# A Method for Calibrating the Transient Storage Model from the Early and Late-Time Behavior of Breakthrough Curves

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Approximation for the Rising Limb of the BTC

#### 2.2. Approximation for the Decreasing Limb of the BTC

## 3. Application

#### 3.1. Numerical Experiment

^{3}s

^{−1}with stream width $b=5.0$ and depth $d=0.4$ m, respectively. The dispersion coefficient is estimated using Fischer’s formula, ${D}_{W}=0.011{U}^{2}{b}^{2}/\left({U}^{*}d\right)=0.735$ m

^{2}s

^{−1}[45], where the shear velocity is ${U}^{*}=\sqrt{gdj}=0.037$ m s

^{−1}and $j$ is the energy slope under the assumption of uniform flow and given a Manning coefficient $n=0.05$ m

^{−1/3}s

^{−1}. The model output is generated using a single compartment TSM with storage area ${A}_{S}=0.1$ m

^{2}, and exchange rate $\alpha ={10}^{-4}$ s

^{−1}at three cross-sections located at ${X}_{1}=500$ m, ${X}_{2}=1000$ m and ${X}_{3}=1500$ m from the injection point. The boundary condition at $x=0$ is a constant concentration ${C}_{0}$ injection of a solute mass $M=192$ g, applied for a period ${T}_{S}=480$ s starting at $t=0$ s.

#### 3.2. Field Experiment

^{3}s

^{−1}.

^{2}. The fitting lines are reported in Figure 3, showing the observed concentration data, normalized by the peak concentration, and the fitting trendlines in linear scale for the rising limb (panels a and b) and semi-log scale for the decreasing limb (panels c and d). The fitting lines provide the values of ${m}^{*}\left(x\right)$ and ${q}^{*}\left(x\right)$ of the linear trendline for the rising part, ${C}_{R}^{*}\left(x,t\right)={m}^{*}\left(x\right)t+{q}^{*}\left(x\right)$, and the values of ${b}^{*}\left(x\right)$ and $n$ of the exponential trendline for the decreasing limb of the curve, ${C}_{D}^{*}\left(x,t\right)={b}^{*}\left(x\right){e}^{n\left(x\right)t}$. The unknown transport parameters $A$, ${A}_{S}$, ${D}_{W}$ and $\alpha $ are obtained by solving Equations (10), (11), (18), and (21) as a system of equations. Note that, here, the flow rate Q is known and therefore the cross-sectional velocity U can be determined from the cross-sectional area A as U = Q/A, whereas the parameter ${T}_{D}$ in Equations (18) and (21) depends on α, A, and A

_{S}, according to the relationship ${T}_{D}=\frac{1}{\alpha}\frac{{A}_{S}}{A}.$ The resulting transport parameters and ${t}_{LIM}^{*}$ are reported in the first two rows of Table 2.

## 4. Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Symbol | Unit | Description |
---|---|---|

$a$ | (kg m^{−3}) | $\mathrm{coefficient}\mathrm{of}\mathrm{the}\mathrm{trendline}\mathrm{for}{C}_{D}^{\delta}$ |

$A$ | (m^{2}) | mean flow area |

${A}_{S}$ | (m^{2}) | transient storage area |

$b,{b}^{*}$ | (kg m^{−3}), (-) | $\mathrm{coefficient}\mathrm{and}\mathrm{normalized}\mathrm{coefficient}\mathrm{of}\mathrm{the}\mathrm{trendline}\mathrm{for}{C}_{D}^{H}$ |

${C}_{0}$ | (kg m^{−3}) | concentration at the injection section |

${C}_{R}^{H}$ | (kg m^{−3}) | approximation for the rising limb of the BTC |

${C}_{R}^{*}$ | (-) | dimensionless linear approximation of the rising limb |

${C}_{D}^{H}$$,{C}_{D}^{\delta}$ | (kg m^{−3}) | approximation for the decreasing limb of the BTC |

${C}_{D}^{*}$ | (-) | dimensionless exponential approximation of the decreasing limb |

