# Numerical Simulation of Water Renewal Timescales in the Mahakam Delta, Indonesia

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## Abstract

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## 1. Introduction

## 2. Study Area

^{2}, with a mean annual river discharge of the order of 3000 m

^{3}/s [32]. The central part of the river, located about 150 km upstream of the delta, is extremely flat. In this area, four large tributaries (Kedang Pahu, Belayan, Kedang Kepala, and Kedang Rantau) contribute greatly to the river flow, and several shallow-water lakes (e.g., Lake Jempang, Lake Melingtang, and Lake Semayang) are also connected to the river via a system of small channels. The water depth in these shallow-water lakes is of the order of 5 m.

## 3. Method

#### 3.1. Hydrodynamics Model

^{1/3}) for the Makassar Strait, (ii) a linearly increasing value in the delta region, from 0.023 (s/m

^{1/3}) in the coastal region to 0.0275 (s/m

^{1/3}) in the region from the delta front to the delta apex, (iii) a constant value of 0.0275 (s/m

^{1/3}) in the Mahakam River and its four tributaries, and (iv) a larger value of 0.0305 (s/m

^{1/3}) in the lakes.

#### 3.2. Water Renewal Timescales

#### 3.2.1. Age of Water

_{e}, is the Mahakam Delta. The water present in it consists of two components, i.e., the original water and the renewing water. The original water is the water present inside the domain at the beginning of the simulation, while the renewing water originates from the environment of the domain and progressively replaces the original water. The renewing water can enter the domain through upstream and downstream boundaries. The exact locations of the domain boundaries for timescale calculations are shown in Figure 3. The upstream boundary Γ

_{u}is placed at the river mouth (i.e., the interface between the river and the delta), while the downstream boundary Γ

_{d}is imposed at the delta front. According to [5], the original and the renewing waters in the delta can be treated as passive tracers and, hence, the different renewal water types can be modeled individually. If the concentrations of water originating from the upstream boundary Γ

_{u}and the downstream boundary Γ

_{d}of the delta are denoted C

_{u}and C

_{d}, respectively, the concentration of renewing water C

_{r}in the delta, consisting of both river and sea waters, is C

_{r}= C

_{u}+ C

_{d}. Note that the concentration of a water type is defined as the water fraction originating from the boundary under consideration. Accordingly, these concentrations are dimensionless functions of time and position.

_{i}and the related depth-mean age concentration α

_{i}of the water type under consideration. These equations read:

**u**is the depth-averaged horizontal velocity vector; κ is the diffusivity coefficient that is parameterized using the Okubo formulation [41], under the form:

_{k}being an appropriate constant, and Δ is the local characteristic length scale of the mesh. A value c

_{k}= 0.018 m

^{0.85}/s, which was calibrated from the best fit to the available salinity at 60 field sampling sites (see Figure 3) [42], is used to evaluate the diffusivity coefficient. The boundary and the initial conditions under which Equations (1) and (2) are to be solved are listed in Table 1. There are no concentration or age concentration fluxes through the impermeable boundaries. Finally, the age of water type i is calculated as follows [17]:

_{0}(t,

**x**) = tC

_{0}(t,

**x**), implying that the age of the original water is equal to the elapsed time, i.e., a

_{0}(t,

**x**) = t [5,40]. This is because the original water can only leave the domain of interest as time progresses, and there is obviously no source of original water. This piece of information is of little diagnostic value. However, it is an additional element supporting the well-foundedness of the partial differential problems laid out above.

#### 3.2.2. Residence Time

_{r}is defined as the time taken by a water parcel initially in the domain of interest to leave it for the first time. It can be obtained by solving, under the initial and the boundary conditions mentioned below, the partial differential equation:

**n**is the outward normal unit vector to the boundary, u

_{n}(=

**u**.

**n**) is the normal velocity, and L

^{*}is the width of the unresolved boundary layer. In [44], it was found to be appropriate to prescribe that the length L

^{*}be equal to the local mesh size.

_{r}(t = T,

**x**), which, unfortunately, is unknown. The reverse time marching is initialized with θ

_{r}(t = T,

**x**) = 0 and, after a spin up period whose duration is equal to a couple of times the typical value of the residence time, a regime solution is attained and, hence, the “initial condition” is no longer important, which is in accordance with Delhez et al. [21].

#### 3.2.3. Exposure Time

_{e}at a given time can re-enter the domain many times before escaping it definitively. To account for this, the concept of exposure time is introduced, that is, the total time spent in the domain. The exposure time θ

_{e}can be numerically computed by solving the following Equation [5]:

_{e}, respectively.

_{e}(Ω

_{e}⊂ Ω). In the present study, the domain Ω is extended significantly both upstream and downstream of the delta, and it is taken to be identical to the computational domain used for the hydrodynamic simulations.

_{e}= 0 is imposed. At upstream boundaries, where the flow is assumed to be always in the downward direction, the expression κ∇θ

_{e}.

**n**= 0 is used [5]. The initial value of the exposure time is set equal to zero in the whole computational domain.

#### 3.2.4. Finite Element Implementation

## 4. Results and Discussion

#### 4.1. Age of Water

#### 4.1.1. River Water

^{3}/s) of the concentration and the age of the river water along the three considered transects (see Figure 3). The river water concentration is equal to unity at 26 km from the river mouth (Figure 4a). Beyond this point, the concentration significantly decreases from unity to zero, resulting in a significant increase of the age of water. As shown in Figure 5a, a similar trend of the concentration is also observed for the high-flow period (during which the daily water discharge at the river mouth varies between 3500 and 6500 m

^{3}/s). The point from which the concentration begins to decrease occurs around km 31. The difference in location of the point from which the concentration starts decreasing in the low and high flow conditions can be explained by differences of water discharges.

