2. METHODS
Candidate RTDs were evaluated for different streambed depths in three
steps. First, a particle tracking technique was used to generate
empirical RTDs associated with bedform pumping through a hyporheic zone
of various depths. Second, each of the four candidate analytical
distributions (EXP, GAM, LN, and FR) were fit to the empirical RTDs, and
parameter sets inferred. Finally, for each condition used to generate
the empirical RTDs the four candidate analytical distributions were
ranked relative to goodness of fit. Details for these three steps are
described next.
2.1 Numerical Generation of the Empirical
RTDs
To mimic the advective flow field associated with hyporheic exchange
through stationary bedforms we adopted the analytical two-dimensional
laminar flow model published by Packman et al. (2000), which is based on
earlier analytical solutions of hyporheic exchange through streambeds by
Elliott and Brooks (1997) and Vaux (1968). These models posit a
sinusoidal pressure variation over the sediment-water interface
(mimicking the static and dynamic pressure variations that develop on
the surface of streambeds in a turbulent overlying flow (Cardenas et
al., 2008), isotropic and homogeneous hydraulic conductivity, constant
sediment porosity and fluid density, a so-called “Toth domain” for the
upper boundary (Frei et al., 2019; Tóth, 1962), and an impermeable lower
boundary at depth \(d_{b}\) below the surface:
\(u^{*}\)= -cos( \(x^{*}\))
[tanh( \(\text{db}^{*}\))
sinh( \(y^{*}\))+cosh( \(y^{*}\))], (1)
\(v^{*}\)= -sin( \(x^{*}\))
[tanh( \(\text{db}^{*}\))
cosh( \(y^{*}\))+sinh( \(y^{*}\))], (2)
\(h_{m}=0.28\ \frac{U^{2}}{2g}\ \left\{\par
\begin{matrix}\left(\frac{\frac{H}{d}}{0.34}\right)^{\frac{3}{8}}\text{\ \ }\frac{H}{d}\ \leq 0.34\\
\left(\frac{\frac{H}{d}}{0.34}\right)^{\frac{3}{2}}\text{\ \ \ \ }\frac{H}{d}\ \geq 0.34\\
\end{matrix}\right.\ \) (3)
\(u_{m}=k\ K_{c}h_{m}\ \tan h(\text{db}^{*})\ \) (4)
\(u^{*}=\frac{u}{u_{m}}\), (5)
\(v^{*}=\frac{v}{u_{m}},\) (6)
\(k=\frac{2\pi}{\lambda}\), (7)
\(x^{*}=k\ X\), (8)
\(y^{*}=k\ Y,\) (9)
\(d_{b}^{*}=k\ d_{b},\) (10)
\(t^{*}=\frac{ku_{m}t}{},\) (11)
here, \(u\) and \(v\) are the horizontal and vertical Darcy fluxes,
respectively, \(H\) is the bedform height, \(d\) is the stream depth,\(K_{c}\) is the hydraulic conductivity, \(u_{m}\) is the maximum Darcy
flux at the bed surface, \(X\) and \(Y\) are the horizontal and vertical
coordinates,\(\ d_{b}^{*}\) is the relative sediment depth, \(\lambda\)is the bedform wavelength, and \(k\) is the wavenumber of the bedforms.
The vertical coordinate, \(Y\), is centred at the SWI (\(Y=0\)) and
oriented upward; i.e., depth into the bed corresponds to negative values
of \(Y\). The range of \(d_{b}^{*}\) was chosen between \(0.1\ \)and\(20\) (Supporting Information) consistent with published experimental
studies performed with dune-like bedforms (Elliott & Brooks, 1997;
Marion et al., 2002; Packman et al., 2000b, 2004; Packman & MacKay,
2003; Rehg et al., 2005; Ren & Packman, 2004).