${C}_{S}$ | (kg m^{−3}) | concentration in the storage area |

${C}_{T}$ | (kg m^{−3}) | elementary solution of the advection–dispersion equation |

${C}_{W}$ | (kg m^{−3}) | concentration in the main flow channel |

${D}_{W}$ | (m^{2} s^{−1}) | longitudinal dispersion coefficient |

$H$ | (-) | Heaviside function |

${I}_{1}$ | (-) | modified Bessel function of the first order and first kind |

$m,{m}^{*}$ | (kg m^{−3} s^{−1}), (s^{−1}) | $\mathrm{slope}\mathrm{and}\mathrm{normalized}\mathrm{slope}\mathrm{of}\mathrm{the}\mathrm{trendline}\mathrm{for}{C}_{R}^{H}$$\mathrm{and}{C}_{R}^{*}$ |

$n$ | (s^{−1}) | $\mathrm{exponent}\mathrm{of}\mathrm{the}\mathrm{trendline}\mathrm{for}{C}_{D}^{H}$$\mathrm{and}{C}_{D}^{*}$ |

$q,{q}^{*}$ | (kg m^{−3}), (-) | $\mathrm{intercept}\mathrm{and}\mathrm{normalized}\mathrm{intercept}\mathrm{of}\mathrm{the}\mathrm{trendline}\mathrm{for}{C}_{R}^{H}$$\mathrm{and}\mathrm{for}{C}_{R}^{*}$ |

