 m in increments of 0.1 m (consistent with the depicted range in Lc/L).

[30] The behavior of the IEZ is different when a pronounced eddy is absent. Consider the case of simulations for dunes with different asymmetry ratios (Lc/L). In these simulations, H/L = 0.05 and flow is always from left to right (Figure 7b). When Lc/L > 0.75, which is more typical of dunes in unidirectional currents, the eddies detach at the crest and reattach further up the downstream stoss face. When Lc/L < 0.75, the eddies are extremely small and are confined to the trough region (Figure 8d); these eddies are not visible at the scale of the images in Figure 7b. This results in minor but important differences in the pressure along the SWI (Figure 8c). For smaller Lc/L's, the pressure distribution is dictated primarily by Bernoulli's Law, the highest pressure is close to the widest area (the trough) and the lowest pressure is at the most constricted area (the crest). This pressure configuration is favorable for forming deeper IEZs (see Figure 7b). For higher Lc/L's, the detachment point becomes co-located with the pressure minimum at the crest and the pressure maximum location moves upstream along the stoss face favoring a shallower IEZ.

3.2. Fluxes and Mean Residence Times

[31] In addition to characterizing the spatial behavior of the IEZ, we investigate how volumetric fluxes through the IEZ change across dynamical settings. Flux through the IEZ is computed by integrating the magnitude of normal fluxes along the SWI and dividing by two, giving the volumetric discharge through the IEZ. We divide by two since the integration does not discriminate between inward and outward flow, which are equal. The flux is then divided by L, to yield an effective flux density based on the bed form length. We normalize the effective flux density by the hydraulic conductivity of the porous media (K = kpρg/μ, K is the hydraulic conductivity) and express it as qint* (the actual flux is then given by LKqint*). The IEZ flux increases with the water column Re via power functions (Figure 6c), with the steeper dunes resulting in less flux for a given Re. For the same Re, a steep dune has smaller flux partly because of our choice for the characteristic length in defining Re(12) which is the bed form height H. Reducing H, and assuming L is constant, would require a water current increase to keep Re constant. Of course, a higher current results in a larger pressure gradient driving more interfacial exchange. Another way to look at this is to reduce H and, keeping velocity and L constant, reduce Re proportionally. Consider the case of H/L = 0.06 and Re = 20,000 in Figure 6c. Reducing H and thus Re by half we can examine the influence of bed form height by comparing to the case of H/L = 0.03 and Re = 10,000. The flux for the lower H is about 7% less. Significantly lower bed form height results in slightly lower flux. In support of our dipole analogy, lower bed form height moves the dipole poles apart but doesn't significantly change the dipole strength.

[32] The volumetric flux is linearly related to the maximum pressure gradient along the SWI. This behavior is depicted in Figure 9, where the difference between the minimum and maximum pressure (pint = pmaxpmin) along the interface is the independent variable. The linear relationship is a natural consequence of Darcy's Law (10). For lower Res, where flow in the water column is laminar, this relationship is not exactly linear due to the changing depth and area of the IEZ [Cardenas and Wilson, 2007]. For fully turbulent flow regimes, where both the eddy and the IEZ no longer grow spatially as Re increases, the qint* vs. (pint/ρgL) plot is linear. The slope of this plot is not equal to K, as a one-dimensional Darcy's Law would predict, since the flow is two-dimensional and the slope also integrates a geometric factor.

Figure 9.

Normalized interfacial exchange zone flux as a function of pint, the difference between the maximum and minimum pressures along the sediment-water interface. L = 1.0 m and Lc/L = 0.9.

[33] The volume of the IEZ is represented by its area (Az) in these two-dimensional simulations. The area is defined by the SWI and streamlines which separate the IEZ from deeper parts of the porous media dominated by underflow (Figure 1d). Figure 6b shows that, like dz, exchange zone area, Az, is relatively constant for fully turbulent flow. Mean residence times of fluids flowing through the IEZ can be readily computed from the volumetric flux and area:

equation image

where tr is the mean residence time. We express tr in dimensionless form, tr* = trK/L, by dividing dimensionless area (Az/L2) by qint*. Since flux follows a power law dependence with Re (Figure 6b), we expect the inverse behavior for tr since Az is more or less constant (Figure 6b). Figure 6c (gray-filled triangles) depicts this and indicates a sharp decrease in residence time with an increase in Re.

