1. INTRODUCTION
The hyporheic zone is defined as the sediment immediately beneath and
adjacent to streams, rivers, and riverine estuaries where surface water
and groundwater interact. It is a hot spot for physical, biological and
biogeochemical processes that control pollutant removal (Beaulieu et
al., 2011; Grant et al., 2014), stream nitrogen cycling (Galloway et
al., 2019), particle transport and mobilization (Stewardson et al.,
2016), pathogen sequestration and mobilization (Grant et al., 2011),
heat budgets (Sawyer et al., 2012; White et al., 1987), oxygen
consumption (Tonina et al., 2015), habitat quality (Baxter & Hauer,
2000; Wu, 2000) and stream health generally (Feminella & Walsh, 2005).
Experimental and modeling studies support the conclusion that small
bedforms, such as ripples and dunes, play an outsized role in the mixing
of water across the sediment-water interface and through the hyporheic
zone (Gomez-Velez et al., 2015). A key characteristic of the exchange
process is the distribution of travel times over which water parcels
cycle from the stream, through the hyporheic zone and back; i.e., the
hyporheic exchange residence times distribution (RTD). RTDs and their
statistical moments are key controls on hyporheic metabolism in the
streambed (Gomez et al., 2012; Harvey & Gooseff, 2015). For instance,
the Damköhler number, a dimensionless number that compares the median
hyporheic residence time and the characteristic biogeochemical reaction
time, is a key predictor of nitrogen removal in streams by
denitrification (Azizian et al., 2015; Grant et al., 2018; Zarnetske et
al., 2012), and the emission of the potent greenhouse gas nitrous oxide
(N2O) (Marzadri et al., 2014; Marzadri et al.,
2017). Gomez-Velez et al. (2015) included the Damköhler Number
in their analysis of N-cycling in the Mississippi River Basin.
Recently, Grant et al. (2020) unified two different descriptions for the
unsteady transport of mass through the hyporheic zone by exchange across
bedforms, namely, an advective pumping model (introduced by Elliott and
Brooks (1997)) and a one-dimensional dispersion model, for which the
dispersion coefficient decays exponentially with depth. In both cases,
key water quality end points (e.g., the time evolution of mass
concentration in the water column and interstitial fluids of the
sediment bed, as well as mass flux across the sediment-water interface)
can be obtained by convolving the time history of solute mass in the
water column with either an RTD (advective model) or Green’s Function
(dispersion model) that describes transport and mixing in the streambed.
Many studies have been performed to investigate the RTD of water
undergoing hyporheic exchange through ripples and dunes. Boano et al.
(2007) utilized the continuous random waking theory (CTRW) to represent
the RTDs in an infinite sediment bed. In such systems, and in the
absence of an imposed groundwater flow, hyporheic exchange across dunes
results in a strongly positively skewed (or “heavy tailed”) RTD,
indicating that most water parcels transit through the hyporheic zone
relatively quickly, while a minority of water parcels linger for a very
long time. Horizontal groundwater flow induced by longitudinal pressure
gradients (so-called underflow) can reduce the RTD’s positive skew and
heavy tail (Bottacin-Busolin & Marion, 2010). Likewise, experimental
(e.g., Fox et al., 2014) and modeling (Azizian et al., 2017) studies
indicate that vertical groundwater flow (in either gaining or losing
configurations) can reduce hyporheic exchange flux and residence times
in the hyporheic zone, and thereby diminish key ecological functions
(such as respiration and nitrogen cycling) in streambed sediments
(Gomez-Velez et al., 2014). Tonina et al. (2016) also demonstrated that
sediment heterogeneity decreases the mean, increases the median, and
increases the positive skew of the RTD.
Impermeable layers that limit the depth of hyporheic exchange also alter
the hyporheic zone’s RTD (e.g., Morén et al., 2017). Although a finite
streambed depth has been considered in previous studies (Packman et al.,
2000a), a systematic assessment of alluvium depth on RTDs in the
hyporheic zone of dune-covered streambeds has not yet been evaluated.
The aim of this work is to address this knowledge gap by identifying
appropriate analytical representations of the hyporheic zone RTD for
various streambed depths. The widely deployed Transient Storage model
(TSM) (Bencala, 1983) implemented in the USGS OTIS package (Runkel,
1998), for example, assumes that the hyporheic zone RTD can be
represented by (Harvey & Gooseff, 2015) an exponential distribution
(EXP). Although the use of an EXP distribution for the hyporheic zone
RTD has been questioned (Knapp & Kelleher, 2020), Zaramella et al.
(2003) claimed that it is a reasonable approximation for shallow beds.
Over the years, other analytical distributions have been suggested for
the hyporheic zone RTD, including the Gamma (GAM) (Kirchner et al.,
2000), Log-Normal (LN) (Cardenas et al., 2008; Wörman et al., 2002), and
Fréchet (FR) (Grant et al., 2020) distributions. In this study we
systematically evaluate the sediment depth ranges under which these four
distributions (EXP, GAM, LN, and FR) apply, and develop a set of
regression formulae for the distribution parameters that are likely to
be useful in practice.