$Q$ | (m^{3} s^{−1}) | discharge |

$t$ | (s) | time |

${t}_{AD}$ | (s) | advective time |

${T}_{D}$ | (s) | mean residence time in the storage area |

${T}_{S}$ | (s) | time duration of a plateau injection |

${t}_{LIM}^{*}$ | (-) | dimensionless time limit |

$U$ | (m s^{−1}) | flow velocity |

$x$ | (m) | longitudinal distance from injection point |

$\alpha $ | (s^{−1}) | exchange rate |

$\delta $ | (s^{−1}) | Dirac delta function |

$\tau $ | (s^{−1}) | dummy variable for time |

$\phi $ | (s^{−1}) | exponential RTD in the dead zones |

## References

- Chatwin, P.C. On the interpretation of some longitudinal dispersion experiments. J. Fluid Mech.
**1971**, 48, 689–702. [Google Scholar] [CrossRef] - Boano, F.; Harvey, J.W.; Marion, A.; Packman, A.I.; Revelli, R.; Ridolfi, L.; Wörman, A. Hyporheic flow and transport processes: Mechanisms, models, and biogeochemical implications. Rev. Geophys.
**2014**, 52, 603–679. [Google Scholar] [CrossRef] - Bottacin-Busolin, A. Non-Fickian dispersion in open-channel flow over a porous bed. Water Resour. Res.
**2017**, 53, 7426–7456. [Google Scholar] [CrossRef] [Green Version] - Kwaw, A.K.; Dou, Z.; Wang, J.; Zhang, X.; Chen, Y. Advancing the knowledge of solute transport in the presence of bound water in mixed porous media based on low-field nuclear magnetic resonance. J. Hydrol.
**2023**, 617, 129059. [Google Scholar] [CrossRef] - Salehin, M.; Packman, A.I.; Wörman, A. Comparison of transient storage in vegetated and unvegetated reaches of a small agricultural stream in Sweden: Seasonal variation and anthropogenic manipulation. Adv. Water Resour.
**2003**, 26, 951–964. [Google Scholar] [CrossRef] - Magazine, M.K.; Pathak, S.K.; Pande, P.K. Effect of Bed and Side Roughness on Dispersion in Open Channels. J. Hydraul. Eng.
**1988**, 114, 766–782. [Google Scholar] [CrossRef] - Jackson, T.R.; Haggerty, R.; Apte, S.V.; O’Connor, B.L. A mean residence time relationship for lateral cavities in gravel-bed rivers and streams: Incorporating streambed roughness and cavity shape. Water Resour. Res.
**2013**, 49, 3642–3650. [Google Scholar] [CrossRef] - Chen, Z.; Tian, Z.; Zhan, H.; Huang, J.; Huang, Y.; Wei, Y.; Ma, X. The Effect of Roughness on the Nonlinear Flow in a Single Fracture with Sudden Aperture Change. Lithosphere
**2022**, 2022, 5775275. [Google Scholar] [CrossRef] - Wallis, S.G.; Young, P.C.; Beven, K.J. Experimental investigation of the aggregated dead zone model for longitudinal solute transport in stream channels. Proc. Inst. Civ. Engin.
**1989**, 87, 1–22. [Google Scholar] - Davis, P.M.; Atkinson, T.C. Longitudinal dispersion in natural channels: 3. An aggregated dead zone model applied to the River Severn, U.K. Hydrol. Earth Syst. Sci.
**2000**, 4, 373–381. [Google Scholar] [CrossRef] - Haggerty, R.; McKenna, S.A.; Meigs, L.C. On the late-time behavior of tracer test breakthrough curves. Water Resour. Res.
**2000**, 36, 3467–3479. [Google Scholar] [CrossRef] [Green Version] - Deng, Z.; Bengtsson, L.; Singh, V.P. Parameter estimation for fractional dispersion model for rivers. Environ. Fluid Mech.
**2006**, 6, 451–475. [Google Scholar] [CrossRef] - Marion, A.; Zaramella, M. A residence time model for stream-subsurface exchange of contaminants. Acta Geophys. Pol.
**2005**, 53, 527. [Google Scholar] - Marion, A.; Zaramella, M.; Bottacin-Busolin, A. Solute transport in rivers with multiple storage zones: The STIR model. Water Resour. Res.
**2008**, 44, W10406. [Google Scholar] [CrossRef] [Green Version] - Boano, F.; Packman, A.I.; Cortis, A.; Revelli, R.; Ridolfi, L. A continuous time random walk approach to the stream transport of solutes. Water Resour. Res.
**2007**, 43. [Google Scholar] [CrossRef] - Young, P.; Garnier, H. Identification and estimation of continuous-time, data-based mechanistic (DBM) models for environmental systems. Environ. Model. Softw.
**2006**, 21, 1055–1072. [Google Scholar] [CrossRef] [Green Version] - Bottacin-Busolin, A. Modeling the effect of hyporheic mixing on stream solute transport. Water Resour. Res.
**2019**, 55, 9995–10011. [Google Scholar] [CrossRef] - Haggerty, R.; Martí, E.; Argerich, A.; von Schiller, D.; Grimm, N. Resazurin as a “smart” tracer for quantifying metabolically active transient storage in stream ecosystems. J. Geophys. Res. Atmos.
**2009**, 114, G03014. [Google Scholar] [CrossRef] [Green Version] - González-Pinzón, R.; Peipoch, M.; Haggerty, R.; Martí, E.; Fleckenstein, J.H. Nighttime and daytime respi-ration in a headwater stream. Ecohydrology
**2016**, 9, 93–100. [Google Scholar] [CrossRef] - Knapp, J.L.A.; González-Pinzón, R.; Haggerty, R. The resazurin-resorufin system: Insights from a decade of “smart” tracer development for hydrologic applications. Water Resour. Res.
**2018**, 54, 6877–6889. [Google Scholar] [CrossRef] - Argerich, A.; Haggerty, R.; Martí, E.; Sabater, F.; Zarnetske, J. Quantification of metabolically active transient storage (MATS) in two reaches with contrasting transient storage and ecosystem respiration. J. Geophys. Res. Atmos.
**2011**, 116, G03034. [Google Scholar] [CrossRef] [Green Version] - Lemke, D.; Liao, Z.; Wöhling, T.; Osenbrück, K.; Cirpka, O.A. Concurrent conservative and reactive tracer tests in a stream undergoing hyporheic exchange. Water Resour. Res.