3.3. Predictive Relationships

[34] It would be ideal for applications if simple predictive relationships were available for determining both the rough spatial configuration of the IEZ and material fluxes through it based on easily measurable parameters. Examples of such parameters are the dune geometry and the Reynolds Number in the fluid column. In the absence of detailed velocity profiles for turbulent flow in the fluid column, the free-stream velocity can be used as surrogate for average velocity keeping in mind that this will result in a slightly higher Re. Simple predictive models are developed and presented here.

[35] When the water column flow is laminar IEZ depths for various cases fall on one curve when the depth is normalized by L; IEZ fluxes can also be described by a common normalized curve [Cardenas and Wilson, 2006a, 2007]. This is not the case for turbulent scenarios (Figures 6b and 6c) even when normalized similarly. Therefore the data presented in Figure 6 needs additional scaling if we are to collapse the data onto a single curve, or at least a more narrow range, so that curve-fitting a single predictive model becomes tenable. It is intuitive to scale the wide-spread values with a variable describing the major differences between the simulations, a geometric parameter. We scale both dz/L and qint* by the dune steepness raised by a scaling power, (H/L)c, manually varying the power c for each until the spread in the data is minimized. The scaled results and fitted equations are presented in Figure 10. The optimal power for scaling dimensionless IEZ depth, dz/L, is c = cd = 0.218. Figure 10a includes a horizontal line that is determined by simply averaging the scaled dz/L values for cases where Re > 5000. The constant value,

equation image

for this line can be used for approximating dz given information about dune geometry and flow conditions. However, this approximation is developed only for the case where Lc/L = 0.9 (typical of angle-of-repose for subaqeous dunes); different constants may be appropriate for bed forms whose asymmetry ratio is very different from 0.9 (such as in ripples).

Figure 10.

(a) Dimensionless exchange zone depths scaled by H/L raised to a power c = cd = 0.218. (b) Dimensionless flux similarly scaled with (H/L)c with c = cq = 2.18. The two fitting parameters for the power-function in (b) are a = 2.74 × 10−12 and b = 2.12. Lc/L = 0.9 in all simulations.

[36] We similarly scale the dimensionless IEZ flux qint* with (H/L)c and find that the power c = cq = 2.18 groups the results on one curve (that cq = 10cd appears to be a coincidence). We then fit a power-function to the scaled curve (Figure 10b) resulting in:

equation image

with a = 2.74 × 10−12 and b = 2.12; the value of power b is suspiciously close to the power cq = 2.18. Fixing b at 2.18 only slightly changes the R2(= 0.99) for the regression and leads to slightly revised predictive model for IEZ fluxes:

equation image

where α = 1.1 × 10−5, β = 1.45 × 10−15, b = 2.18, and ReL is a new Reynolds Number based on dune length:

equation image

The characteristic length scale for fluxes changes from H to L, and the flux (16) no longer has any dependence on H. Equations (15) and (16) can be used in the prediction for IEZ fluxes, at least for dunes with shapes similar or close to those considered in the sensitivity analysis (H/L = 0.03 to 0.075 and Lc/L = 0.9). Of course, this is only valid for fully turbulent flow.

3.4. Toward Fully Coupled Models and Determining the Nature of Coupling

[37] Our simulations are coupled sequentially. The full or simultaneous coupling between free fluid flow and porous media flow is an active field of research. There have been major advances regarding our knowledge of how such systems can be simultaneously coupled but severe limitations have been noted. Typical simultaneous coupling schemes for laminar flow in the water column and porous media flow involve a Brinkman-type equation and modifications thereof [Brinkman, 1947; Durlofsky and Brady, 1987; Shavit et al., 2002], or different versions of the Beavers-Joseph-type equations [Beavers and Joseph, 1976]. Recently, it has been shown that the coupled problem based on the Beavers-Joseph formulation can be solved through non-coupled steps [Layton et al., 2002]. Most Brinkman-type equations have been shown to be valid only for porous media with very high porosity [Durlofsky and Brady, 1987] and, to our knowledge, virtually no models have been presented for coupled detached turbulent flow-porous media flow. Modifications to the governing equations need be considered, especially for near-boundary effects along the interface. For example, the logarithmic law for velocity along a flat wall is modified by bed suction or injection [Cheng and Chiew, 1998; Chen and Chiew, 2004] and the area within the porous bed and adjacent to the SWI should consider higher-order momentum loss terms (e.g. Darcy-Brinkman-Forcheimer equations) [Zhou and Mendoza, 1993]. This has been investigated both theoretically and in laboratory experiments for simple flow conditions, that is, flat beds and no eddies.

