Predicting the effective diffusivity across the sediment–water interface in rivers

https://doi.org/10.1016/j.jclepro.2021.126085Get rights and content

Highlights

  • A genetic programming-based effective diffusion coefficient prediction model is established.

  • The channel Reynolds number and permeability Reynolds number together characterize the mass transport across the SWI.

  • The depth-averaged velocity and friction velocity together reflect fluid flow conditions within the hyporheic zone.

Abstract

Hyporheic exchange directly controls and regulates the transport of nutrients, heat, and organic matter across the sediment–water interface (SWI), thereby affecting the biochemical processes in rivers, which is critical for maintaining the health of aquatic ecosystems. The interface exchange is controlled by multiple processes, including physical, chemical, and biological processes, which can be modeled by the effective diffusion model using an effective diffusion coefficient, Deff, to quantify the hyporheic exchange rate. In this study, genetic programming (GP), a machine learning (ML) technique based on natural selection, is adopted to search for a robust relationship between the effective diffusion coefficient and surface flow conditions, bedforms, and sediment characteristics on the basis of published broad interfacial mass exchange flux measurements. By utilizing a data set covering a wide range of environmental condition parameters, the effective diffusion coefficient prediction models for the SWI with and without bedforms are developed. Results show that the dimensionless effective diffusion coefficient is not only related to the permeability Reynolds number, ReK, but also to the channel Reynolds number, Re. Compared with the flat bed, ReK has a greater effect on the hyporheic exchange when bedforms present at the SWI by affecting the pumping advection strength. The new Deff predictor with a relatively concise form exhibits considerable improvements with regard to prediction ability and is physically sound relative to the existing predictors.

Introduction

Surface water and groundwater are two important parts of the earth’s continental water cycle. They have complex interactions between them, particularly in rivers. The river system is composed of overlying water body, sediment–water interface (SWI), and part of the groundwater, forming an organic whole together. The area where surface water and subsurface water exchange, mix, and interact is called the hyporheic zone, which is also the transition layer between surface water and groundwater (Hester and Gooseff, 2010). The hyporheic zone is the enrichment area of biogenic substances, mineral particles, and microbial communities in the river ecosystem and is also the key area of material and energy exchange between surface water and pore water within the riverbed, which plays a critical role in biogeochemical cycles and biological habitats (Chandler et al., 2016; Gooseff, 2010; Grant et al., 2018b). Given its unique hydrologic mechanism, the hyporheic zone has significant hydrological regulation, environmental buffer, and ecological protection. In addition, given its high biochemical gradient of dissolved oxygen (DO) and nutrients, the area near the SWI acts as an active bioactive reaction zone, and its growth and extinction affect the ecological health level of the river system on multiple spatial and temporal scales (Hester et al., 2017; Hinkle et al., 2001; Malard et al., 2002; Pinay et al., 2009). The exchange of substances and energy that occurs across the SWI (referred to as hyporheic exchange) controls and regulates the migration and transformation of DO, nutrients, heavy metals, and organic pollutants in the environmental water body, thereby affecting the physical, chemical, and biological processes in rivers, such as algal growth, invertebrate composition, and litter decomposition (Boano et al., 2014; Grant et al., 2018a; O’Connor and Harvey, 2008). A comprehensive understanding of the hyporheic exchange flux and its influencing factors is of great significance in the evaluation and treatment of endogenous pollution of riverbed sediment, prediction and prevention of water eutrophication, and risk assessment and restoration of aquatic ecosystem.