**2013**, 49, 3024–3037. [Google Scholar] [CrossRef] - Bottacin-Busolin, A.; Dallan, E.; Marion, A. STIR-RST: A Software tool for reactive smart tracer studies. Environ. Model. Softw.
**2020**, 135, 104894. [Google Scholar] [CrossRef] - Haggerty, R.; Argerich, A.; Martí, E. Development of a “smart” tracer for the assessment of microbiological activity and sediment-water interaction in natural waters: The resazurin-resorufin system. Water Resour. Res.
**2008**, 44, W00D01. [Google Scholar] [CrossRef] - Lemke, D.; González-Pinzón, R.; Liao, Z.; Wöhling, T.; Osenbruck, K.; Haggerty, R.; Cirpka, O.A. Sorption and transformation of the reactive tracers resazurin and resorufin in natural river sediments. Hydrol. Earth Syst. Sci.
**2014**, 18, 3151–3163. [Google Scholar] [CrossRef] [Green Version] - Chen, J.L.; Steele, T.W.J.; Stuckey, D.C. Metabolic reduction of resazurin; location within the cell for cytotoxicity assays. Biotechnol. Bioeng.
**2017**, 115, 351–358. [Google Scholar] [CrossRef] [PubMed] - Dallan, E.; Regier, P.; Marion, A.; González-Pinzón, R. Does the Mass Balance of the Reactive Tracers Resazurin and Resorufin Close at the Microbial Scale? J. Geophys. Res. Biogeosci.
**2020**, 125, e2019JG005435. [Google Scholar] [CrossRef] - Bencala, K.E.; Walters, R.A. Simulation of solute transport in a mountain pool-and-riffle stream: A transient storage model. Water Resour. Res.
**1983**, 19, 718–724. [Google Scholar] [CrossRef] - Mulholland, P.J.; Marzolf, E.R.; Webster, J.R.; Hart, D.R.; Hendricks, S.P. Evidence of hyporheic retention of phosphorus in Walker Branch. Limnol. Oceanogr.
**1997**, 42, 443–451. [Google Scholar] [CrossRef] - Valett, H.M.; Morrice, J.A.; Dahm, C.N.; Campana, M.E. Parent lithology, surface-groundwater exchange, and nitrate retention in headwater streams. Limnol. Oceanogr.
**1996**, 41, 333–345. [Google Scholar] [CrossRef] - Briggs, M.A.; Gooseff, M.N.; Arp, C.D.; Baker, M.A. A method for estimating surface transient storage parameters for streams with concurrent hyporheic storage. Water Resour. Res.
**2009**, 45, W00D27. [Google Scholar] [CrossRef] - Bottacin-Busolin, A.; Marion, A.; Musner, T.; Tregnaghi, M.; Zaramella, M. Evidence of distinct contaminant transport patterns in rivers using tracer tests and a multiple domain retention model. Adv. Water Resour.
**2011**, 34, 737–746. [Google Scholar] [CrossRef] - Kerr, P.; Gooseff, M.; Bolster, D. The significance of model structure in one-dimensional stream solute transport models with multiple transient storage zones–competing vs. nested arrangements. J. Hydrol.
**2013**, 497, 133–144. [Google Scholar] [CrossRef] - Marion, A.; Zaramella, M.; Packman, A.I. Parameter Estimation of the Transient Storage Model for Stream–Subsurface Exchange. J. Environ. Eng.
**2003**, 129, 456–463. [Google Scholar] [CrossRef] [Green Version] - Zaramella, M.; Packman, A.I.; Marion, A. Application of the transient storage model to analyze advective hyporheic exchange with deep and shallow sediment beds. Water Resour. Res.
**2003**, 39, 1198. [Google Scholar] [CrossRef] [Green Version] - Marion, A.; Zaramella, M. Diffusive Behavior of Bedform-Induced Hyporheic Exchange in Rivers. J. Environ. Eng.
**2005**, 131, 1260–1266. [Google Scholar] [CrossRef] - Nordin, C.F.; Troutman, B.M. Longitudinal dispersion in rivers: The persistence of skewness in observed data. Water Resour. Res.
**1980**, 16, 123–128. [Google Scholar] [CrossRef] - González-Pinzón, R.; Haggerty, R.; Dentz, M. Scaling and predicting solute transport processes in streams. Water Resour. Res.
**2013**, 49, 4071–4088. [Google Scholar] [CrossRef] [Green Version] - Gooseff, M.N.; Briggs, M.A.; Bencala, K.E.; McGlynn, B.L.; Scott, D.T. Do transient storage parameters directly scale in longer, combined stream reaches? Reach length dependence of transient storage interpretations. J. Hydrol.
**2013**, 483, 16–25. [Google Scholar] [CrossRef] - Haggerty, R.; Wondzell, S.M.; Johnson, M.A. Power-law residence time distribution in the hyporheic zone of a 2nd-order mountain stream. Geophys. Res. Lett.
**2002**, 29, 1640. [Google Scholar] [CrossRef] [Green Version] - Knapp, J.L.; Kelleher, C. A Perspective on the Future of Transient Storage Modeling: Let’s Stop Chasing Our Tails. Water Resour. Res.
**2020**, 56, e2019WR026257. [Google Scholar] [CrossRef] [Green Version] - Davis, P.M.; Atkinson, T.C.; Wigley, T.M.L. Longitudinal dispersion in natural channels: 2. The roles of shear flow dispersion and dead zones in the River Severn, U.K. Hydrol. Earth Syst. Sci.
**2000**, 4, 355–371. [Google Scholar] [CrossRef] [Green Version] - Olver, F.W.J. Asymptotic and Special Functions; Academic Press: New York, NY, USA, 1974. [Google Scholar]
- Abramowitz, M.; Stegun, I.A. Handbook of Mathematical Functions; National Bureau of Standards Applied Mathematics Series-55; US Government Printing Office: Washington, DC, USA, 1972.
- Fischer, H.B. Discussion of “Simple method for predicting dispersion in stream”. J. Environ. Eng. Div.
**1975**, 101, 453–455. [Google Scholar] [CrossRef]