[38] For the case of turbulent recirculatory flow, our sequential coupling of a robust two-equation turbulence model (k-ω model) is a useful tool for investigating the macroscopic behavior of the coupled domains. Although the numerical algorithms we use allow full coupling of the equations governing the two domains, the physics may not be amenable to this, especially since the equations solved are not modified to address the complicated and poorly understood fluid physics at the interface. We ran fully coupled simulations with a Brinkman-type model, but were not able to generate eddies in the free-flowing fluid. Ideally there should be feedback between the free fluid and the porous media which is missing in our sequentially coupled simulations. This feedback, however, is limited for the systems on which we focus and for other scenarios where the permeability and porosity are small enough such that porous flow becomes Darcian at distances very close to the interface, there is no or minimal penetration of turbulent flow into the porous media, and the flux of fluid across the interface is small compared to the total flux in the free-flowing fluid.

[39] A consequence of a sequential formulation is suggested by Figures 4, 7 and 8. Figure 4a shows that Elliott and Brook's [1997a] observed streamlines diverge from a point on the stoss side of the bed form which is slightly upstream of the simulated divergence point. Figures 7 and 8 show that the simulated eddy reattachment and maximum pressure points do not coincide; the reattachment point is located upstream of the maximum pressure point. This is expected for turbulent flow over impermeable surfaces, with the eddy reattachment and maximum pressure points becoming more separated with increasing skin roughness (i.e., larger grains) [Yoon and Patel, 1996]. In the case of an impermeable rough bed such as in a flume where the impermeable bed is coated with a layer of gravel, one would expect an eddy reattaching further upstream of the maximum pressure, and closer to the trough. When the bed is porous, permeable, and consists of larger grains, say gravel instead of sand, the turbulent flow in the free-flowing fluid is more likely to penetrate into the bed (e.g., Packman et al. [2004]), and feedback becomes more important. For a permeable gravel bed, one would expect the main in-flow point to the IEZ, from which porous media flow streamlines diverge, to occur right where the eddy is reattaching since velocity normal to the wall is largest there. In this case, the eddy is expected to reattach at the in-flow point through the SWI. This presents a paradox. Our simulations suggest that the main in-flow (or divergence) point in the porous media corresponds to the maximum pressure location. Will the maximum pressure be co-located with the eddy reattachment point? The answers may contradict what one would expect for the case where the gravelly surface is impermeable and the eddy reattachment and maximum pressure points are separated [Yoon and Patel, 1996]. To our knowledge, no laboratory experiments have been implemented that address these questions; nor is our methodology entirely appropriate to address them. Prinos et al. [2003] conducted experiments to analyze the hydrodynamics of turbulent flow over and through bundles of cylinders. Their work clearly showed that flow through porous media may become non-Darcian under turbulent conditions and that momentum transport through the porous media is affected by turbulent transport from the open channel, thereby also affecting the open channel flow field. Such feedback mechanisms will clearly be important for interfacial flows where the medium is very permeable. But for natural systems, Prinos et al.'s [2003] results may not be applicable; their permeability and porosity values range from 5.5 × 10−7 to 4.1 × 10−4 m2 and 0.44 to 0.83, respectively. At the very least, their results represent an extreme end member of high permeability and porosity, but nonetheless illustrate potential difficulties when dealing with high permeability sediments. One difficulty when dealing with most natural sediment-water interfaces is that the effects of turbulence on the porous media is confined to a narrow zone near the sediment-water interface; a transition zone from turbulent to non-Darcian to Darcian flow develops. From a numerical modeling perspective, solving three-dimensional Navier-Stokes equations via direct numerical simulation (DNS) of a fluid-continuum domain, where the solid components of the porous media are explicitly represented, seems to be the only fully-coupled method that can capture the transition from fully turbulent flow above the interface, through turbulent flow in the upper portion of the porous media, to non-Darcian (high inertial forces) and then Darcian flow below. Currently available computing resources may not be able to handle such DNS simulations for domains of a meter scale or even tens of centimeters. A Lattice-Boltzmann approach may make computation easier. Until alternative robust computational methods, which appropriately honor the underlying physics, are employed, and until optimally designed laboratory experiments for validation are implemented, our sequential approach provides a useful tool for investigating coupled turbulent fluid flow and underlying porous media flow in natural systems.