_{d}, the concentration of the river water (C

_{u}) and its age concentration (α

_{u}) are both prescribed to be zero, implying that the corresponding age (a

_{u}= α

_{u}/C

_{u}) is an indeterminate form, i.e., 0/0. In the absence of an analytical solution, explicitly evaluating this limit is impossible. However, this issue may be successfully addressed by first deriving the equation for the age. This is achieved by manipulating concentration and age concentration Equations (1) and (2) and casting the resulting relation into the appropriate form, eventually yielding:

_{d}, the concentration is prescribed to be zero. Therefore, Equation (14) simplifies to ${\left[\nabla {a}_{u}.\nabla {C}_{u}\right]}_{\mathrm{x}\in {\mathsf{\Gamma}}_{d}}=0$. In addition, since C

_{u}is zero on Γ

_{d}, its gradient is parallel to

**n**, the unit normal vector to the boundary. As a consequence, the age satisfies boundary condition ${\left[\nabla {a}_{u}.\mathbf{n}\right]}_{\mathrm{x}\in {\mathsf{\Gamma}}_{d}}=0$, i.e., the derivative of the age in the direction normal to the boundary must be zero, which is why the age should have a finite value on the boundary. To the best of our knowledge, this theoretical result is novel. The simulated age profiles displayed on Figure 4 and Figure 5 are in agreement with it.

#### 4.1.2. Total Renewing Water

#### 4.2. Residence Time

#### 4.3. Exposure Time

#### 4.4. Return Coefficient

_{e}and θ

_{r}are exposure and residence time, respectively. According to this definition, the return coefficient varies between zero and one. The lower limit of the return coefficient (rc = 0) occurs when the exposure time equals the residence time or when no water parcels re-enters the delta. In contrast, the upper limit of the return coefficient (rc = 1) is reached if the residence time is much smaller than the exposure time.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Grid of the computational domain: (

**a**) whole computational mesh and (

**b**) zoom on upstream domain and delta, showing the connection between the 1D and the 2D sub-domains (dashed-blue lines) and upstream boundary locations (black dots).

**Figure 3.**Bathymetry in the delta and the delta front, with field sampling sites of salinity (blue dots), upstream and downstream boundaries (black dashed lines) for computing the water renewal timescales, the three considered transects (blue lines) and three points (red squares).

**Figure 4.**Calculation results of: (

**a**) concentration and (

**b**) water age during the low flow period, along the three considered transects (see Figure 3) for the case of the river water.

**Figure 5.**Calculation results of: (

**a**) concentration and (

**b**) water age during the high flow period, along the three considered transects (see Figure 3) for the case of the river water.

**Figure 6.**Calculation results of the age of the total renewing water during the: (

**a**) low flow period and (

**b**) high flow period, along the three considered transects (see Figure 3).

**Figure 7.**Distribution of the age of the total renewing water for the: (

**a**) low and (

**b**) high flow period. The color bar is cropped at 8 days in order to focus on the variation of the age in deltaic channels, and the unit is day.

**Figure 10.**Simulated values of the residence time at the: (

**a**) P

_{1}; (

**b**) P

_{2}; (

**c**) P

_{3}(see Figure 3) for the low period and simulated values of the residence time at the: (

**d**) P

_{1}; (

**e**) P

_{2,}(

**f**) P

_{3}for the high flow period. In each panel, the back line indicates the instantaneous value, while the red curve presents the averaged value over 24 h taken to filter out tidal oscillation.

**Figure 11.**Distribution of the residence time in the whole delta for the: (

**a**) low; (

**b**) high flow period. The color bar is cropped at 13 days in order to focus on the variation of the residence time in deltaic channels, and the unit is day.

**Figure 12.**Distribution of the exposure time in the whole delta for the: (

**a**) low; (

**b**) high flow period. The color bar is cropped at 13 days in order to focus on the variation of the exposure time in deltaic channels and the unit is day.

Original Water | Renewing Water | |||||
---|---|---|---|---|---|---|

River Water | Total Renewing Water | |||||

t = 0 | C_{o} = 1 | α_{o} = 0 | C_{u} = 0 | α_{u} = 0 | C_{r} = 0 | α_{r} = 0 |

x ∈ Γ_{u} | C_{o} = 0 | α_{o} = 0 | C_{u} = 1 | α_{u} = 0 | C_{r} = 1 | α_{r} = 0 |

x ∈ Γ_{d} | C_{o} = 0 | α_{o} = 0 | C_{u} = 0 | α_{u} = 0 | C_{r} = 1 | α_{r} = 0 |

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## Share and Cite

**MDPI and ACS Style**

Pham Van, C.; De Brye, B.; De Brauwere, A.; Hoitink, A.J.F.; Soares-Frazao, S.; Deleersnijder, E.
Numerical Simulation of Water Renewal Timescales in the Mahakam Delta, Indonesia. *Water* **2020**, *12*, 1017.
https://doi.org/10.3390/w12041017

**AMA Style**

Pham Van C, De Brye B, De Brauwere A, Hoitink AJF, Soares-Frazao S, Deleersnijder E.
Numerical Simulation of Water Renewal Timescales in the Mahakam Delta, Indonesia. *Water*. 2020; 12(4):1017.
https://doi.org/10.3390/w12041017

**Chicago/Turabian Style**

Pham Van, Chien, Benjamin De Brye, Anouk De Brauwere, A.J.F. (Ton) Hoitink, Sandra Soares-Frazao, and Eric Deleersnijder.
2020. "Numerical Simulation of Water Renewal Timescales in the Mahakam Delta, Indonesia" *Water* 12, no. 4: 1017.
https://doi.org/10.3390/w12041017