# Age of Water Particles as a Diagnosis of Steady-State Flows in Shallow Rectangular Reservoirs

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## Abstract

**:**

## 1. Introduction

- Persson [27] applied a 2D depth-averaged model to estimate the residence time distribution in ponds of various layouts, based on the advection–diffusion equation for a tracer. The results revealed that a subsurface berm or an island placed in front of the inlet reduces short-circuiting, and improves the effective volume and degree of mixing;
- Sonnenwald et al. [6] used a 3D computational model to obtain residence time distributions for vegetated storm water treatment ponds, showing that the presence of vegetation results in residence times close to those of plug flow conditions;
- By means of tracer studies with laboratory-scale models, Guzman et al. [3] evaluated the residence time distribution for 54 topographies of storm water detention ponds and treatment wetlands, to compare the hydraulic performances of the various designs.

## 2. Data and Method

#### 2.1. Characteristics of the Considered Reservoirs

^{0.6}b

^{0.4}). Critical values of SF were determined by Dufresne et al. [14] based on laboratory experiments. A detached jet flowing straight from the inlet to the outlet is observed for relatively short reservoirs (SF < 6.2). In contrast, for longer reservoirs (SF > 6.8), the jet reattaches on either of the side-walls. In the case of particularly long reservoirs, a flow pattern close to plug flow is observed in the downstream portion of the reservoir. For 6.2 ≤ SF ≤ 6.8, a transition zone was reported, in which both detached and reattached flow patterns may occur.

#### 2.2. Flow Model and Computed Flow Fields

**u**, and the flow depth, H. These variables depend on the horizontal position vector

**x**= (x, y), where x and y are Cartesian coordinates. The horizontal eddy viscosity and diffusivity are evaluated by means of a two-length-scale depth-averaged k–ε turbulence model, as described by Camnasio et al. [18].

#### 2.3. Water Age Distribution Function

^{in}, Γ

^{out}and Γ

^{imp}, respectively.

**x**) = δx δy H (

**x**). The age distribution function is denoted as c (

**x**, τ), where the independent variable τ is the age (0 ≤ τ ≤ ∞). In accordance with Delhez et al. [28], the following definition is adopted: in the abovementioned elemental control domain, the mass of the water whose age lies in the interval [τ, τ + δτ] tends to ρ c (

**x**, τ) δV δτ in the limit δV, δτ → 0 (with ρ as the water density). The physical dimension of the age distribution function c (

**x**, τ) is time

^{−1}.

**K**(

**x**) denotes the diffusivity tensor, which must be symmetric and positive definite. Equation (1) is of a parabolic nature. Formally, Equation (1) is similar to an “evolution” equation, in which the independent variable τ plays a role equivalent to that of time in a classical advection–diffusion equation. A detailed derivation of this equation, as well as some of the mathematical properties of its solution, may be found in Deleersnijder and Dewals [35].

^{in}. The incoming-flux condition reads [36]:

#### 2.4. Mathematical Properties of the Mean Water Age

**x**, τ), namely c (

**x**, τ) is non negative, τ c (

**x**, τ) tends to zero in the limit τ→ ∞, and c (

**x**, τ) satisfies the following integral constraint expressing that the concentration of water C(

**x**) is equal to unity at any location in the domain [28]:

**x**is defined as the first-order moment of the water age distribution function:

**x**, τ). The obtained values of a(

**x**) exhibit the following properties [35]:

- The solution is unique;
- a(
**x**) is non-negative; - The value of the mean age on the incoming boundary is not zero, unless diffusion is zero. Indeed, on Γ
^{in}, there is a mixture of water particles that are entering the domain and particles that have been moving for some time in the domain and were brought back to the incoming boundary by diffusion; - The maximum of the mean age is not necessarily on the outgoing boundary Γ
^{out}, but it may be located inside the domain of interest. This contrasts with a one-dimensional setting, in which the maximum of mean age occurs at the outgoing boundary because there is only one path from the reservoir inlet to the outlet; - On the outgoing boundary Γ
^{out}, the average value of the mean age satisfies:

#### 2.5. Computational Procedure

**x**, τ) from the partial differential problem (5)–(9) is unlikely to be the optimal method due to the Dirac delta impulse prescribed as incoming boundary condition. Therefore, as proposed earlier [37], an alternate approach was set up here. In line with the theory of linear system dynamics, stating that the impulse response can be evaluated as the derivative of the step response, i.e., by prescribing a step function as inflowing boundary condition instead of the Dirac delta impulse. The step response is of course much easier to compute numerically than the impulse response.