4. Conclusions

[40] Fluid flow along and across interfaces between a water column and an underlying porous media are ubiquitous in nature and play a determining role in the thermal, chemical and biological dynamics of such systems. However, much is left to be learned about these coupled physical processes. We present a sequential numerical simulation methodology that accurately reproduces both the turbulent flow in the water column and Darcy flow in the porous media, where the interface between these two domains is composed of dunes (triangular roughness elements). Using this approach, we are able to investigate the fundamental dynamics of such processes and develop some simple predictive relationships. This study builds on previous work for laminar flow conditions in the water column [Cardenas and Wilson, 2006a, 2007].

[41] A salient feature of water column flow over asymmetric triangular topography is an eddy detaching at or near the crest and reattaching on the stoss side of the succeeding dune. This recirculatory flow modifies the pressures along the interface between the water column and the porous media. Bernoulli's Law predicts a pressure minimum at the crest and a maximum at the trough. This is replicated for cases where there is no pronounced eddy, such as flow over more symmetric dunes or dunes with a small height-to-length ratio. When a large eddy is present, it detaches at or near the pressure minimum at the crest and reattaches just upstream of the pressure maximum. In these cases, the pressure maximum is no longer located close to the trough but migrates upwards along the stoss side of the following dune. The pressure distribution along the interface, which is tied to the eddy, is essentially represented by the location of the pressure maxima and minima. These pressures dominantly drive flow through the porous media and determine both the spatial configuration of the interfacial exchange zone and volumetric flux through the zone. Generally, the farther upstream the pressure maxima, the deeper and larger is the exchange zone. Flux through the zone is linearly related to the gradient between these two points, consistent with Darcy's Law. The pressure or head gradient in turn is related to the Reynolds Number of the turbulent flow via a power function.

[42] Eddies scale with the height of the bed form, with the eddy length being ∼4–6 times the height of the dune for fully turbulent flow across a range of Reynolds Numbers. This has consequences for the dynamics of the interfacial exchange zone since the critical pressure points are related to the eddy; the interfacial exchange zone depths and volumes are similarly confined to a narrow range. Steeper dunes (larger height-to-length ratios) generally result in longer eddies which result in shallower exchange zones. The exchange zone depths therefore are sensitive to and vary with dune steepness. We were able to find a simple predictive expression (14) for exchange zone depth as a function of Reynolds number by scaling the depth by the dune steepness raised to a power.

[43] Flux through the exchange zone is also dependent on Reynolds Number via a power function. For Reynolds Numbers with dune height as the characteristic length, each dune shape results in a unique flux-Reynolds Number power relationship with steeper dunes having smaller flux for a given Reynolds Number. Scaling instead by a new Reynolds Number, with dune length as the characteristic length, condenses all of the data to a single power function, permitting prediction of exchange zone fluxes (16) from dune length, free-flowing fluid velocity, porous media permeability, and fluid viscosity.

[44] The exchange zone volume, or area in the case of our two-dimensional simulations, is directly related to the depth of the exchange zone and for turbulent flow also remains constant across a broad range of Reynolds Numbers. The mean residence or turnover times for fluids flowing through the exchange zones (volume/volumetric flux) is therefore just an inverted form of the flux-Reynolds Number relationship. Residence time initially decreases dramatically with an increase in Reynolds Number, and eventually becomes less sensitive to further increases in Reynolds Number.

[45] This study focuses on the coupled system where the fluid is water and the underlying porous media are sandy sediments, but the results are directly transferable to other natural environments that are geometrically and dynamically similar with our studied system.


[46] This research was funded by an American Geophysical Union Horton Research Grant and a New Mexico Water Resources Research Institute Student Grant awarded to MBC. MBC was supported by the Frank E. Kottlowski Fellowship of the New Mexico Bureau of Geology and Mineral Resources at the New Mexico Institute of Mining and Technology (NMIMT) throughout the duration of this study.