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.