The exchange of solute (conservative solute and reactive solute) across the hyporheic zone, that is, hyporheic exchange process, can occur over a wide range of spatial and temporal scales, from particle scale to river scale and even to watershed scale in space, and from minutes to months in time (Gooseff et al., 2003; Grant et al., 2012; Grant and Marusic, 2011; Haggerty, 2002; Stonedahl et al., 2010). Meanwhile, the hyporheic exchange is controlled by multiple processes, including physical processes (streams, groundwater), chemical processes (chemical deposition), and biological processes (biofilms) (O’Connor and Harvey, 2008). Nonetheless, given its rapid mixing rate and wide area over which it acts, the transport behavior of small-scale hyporheic exchange, typically from gravel scale to patch scale (<10m), generates significant interfacial transport fluxes (Caruso et al., 2017; Huettel et al., 2014; Santos et al., 2012; Tomasek et al., 2018). At these scales, hyporheic exchange is dominated by a diverse range of phenomena. In lake, reservoir, and estuarine ecosystems, the bottom flow velocity is usually slow, and the bed sediment is composed of fine particle-sized sand, that is, the bed is smooth, and its permeability is negligible; the dominant transport mechanism for mass exchange is molecular diffusion (Lorke et al., 2003). However, in river systems, the flow velocity near the riverbed is relatively high, and the bed is rough and permeable. At present, molecular diffusion will no longer be the dominant behavior of hyporheic exchange. As shown in Fig. 1 (a), the stream flows over a flat river bed composed of coarse grain. Near the bed surface, subjected to the frictional resistance of the sediment bed, the stream flow velocity decreases sharply. Combined with the low seepage velocity field of sediment pore water, a “mixing layer” type flow velocity distribution pattern is formed across the SWI, characterized by an inflection point at the SWI (Fang et al., 2018; Grant et al., 2018b; He et al., 2019; Reidenbach et al., 2010; Voermans et al, 2017, 2018a). Then, Kelvin–Helmholtz (K–H) instability occurs, leading to the development of coherent structures (red vortices). The dimensions of these vortices are comparable to the length scale of the mixing layer. Meanwhile, when the overlying water is not deep, large vortices are found at flow depth scale (grey vortices). Such vortices combined with K–H coherent vortices penetrate into the sediment bed and contribute to the interfacial exchange through turbulent transport (referred to as “turbulent penetration”) (Packman et al., 2004; Reidenbach et al., 2010; Voermans et al., 2018a). These spatially coherent structures will generate pressure waves (blue wavy dashed line) traveling downstream along the SWI, thereby driving oscillating flow across the SWI and resulting in mass exchange, which is known as “turbulent pumping” (Boano et al., 2011; Roche et al., 2018). Last but not least, there are small vortices at pore scales and grain scales in the pores of the interface sediments (small black vortices). These small vortices have a high frequency, that is, a fast mixing rate, which will produce significant interface mass fluxes (Huettel et al., 2014; Santos et al., 2012). Several studies have shown that turbulent transport is the dominant mechanism of hyporheic exchange in rivers with flat beds (Packman et al., 2004; Voermans et al., 2018a, 2018b). However, if the bed surface has topography, such as ripples and dunes, other solute transport mechanisms will be observed in addition to turbulent transport as in the case of flat bed. For streams flowing over the bed geometry, the interaction between the bedforms at the SWI and the overlying water column causes the bedform to have high pressure on its facing water side and low pressure on its back water side, which is the result of the separation and reattachment of the bottom boundary layer (Fig. 1 [b]) (Cardenas and Wilson, 2007a; Clark et al., 2019; Elliott and Brooks, 1997a; Packman et al., 2004; Salehin et al., 2004). This pressure distribution will drive the flow into and out of the sediment bed, which is referred to as “bedform pumping” (Grant et al., 2012, 2018b). Through flume experiments, Packman et al. (2004) compared the hyporheic exchange across flat bed and formed bed. It was found that the existence of bedforms significantly enhanced the mass exchange rate, due to the advective transport of pumping mechanism. In other words, advective pumping is the dominant mechanism of interface mass exchange when bedforms present at the SWI. For the movable riverbed, the movement of bedforms will intermittently cause the interception and release of pore water, resulting in “turnover exchange” (Elliott and Brooks, 1997a, 1997b; Packman and Brooks, 2001; Zheng et al., 2019).

The hyporheic exchange of conservation solute across the SWI can be modeled as an effective diffusion process utilizing Fick’s second law (Grant et al., 2012):θCt=z(θDeffCz)

Accordingly, the interfacial solute transport flux can be expressed as follows:J=-DeffθCzz=0where C is the interface solute concentration after being time-averaged in time and space-averaged in the horizontal plane, kg/m3; t is time, s; θ is a dimensionless parameter representing the porosity of the sediment bed; z is the surface flow depth coordinate (z = 0 at the SWI), m; Deff is referred to as the effective diffusion coefficient, which represents the collective contribution of all transport mechanisms described above, m2/s.Deff=Dm+Dt+Dd+DbD′m is the molecular diffusion coefficient, considering the tortuosity of the sediment pore caused by Brownian motion, Dm=Dm/(4+3θ), in which Dm is the molecular diffusion coefficient of solute in clear water; Dt is the turbulent diffusion coefficient, related to the turbulent component of vertical velocity and the instantaneous turbulent concentration field (Li and Katul, 2020); Dd is the dispersion coefficient, related to the spatial dispersion component of time-averaged vertical velocity and the local mean solute concentration; Db represents the biological diffusion, which is not considered in this study. The penetration of the abovementioned spatial coherent vortex structures, turbulent pumping, and bedform pumping contributes directly to Dt and Dd (Voermans et al., 2018a, 2018b).