**x**, τ), was obtained by solving the following partial differential problem, which is akin to a standard advection–diffusion problem (in which τ plays a role equivalent to that of the time):

**K**= κ

**I)**was used, with the value of κ taken as equal to the eddy viscosity.

**x**, τ) was computed by evaluating the derivative of b (

**x**, τ):

**x**, τ) tends to unity for τ → ∞. Finally, once the age distribution function is computed, the mean water age a(

**x**) can be evaluated thanks to Equation (8). Note that an alternate approach consists in simulating the problem in Laplace space [38].

## 3. Results

#### 3.1. Model Verification: Prismatic Channel

#### 3.2. Distribution of Water Age at Reservoir Outlet

- In the nearly-plug-flow configurations (Tests 8, 9 and 10), the water age distribution at the outlet is unimodal, but strongly skewed towards the higher values of τ;
- Looking sequentially at the results obtained for Tests 7, 6 and 5, all corresponding to configurations with a reattached jet, it appears that the water age distribution at the outlet shifts gradually from a unimodal to a bimodal distribution;
- In all cases with a detached jet, the water age distribution at the outlet is bimodal (Tests 4, 3 and 2), or even multimodal in the case of Test 1.

- V/Q, which would correspond to the transit time through the reservoir in the case of a perfect plug flow all over the reservoir;
- L/(Q/S
^{in}), would be equal to the transit time through the reservoir if the jet at the inlet was not diffused at all in the crosswise direction.

^{in}) tend to provide an envelope for the mode, median and mean ages. In the case of nearly-plug-flow, the characteristic time V/Q is found to approximate remarkably well the mean water age, with overestimations of no more than 5%. Furthermore, in one case of a reattached jet in a particularly elongated reservoir (Test 7), V/Q matches the mean age with a difference of less than 1%. In all other configurations, V/Q remains a reasonable approximation for the mean age, with overestimations of no more than 22%. Only in the case corresponding to a detached jet in a relatively wide basin (Tests 1), V/Q overestimates the mean age by a factor 2.5.

^{in}), is generally between 1.5 and 2.5 times smaller than the median age and the mode of the age distribution. Only in the configurations with a detached jet (Tests 1 and 2), L/(Q/S

^{in}) approximates well the mode and the median age, with an underestimation of the order of 10% to 20% only. Note that only for Test 1, a small portion of the water particles show a transit time even smaller than the characteristic value L/(Q/S

^{in}). This results from the velocity profile over the jet cross-section. Indeed, the central part of the jet at the inlet has a velocity higher than Q/S

^{in}. In all other cases, the characteristic time L/(Q/S

^{in}) substantially underestimates the actual mode, median and mean values of the age distribution.

^{in}) appears as a useful time scale for the configurations with a detached jet flowing directly from the inlet to the reservoir outlet (Tests 1 and 2), whereas the characteristic time scale V/Q approximates well the mean age at the outlet in the case of elongated reservoirs with nearly-plug-flow conditions (Tests 8, 9 and 10) or close to (Tests 6 and 7).

#### 3.3. Distribution of Water Age within the Reservoirs

#### 3.4. Mean Age in the Reservoirs

^{in}) than V/Q.

## 4. Conclusions

^{in}) (with L the reservoir length and S

^{in}the inlet cross-section), and their relative merits are discussed as a function of the reservoir dimensions. The former time scale provides an estimate of practical relevance for the mean water age at the outlets of elongated reservoirs, whereas the latter is a useful proxy for the mode and the median age at the outlets of relatively short reservoirs. The spatial distribution of the mean age within the reservoir is shown to vary strongly with the reservoir’s geometry, and these variations are attributed to characteristics of the flow fields.

## Supplementary Materials

^{2}s

^{−1}) computed by Camnasio et al. (2014), Figure S3: Step response b(

**x**,τ) computed for various values of τ in the reservoir corresponding to Test 5, Figure S4: Step response b(

**x**, τ) computed for various values of τ in the reservoir corresponding to Test 6, Figure S5: Comparison of age statistics and characteristic times (linear scale), Figure S6: Comparison of age statistics and characteristic times (log scale), Figure S7: Water age distribution at six locations in the reservoir for Test 7: (

**a**) considered locations A, B, C, D, E and F, (

**b**) cumulated distributions b and (

**c**) water age distributions c. The blue (–), green (–) and red (–) vertical lines show the mode, median and mean of the age distribution, respectively. The plain (–) and dashed (--) black vertical lines represent the characteristic times V/Q and L/(Q/S

^{in}), respectively.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Geometry of the rectangular shallow reservoirs and main notations. Ω is the domain of interest, while Γ

^{in}, Γ

^{out}and Γ

^{imp}refer to the inlet, outlet and impermeable boundaries, respectively.