**Figure 1.**Comparison between the modified Bessel function ${I}_{1}$ and its approximation (Equation (22)) for four different values of ${t}_{LIM}^{*}$ : (

**a**) ${t}_{LIM}^{*}=1.5$, (

**b**) ${t}_{LIM}^{*}=1.10$, (

**c**) ${t}_{LIM}^{*}=1.05$, (

**d**) ${t}_{LIM}^{*}=1.02$.

**Figure 2.**BTCs generated by the TSM model at ${X}_{1}=500$ m, ${X}_{2}=1000$ m and ${X}_{3}=1500$ m and approximations for (

**a**) the decreasing limb in semi-log scale and for (

**b**) the rising limb in linear scale.

**Figure 3.**Normalized observed concentration data (“Cobs”, circles), the approximating trendlines (solid black lines), the simulated BTCs from the approximations (“Csim”, dashed-dot blue lines) and the optimized simulated BTCs (“Copt”, dashed red lines), for section X1 (panels (

**a**) and (

**c**) and section X2 (panels (

**b**) and (

**d**)). Panel (

**a**) and (

**b**) refer to the approximation of the rising limb of the BTC. Panel (

**c**) and (

**d**) are in semi-log scale and refer to the approximation of the decreasing part of the BTC.

**Table 1.**TSM parameters $A$, ${A}_{S}$, ${D}_{W}$, and $\alpha $ are used for the concentration curves in Figure 2, coefficients ${m}^{*}$, ${q}^{*}$, ${b}^{*}$, and $n$ of the approximation trendlines for the decreasing and rising parts at the three output sections ${X}_{i}$, and peak concentration ${C}_{p}$ at each section used for normalizing the BTC in Figure 2.