The development of flow measurement equipment and computational fluid dynamics (CFD) technology allows in-depth understanding and quantification of the interfacial mass transport process. For example, utilizing refractive index-matched particle tracking velocimetry, Voermans et al. (2017, 2018a) measured the flow field within the hyporheic zone with a flat bed and developed a mechanistic model to describe the hyporheic exchange. Han et al. (2018) used large eddy simulation to model the transport process of solute from surface water to the sediment bed. Chen et al. (2015, 2018) simulated the bedform-driven hyporheic exchange by coupling the Reynolds-averaged Navier–Stokes equations describing the surface flow and the groundwater flow equation and explored the sensitivity of the interfacial exchange flux to the bedform geometry. However, at present, the direct measurement of flow field in the hyporheic zone is still limited to the riverbed composed of large-size sand and gravel, and it is still powerless to the sediment bed composed of fine particles. Moreover, the simultaneous measurement of the flow and concentration fields across the SWI is still quite difficult (Voermans et al., 2018a). Meanwhile, the CFD method solves the overlying flow and pore flow fields by defining a sharp boundary at the SWI, which indicates that the turbulent penetration is not considered and the effect of subsurface flow on the surface flow is also ignored (Cardenas and Wilson, 2007b; O’Connor and Harvey, 2008). Furthermore, the flow field of groundwater is described using Darcy’s law (Cardenas and Wilson, 2007b). However, in partial area immediately below the SWI, the pore flow is non-Darcy seepage (i.e., turbulent seepage). Considering the abovementioned factors, the effective diffusion model presented previously is still a powerful and simple model to describe the interfacial mass transport flux because of its concise physical basis.

The key to effective diffusion model is the quantification of the effective diffusion coefficient, Deff, which can be directly obtained from tracer experiments in several ways. All experimental approaches are to obtain Deff through establishing initial unbalanced solute concentration between the surface water and the sediment pore water and then monitoring the response process. Here we only briefly review the two commonly used experimental approaches. The first approach is carried out in a closed experimental system, such as recirculating flume and stirring tank (e.g., Chandler et al., 2016; Packman et al., 2004). Initially, the overlying water body has uniform solute concentration Cw,0≠0, whereas the solute concentration in the sediment pore water is 0 (Cs,0 = 0). With the start of the experiment, the tracer in the surface water will be transported across the SWI into the sediment bed, accompanied by the decrease of the solute concentration in surface water (Cw(t)) with time. Then, by monitoring Cw(t), Deff can be calculated as follows:Deff=π4(VwAsθdCdt)2where C∗ = Cw(t)/Cw,0; dC/dt is the initial slope; Vw is the total volume of overlying water; As is the planar surface area of SWI. The second approach can be performed out in a closed system or an open system. Initially, in contrast to the first approach, the sediment bed has non-zero concentration Cs,0≠0, whereas the concentration of the surface water is 0 (Cw,0 = 0). As the experiment progress, the tracer will be transferred into the surface water column from the sediment bed. Through monitoring the change of the cumulative mass of tracer transported into the overlying water, Mw(t), with time, Deff can be calculated as follows:Deff=π4(dMwθAsCs,0dt)2where dMw/dt is also the initial slop. More experimental approaches to obtain Deff can be found in Grant et al. (2012).

Tracer experiments provide a way to quantify the effective diffusion coefficient directly; however, when the tracer tests are not available, the only way to obtain Deff is to establish its prediction model. Flow conditions and riverbed morphology in the natural system are variable in time and space, which will affect the hyporheic exchange process (particularly turbulent penetration, bedform pumping, and turnover exchange) (Fox et al., 2014; Mojarrad et al., 2019) and then Deff. Consequently, previous researches (Chandler et al., 2016; Elliott and Brooks, 1997b; Grant et al., 2012; O’Connor and Harvey, 2008; Packman and Salehin, 2003; Richardson and Parr, 1988; Voermans et al., 2018a) have attempted to scale the Deff value directly to the hydrodynamic conditions of the overlying flow and the characteristics of the sediment bed. Several process-based predictors (Elliott and Brooks, 1997b; Voermans et al., 2018a) and data-driven (for example, multiple linear regression, meta-analysis, and curve fitting) empirical models (Grant et al., 2012; O’Connor and Harvey, 2008) have been developed. Some of these models have attempted to relate Deff to a certain characteristic velocity scale and length scale. For example, Richardson and Parr (1988) and Voermans et al. (2018a) selected the velocity scale as the friction velocity, u, and the length scale as K (K is the permeability of the sediment bed). By contrast, Packman and Salehin (2003) and Packman et al. (2004) selected the depth-averaged velocity, U, and the water depth, H, of the surface flow. Therefore, the variables that best characterize the overlying flow and sediment bed are still uncertain. Although previous studies have provided insights into the dominant parameters that affect interface exchange, the existing effective diffusion coefficient predictive models can only be applied to narrow environmental conditions and with quite low accuracies, which will lead to large errors of the interfacial exchange flux.

