# 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 |

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**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 |

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**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