Section | Distance | ${\mathit{D}}_{\mathit{W}}$ | $\mathit{A}$ | ${\mathit{A}}_{\mathit{s}}$ | $\mathit{\alpha}$ | ${\mathit{t}}_{\mathit{L}\mathit{I}\mathit{M}}^{*}$ | $\mathit{n}$ | ${\mathit{b}}^{*}$ | ${\mathit{m}}^{*}$ | ${\mathit{q}}^{*}$ | ${\mathit{C}}_{\mathit{p}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|

(m) | (m^{2} s^{−1}) | (m^{2}) | (m^{2}) | (s^{−1}) | (-) | (s^{−1}) | (-) | (s^{−1}) | (-) | (g m^{−3}) | |

${X}_{1}$ | 500 | 0.735 | 2.00 | 1.0 × 10^{−1} | 1.0 × 10^{−4} | 1.400 | −1.75 × 10^{−3} | 4.63 × 10^{1} | 2.04 × 10^{−3} | −4.32 | 0.502 |

${X}_{2}$ | 1000 | 1.100 | −1.50 × 10^{−3} | 2.22 × 10^{3} | 1.63 × 10^{−3} | −7.27 | 0.347 | ||||

${X}_{3}$ | 1500 | 1.044 | −1.25 × 10^{−3} | 2.00 × 10^{4} | 1.31 × 10^{−3} | −8.96 | 0.274 |

**Table 2.**Coefficients ${m}^{*}$ , ${q}^{*}$, ${b}^{*}$, $n$ of the approximation trendlines in Figure 3; TSM parameters $A$, ${A}_{S}$, ${D}_{W}$, $\alpha $ obtained from the trendline approximations (“Approx.”) and from a numerical optimized fit (“Optim.”) of the BTCs; estimation of ${t}_{LIM}^{*}$ for each section and type of fitting, and peak concentration ${C}_{p}$ at each section used for normalizing the BTC in Figure 3.

Section | Distance | Fit | $\mathit{n}$ | ${\mathit{b}}^{*}$ | ${\mathit{m}}^{*}$ | ${\mathit{q}}^{*}$ | ${\mathit{D}}_{\mathit{W}}$ | $\mathit{A}$ | ${\mathit{A}}_{\mathit{s}}$ | $\mathit{\alpha}$ | ${\mathit{t}}_{\mathit{L}\mathit{I}\mathit{M}}^{*}$ | ${\mathit{C}}_{\mathit{p}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|

(m) | (s^{−1}) | (-) | (s^{−1}) | (-) | (m^{2} s^{−1}) | (m^{2}) | (m^{2}) | (s^{−1}) | (-) | (mg m^{−3}) | ||

${X}_{1}$ | 262 | Approx. | −2.11 × 10^{−3} | 98.5 | 2.21 × 10^{−3} | −2.278 | 0.279 | 0.202 | 2.93 × 10^{−2} | 4.00 × 10^{−4} | 1.33 | 48.12 |

Optim. | - | - | - | - | 0.243 | 0.200 | 3.72 × 10^{−2} | 5.65 × 10^{−4} | 1.21 | |||

${X}_{2}$ | 567 | Approx. | −1.80 × 10^{−3} | 4252.9 | 8.86 × 10^{−4} | −2.582 | 0.090 | 0.244 | 1.86 × 10^{−2} | 4.55 × 10^{−4} | 1.02 | 42.8 |

Optim. | - | - | - | - | 0.535 | 0.270 | 1.86 × 10^{−2} | 1.37 × 10^{−4} | 1.16 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Dallan, E.; Bottacin-Busolin, A.; Zaramella, M.; Marion, A.
A Method for Calibrating the Transient Storage Model from the Early and Late-Time Behavior of Breakthrough Curves. *Water* **2023**, *15*, 979.
https://doi.org/10.3390/w15050979

**AMA Style**

Dallan E, Bottacin-Busolin A, Zaramella M, Marion A.
A Method for Calibrating the Transient Storage Model from the Early and Late-Time Behavior of Breakthrough Curves. *Water*. 2023; 15(5):979.
https://doi.org/10.3390/w15050979

**Chicago/Turabian Style**

Dallan, Eleonora, Andrea Bottacin-Busolin, Mattia Zaramella, and Andrea Marion.
2023. "A Method for Calibrating the Transient Storage Model from the Early and Late-Time Behavior of Breakthrough Curves" *Water* 15, no. 5: 979.
https://doi.org/10.3390/w15050979