**Figure 2.**Examples of computed flow fields: (

**a**) detached jet, (

**b**) detached jet, near transition, (

**c**) reattached jet, and (

**d**) reattached jet with plug flow in the downstream part of the reservoir [18]. The color scale represents the velocity magnitude (m/s).

**Figure 3.**(

**a**) Computed (symbols) and analytical (solid lines) profiles of water mean age a(x) in a one-dimensional prismatic channel, and (

**b**) corresponding age distribution functions.

**Figure 4.**Water age distribution at the reservoir outlet in all configurations: (

**a**) cumulated distributions b and (

**b**) distributions c. Abbreviations “D”, “NT”, “R” and “R-NPF” refer respectively to detached jet, near-transition, reattached and reattached with nearly-plug-flow. The blue (

**−**), green (

**−**) and red (

**−**) vertical lines (

**−**) show the mode, median and mean of the age distribution, respectively. The plain (

**−**) and dashed (--) black vertical lines represent the characteristic times V/Q and L/(Q/S

^{in}), respectively.

**Figure 5.**Water age distribution at six locations in the reservoir for Test 5: (

**a**) considered locations A, B, C, D, E and F, (

**b**) cumulated distributions b and (

**c**) water age distributions c. The blue (

**−**), green (

**−**) and red (

**−**) vertical lines show the mode, median and mean of the age distribution, respectively. The plain (

**−**) and dashed (--) black vertical lines represent the characteristic times V/Q and L/(Q/S

^{in}), respectively.

**Figure 6.**Computed mean age a, scaled by the characteristic time V/Q. Contours are represented only for values of a/(V/Q) ≥ 4, i.e., beyond the range of the color scale. Subfigures (

**a**–

**j**) correspond to the 10 reservoir configurations.

**Figure 7.**Histograms of the non-dimensional mean water age a/(V/Q) across the 10 considered reservoirs. Subfigures (

**a**–

**j**) correspond to the 10 reservoir configurations.

Persson [27] | Sonnenwald et al. [6] | Guzman et al. [3] | Zhang et al. [11] | Present Study | |
---|---|---|---|---|---|

Modeling approach | 2D | 3D | Lab | 3D | 2D |

Considered layouts | 13 | 4 × 3 | 45+ | 3 | 10 |

Full domain considered (i.e., no a priori assumption of symmetric flow) | ✓ | ✓ | ✓ | ✓ | |

Residence time distribution (at the outlet) | ✓ | ✓ | ✓ | ✓ | |

Mean water age throughout the reservoir | ✓ | ✓ | |||

Age distribution throughout the reservoir | ✓ |

Test ID | L (m) | B (m) | SF (−) | Type of Flow Pattern | Test ID in Camnasio et al. [18] |
---|---|---|---|---|---|

1 | 3 | 4 | 3.6 | Detached jet (D) | 5 |

2 | 4 | 3 | 5.8 | Detached jet (D) | 6 |

3 | 5 | 4 | 6.0 | Near-transition (NT) | 4 |

4 | 5.8 | 4 | 6.9 | Near-transition (NT) | 2 |

5 | 6 | 4 | 7.2 | Reattached jet (R) | 1 |

6 | 4 | 2 | 7.6 | Reattached jet (R) | 7 |

7 | 4 | 1 | 12.5 | Reattached jet (R) | 8 |

8 | 6 | 1 | 18.8 | Reattached nearly-plug-flow (R-NPF) | 9 |

9 | 6 | 0.75 | 24.0 | Reattached nearly-plug-flow (R-NPF) | 10 |

10 | 6 | 0.6 | 29.7 | Reattached nearly-plug-flow (R-NPF) | 11 |

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## Share and Cite

**MDPI and ACS Style**

Dewals, B.; Archambeau, P.; Bruwier, M.; Erpicum, S.; Pirotton, M.; Adam, T.; Delhez, E.; Deleersnijder, E. Age of Water Particles as a Diagnosis of Steady-State Flows in Shallow Rectangular Reservoirs. *Water* **2020**, *12*, 2819.
https://doi.org/10.3390/w12102819

**AMA Style**

Dewals B, Archambeau P, Bruwier M, Erpicum S, Pirotton M, Adam T, Delhez E, Deleersnijder E. Age of Water Particles as a Diagnosis of Steady-State Flows in Shallow Rectangular Reservoirs. *Water*. 2020; 12(10):2819.
https://doi.org/10.3390/w12102819

**Chicago/Turabian Style**

Dewals, Benjamin, Pierre Archambeau, Martin Bruwier, Sebastien Erpicum, Michel Pirotton, Tom Adam, Eric Delhez, and Eric Deleersnijder. 2020. "Age of Water Particles as a Diagnosis of Steady-State Flows in Shallow Rectangular Reservoirs" *Water* 12, no. 10: 2819.
https://doi.org/10.3390/w12102819