Temporal moment analysis of tracer discharge in streams: Combined effect of physicochemical mass transfer and morphology

Abstract

Abstract. In environmental applications it is often of interest to predict the rates at which contaminant mass is discharged at a given cross section of streams and rivers. We present a Lagrangian methodology for evaluating tracer discharge (mass per unit time) at specified control cross sections (CCS) of streams. The key transport processes included in the analysis are advection, degradation/decay, and kinetically controlled mass transfer in storage zones and in bed sediment. The transport in the bed sediment is described as a diffusion process, where the tracer may sorb onto the sediment. We have derived a general solution for tracer discharge in the Laplace domain wherefrom temporal moments are computed. The derived solutions may account for deterministic changes in morphological characteristics along stretches of streams. The results are illustrated for zeroth and first two moments where we show the combined effect of advection, degradation, physicochemical mass transfer, and morphology. For illustrative purposes, we assume morphology to change downstream following power laws suggested by Langbein [1947] and Leopold and Maddock [1953]. The moments depend nonlinearly on the downstream distance, following power laws that reflect the power laws for the hydraulic geometry. We define two main dimensionless parameters, namely, kinetic storage parameter a* and bed parameter M, that control the amount of tracer mass eventually discharged at any given CCS. For M •-• 0.3 the most dominant mechanism that controls the amount of ultimately discharged tracer mass is the exchange with the bed sediment. Once estimated from field data for stretches of specified streams, the dimensionless parameters can be used in the derived expressions for predictive purposes.

Hyporheic flow and transport processes: Mechanisms, models, and biogeochemical implications

“…The solute traveltime distribution for the watershed φ W ( t ) can be expressed using a form similar to equation , where φ ( x , t ) is the PDF of the solute mass transit times at the watershed effluent that are associated with a travel path length x , i.e., the unit‐solute‐mass response function. This integration over the pathway lengths can be expressed also in a Lagrangian framework in terms of the solute residence time (compare equations and presented by Gupta and Cvetkovic []).…”

Solute transport through the integrated groundwater‐stream system of a catchment

“…In the following we derive a process‐based solute transport model for quantifying the solute breakthrough curve (BTC) that results at an arbitrary observation/control plane in the stream from tracer input distributed over the entire catchment soil surface. For this purpose, we use a Lagrangian stochastic advective‐sorptive (LaSAS) travel time approach, in analogy with many previous studies reported in the scientific literature, which have so far considered separately the solute transport‐sorption processes through the subsurface [e.g., Cvetkovic and Shapiro , 1990; Destouni and Cvetkovic , 1991; Cvetkovic and Dagan , 1994; Destouni et al , 1994; Ginn et al , 1995; Simmons et al , 1995; Destouni and Graham , 1995, 1997; Berglund and Cvetkovic , 1996; Cvetkovic and Dagan , 1996; Cvetkovic and Haggerty , 2002] and the surface [ Gupta and Cvetkovic , 2000, 2002; Haggerty et al , 2002] water systems of a catchment. Such LaSAS modeling approaches may explicitly express the large‐scale physical solute spreading that results from random flow heterogeneity in terms of nonsorptive (i.e., purely advective) probability density functions (pdfs) of solute travel time.…”

“…By explicitly representing large‐scale solute spreading from pure and independently determined advection variability, in terms of an advective travel time pdf, one does not risk confusion with other types of physical or chemical solute spreading mechanisms, such as diffusive (i.e., physical rather than chemical) nonequilibrium sorption‐desorption, or mass transfer, between mobile and immobile water zones [see, e.g., Cvetkovic and Shapiro , 1990; Destouni and Cvetkovic , 1991; Cvetkovic and Dagan , 1994; Destouni et al , 1994]. In order to include the additional solute spreading effects of kinetic, linear or nonlinear, single‐rate or multirate sorption‐desorption/mass transfer, LaSAS modeling approaches may explicitly couple the purely advective solute travel time pdf with a relevant mathematical representation of the considered kinetic sorption/mass transfer process and derive a resulting convoluted advective‐sorptive travel time pdf or at least the first statistical travel time moments of this advective‐sorptive pdf; such process‐based derivation methods for the resulting convoluted advective‐sorptive travel time variability (expressed in terms of either an entire travel time pdf or its first statistical moments, mean, and variance of travel time) have been reported separately for subsurface [e.g., Cvetkovic and Shapiro , 1990; Destouni and Cvetkovic , 1991; Cvetkovic and Dagan , 1994; Destouni et al , 1994; Ginn et al , 1995; Simmons et al , 1995; Destouni and Graham , 1995, 1997; Berglund and Cvetkovic , 1996; Cvetkovic and Dagan , 1996; Cvetkovic and Haggerty , 2002] and surface [e.g., Gupta and Cvetkovic , 2000, 2002; Haggerty et al , 2002] water systems. Alternatively, it is also possible to relate an implicitly coupled advective‐sorptive travel time pdf directly to solute transport observations in a water system [e.g., Haggerty et al , 2002], however risking then possible model confusion between the two different spreading mechanisms of variable‐advection and sorption‐desorption/mass transfer between mobile and immobile water [ Zinn and Harvey , 2003].…”

“…We can then use previous LaSAS results for the separate representations of solute transport in the subsurface [ Cvetkovic and Dagan , 1994; Destouni and Graham , 1995] and the surface [ Gupta and Cvetkovic , 2000] water systems of a catchment and the methodology proposed by Destouni and Graham [1995] for coupling a series of different hydrological subsystems, in order to express the resulting total mass flow rate trough the CP at location x and time t as [ Gupta , 2000; Simic , 2001] where S 1 is the solute mass flow rate component from main pathway 1 and S 2 the component from main pathway 2. M 0 is the total injected solute mass at the catchment soil surface, the factor 0 ≤ α s ≤ 1 quantifies the fraction of M 0 entering the stream directly via pathway 1, and T s ( x ) is solute travel time in the stream to x , implying that the solute travel time from the soil surface to the groundwater‐stream interface along pathway 2 must then be T gw = t − T s .…”

A software tool to assess uncertainty in transient-storage model parameters using Monte Carlo simulations

“…For flexible comparisons with these studies, OTIS-MCAT calculates a suite of commonly used objective functions (Gupta and Cvetkovic 2000, Schmid 2003, Mason et al 2012, Ward et al 2013b, including RSS, the singular objective function implemented within OTIS-P. All objective functions, unless otherwise noted, are calculated for both observed and normalized time series of concentration. Time series were normalized by their 0 th temporal moments as:…”