The volumetric water flux predicted by equation (\(2\)) varies
sinusoidally with horizontal distance along the SWI, forming
well-defined upwelling and downwelling regions that are fully
characterized by a repeating unit cell, one of which occurs over the
domain, \(x^{*}\in\ [-\pi,\pi]\) (see figure 1b in Grant
et al. (2020)). Thus, the RTD associated with Packman et al.’s hyporheic
exchange flow field can be fully characterized by tracking the arrival
times of particles released in the downwelling zone of a single unit
cell. Accordingly, we released \(10000\) particles in the downwelling
zone (\(0\leq\ x^{*}\leq\frac{\pi}{2})\) of the unit cell centered
on \(x^{*}=0\), with a flux weighting scheme that added particles in
proportion to the local downwelling flux (to assure that roughly the
same number of particles entered the hyporheic zone along every
streamline). The RTD for different choices of \(d_{b}^{*}\) in Packman
et al.’s flow model was obtained by fixing the length of each particle
step within the sediment domain (\({s}^{*}=ks)\) to be\({5*10}^{-3}\). Then, the i -th time step (\({t}^{*}\)) was
calculated as:
\({t}_{i}^{*}=\frac{{s}^{*}}{\sqrt{{u_{i}^{*}}^{2}+{v_{i}^{*}}^{2}}}\), (12)
where \(u_{i}^{*}\) and \(v_{i}^{*}\) denote the velocity components at
the particle location at the end of i -th step. For each particle
the time at the end of the i -th step (\(t_{i}^{*}\)) within the
sediment bed domain (\(y^{*}<0)\) is:
\(t_{i}^{*}=t_{i-1}^{*}+{t}_{i}^{*},\ \ \ i=1,\ldots,\ N\), (13)
where \(t_{0}^{*}=0\), N is the number of steps undertaken by each
particle and the corresponding horizontal and vertical particle
displacements (\({x}_{i}^{*}\ \)and \({y}_{i}^{*}\), respectively) of
the i -th step are:
\({x}_{i}^{*}=\ u_{i}^{*}\frac{{t}_{i}^{*}}{}\) (14)
\({y}_{i}^{*}=\ v_{i}^{*}\frac{{t}_{i}^{*}}{}\ \)(15)
where is the sediment effective porosity. For each \(d_{b}^{*},\)cumulative distribution function (CDF) and probability density function
(PDF) forms of the RTD were calculated from the observed residence times
of the 10000 particles.
2.2 Analytical Distribution Parameter
Inference.
Separate particle tracking RTDs were generated for \(125\) dimensionless
bed depths ranging from \(d_{b}^{*}=0.1\) to \(20\). Each of these
RTDs was fit to the four analytical distributions (EXP, GAM , LN, and
FR, see Table \(1\)) described earlier using Maximum Likelihood
Estimation (Mathematica, Wolfram). EXP is characterized by a decreasing
monotonic function and parametrized by a single parameter (\(\rho\)).
The parameters for the GAM distribution, \(\alpha\) and \(\beta\),
control the shape and scale characteristics of the distribution,
respectively. The parameters of LN, \(\mu\) and σ, determine the mean of
the log-transformed random variable and its standard deviation,
respectively. The FR distribution is a three-parameter distribution
(shape parameter \(s\), scale parameter \(q\), and location parameter\(m\)), but for the sake of parsimony and for consistency with the other
distributions considered in this study, the FR distribution with two
parameters was chosen by fixing \(s=1\), as assumed by Grant et al.
(2020). To determine which of these distributions best represents the
empirical RTD at each dimensionless depth \(d_{b}^{*}\), the candidate
analytical distributions were ranked using the Kolmogorov-Smirnov (KS)
test, which measures the difference in the CDF between the particle
tracking RTD and the assumed distribution:
\begin{equation}
KS=\sup_{t}\left|\hat{F}\left(t^{*}\right)-F\left(t^{*}\right)\right|\text{\ \ }\left(16\right)\nonumber \\
\end{equation}Where \(\sup_{t}\) is the supremum over \(t\),\(\hat{F}\left(t^{*}\right)\) is the CDF form of the empirical RTD for
a certain \(d_{b}^{*}\), and \(F\left(t^{*}\right)\) is the CDF form
of the candidate analytical distribution. A set of regression formulae
for each distribution was then prepared to correlate the analytical
distribution parameters with \(d_{b}^{*}\), using the “Curve Fitting”
tool in Matlab.