Recently, an advanced data-driven approach, machine learning (ML), allows the exploration of all possible relationships among variables in complex phenomena on the basis of the provided database. The motivation of the current research is to adopt genetic programming (GP), an ML technology based on the principle of “natural selection,” to obtain the response relationship between Deff and overlying flow’s hydrodynamic conditions, riverbed morphology, and sediment characteristics and to develop a new Deff prediction model on the basis of published broad hyporheic exchange flux measurements. Compared with the traditional data-driven methods, such as the multiple regression analysis used by Grant et al. (2012) and the meta-analysis used by O’Connor and Harvey (2008), GP does not require researchers’ unnecessary subjective intervention during predictor establishment (Liu et al., 2020; Tinoco et al., 2015; Wang et al., 2017). For example, in the meta-analysis of O’Connor and Harvey (2008), the dimensionless parameter U/u was subjectively dropped because of its “relatively weak” correlation with the effective diffusion coefficient. GP is based on the principle of natural selection to automatically explore all possible dependencies of the dependent variable on independent variables and only retains the predictors with high accuracy during evolution. This means that selecting the characteristic velocity scale and length scale is not required. Consequently, the solution given by GP is always concise in form and is physically sound, thereby having excellent performance.

In the current research, a fully tested GP software, Eureqa, developed by Schmidt and Lipson (2013), which has been successfully applied in hydraulics and hydrology (Goldstein and Coco, 2014; Li et al., 2015, 2019; Liu et al., 2020; Shi et al., 2019; Tinoco et al., 2015), is utilized to develop a new Deff predictor.

Section snippets

Data pre-processing

In this study, a total of 102 measured data points from 12 sets of experiments were collected, covering a wide range of surface flow conditions and bed sediment characteristics (Table 1) (Elliott and Brooks, 1997a; Fan et al., 2020; Lai et al., 1994; Marion et al., 2002; Nagaoka and Ohgaki, 1990; Packman et al., 2000, 2004; Packman and MacKay, 2003; Rehg et al., 2005; Ren and Packman, 2004; Richardson and Parr, 1988; Tonina and Buffington, 2007). dg is the average particle size of the bed

Flat bed

After evaluating 1.1 × 1010 formulas, only 14 formulas survived with a minimum complexity of 1 and a maximum complexity of 35. The longer running time of Eureqa showed that no evident difference was observed in the survival formulas; only the coefficient values varied slightly. The solution series is shown in Table 2. Considering that formulas with complexity greater than 17 were too complex, only formulas with complexity less than or equal to 17 were listed here. Pareto front is shown in Fig. 3

Comparison of model performance

The model developed in the current research was compared with the models established by previous traditional regression methods and process-based models to evaluate the performance of the GP model.

O’Connor and Harvey (2008) scaled the effective diffusion coefficient using meta-analysis based on existing hyporheic exchange data sets (abbreviated as OH08):DeffDm=5×104RePeK1.2(4+3θ)1.2

Grant et al. (2012), after correcting the errors in calculating Deff from tracer experiments (i.e., Eqs. [4,

Summary and conclusion

Hyporheic exchange is the integration of multiple transport processes and controlled by a series of variables of surface flow, sediment, and geometry of the sediment–water interface (SWI). Based on the previously published experimental measurements of interfacial mass exchange, the genetic programming (GP), a machine learning (ML) technology, is adopted to develop a new predictive model of the effective diffusion coefficient for the SWI with and without bedforms. The final model is obtained by

CRediT authorship contribution statement

Meng-Yang Liu: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Writing - original draft, Writing - review & editing, Visualization. Wen-Xin Huai: Conceptualization, Methodology, Writing - review & editing, Supervision, Funding acquisition. Bin Chen: Conceptualization, Methodology, Formal analysis, Writing - review & editing.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work was supported by the National Natural Science Foundation of China [grant numbers 52020105006, 11872285].

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