The pollutants discharged into semi-enclosed bays are mixed with the receiving water through physical processes such as convection transport and dilution diffusion [1]. In the process of exchanging water between the inner and outer bays, the concentration of contaminants gradually decreases, which improves water quality. The exchange and mixing processes are closely related to self-purification capacity and the transport of nutrients, among other factors, which has aroused the interest of researchers, increasingly frequently in studies of water quality in estuaries and coastal areas [1-6]. As a concept related to water exchange, transport time scales such as water age, flushing time, residence time and renewal time are often used as parameters to represent the time scale of physical transport processes [1]. These time scales can describe how some water quality parameters (e.g. eutrophication) change under different conditions [1,7]. The exchange of water in a bay is the result of the complex interaction of factors such as geographical features, waves, tides, winds and currents, and therefore knowledge of the scale of transport time is of great importance for protecting the marine environment and maintaining the ecological balance of coastal areas [1]. Residence time has marked effects on the ecological quality of water bodies in semi-enclosed bays and estuaries [8]. In semi-enclosed domains such as Buenaventura Bay the water renewal capacity is mainly controlled by the exchange flow between the outer and inner domains as well as by the inner circulation [8]. Internal circulation is affected by bathymetry, among other aspects. This parameter plays a very important role in the vulnerability of water bodies to eutrophication and pollution [9]

In Buenaventura Bay (Fig. 1), Colombia's main port terminal on the Pacific Ocean operates. It is located between latitudes 3° 44' and 3° 56' N and longitudes 77° 1' and 77° 20' W. It is 21 km long and its width varies between 3.4 km at the sea exit and 5.5 km in the internal part. It is narrow and elongated, follows an orientation SO to NE with an approximate extension of 682 km² and average depths of 5 m. has a single entrance known as La Bocana, formed by a strait of 1.6 km [10]. Its interior is an estuarine system due to the combination of the salty water of the Pacific Ocean with the fresh water of the rivers and estuaries that flow into it, such as Dagua (66 m³/s), Achicayá (99 m³/s), Humane (30 m³/s), Gamboa (30 m³/s), Aguacate (10 m³/s), San Antonio (20 m³/s), Hondo (10 m³/s) and Agua Dulce (80 m³/s). The values of tributaries discharges were obtain from [11]. The external sector has direct communication with the open sea and receives all its influence [10].

Figure 1 Source: The Authors.

In order to improve competitiveness, the port terminals carry out dredging activities of their access channel that seek to take it to a final depth of 17.7 m in the inner part of the bay where the city of Buenaventura is also located, officially declared by the Congress of the Republic of Colombia as a Special, Industrial, Port, Biodiverse and Ecotourism District in 2013. Through this port, Colombia exports 80% of its coffee and 8.7% of all its international maritime trade.

To study the change in residence time in Buenaventura Bay due to the dredging process, a combination of a two-dimensional hydrodynamic model and a particle trajectory model was used.

A two-dimensional finite element model RMA2 [12] was used to provide the values of the water surface elevation and the components of the horizontal velocity vector. It computes water surface elevations and horizontal velocity components for subcritical, free-surface flow in two-dimensional flow fields [13]. The RMA2 model is frequently used in physical oceanography to represent processes such as tides and currents in coastal regions and estuaries, its use has been documented by [14-21]. RMA2 solves the depth-averaged equations of fluid mass and momentum conservation in two horizontal directions. These equations can be written as follows:

Momentum in X-direction equation:

Y-direction momentum equation:

Continuity equation:

where, h is depth, u, v are velocities in the Cartesian directions, ρ is density of fluid, E is eddy viscosity coefficient, g is acceleration due to gravity, n is Manning’s roughness, δ is empirical wind shear coefficient, v _α is wind speed, φ is wind direction, ω is rate of earth’s angular rotation and 𝜙 is local latitude.

The Lagrange trajectory model used determines the movement of discrete particles from an Euler field of velocities provided by the RMA2 model, for which the following difference equation is solved.

Where, x,i are the values of the particle position at time t, y u,i corresponds to the velocity field. This equation is solved numerically using the Runge-Kutta method of fourth order, to minimize error [8]. The particles were virtually released two days after the RMA2 model was initiated to avoid initial instability due to their starting or heating period and their trajectory was determined until the end of the hydrodynamic simulations.

The bottom roughness was the main calibration parameter of the hydrodynamic model proposed for Buenaventura Bay. The RMA2 model allows assigning discrete bottom roughness values to different areas of the simulation domain, for this specific case 7 Manning coefficient values were applied to the same number of element types. The types of elements were differentiated by depth, proximity to open borders and to the coast. The maximum variation of the RMS for both calibration and validation with water level data was between 9.4 and 10.7 % while the correlation coefficient was higher than 0.96 and the minimum predictive ability was 0.94. These values are adjusted in the range suggested by [24]. The graphical comparison between the measured data and the results of the hydrodynamic model for the tide are presented in Table 3 and Fig. 6, where the fit between the time series can be seen confirming what was shown by the statistical valuation techniques.

Table 3

Reference	Water level				Actual speed
	RMS - % Maximum variation	Correlation coefficient	Skill	RMS - % Maximum variation	Correlation coefficient	Skill
T1 (WRL7)	9.4	0.975	0.97	-		-
T2(RCM9)			-	10.5	0.94	0.89-
T1 (RCM9)		-	-	12.3-	0.84-	0.81
T2(WRL7)	10.7	0.961	0.94

Source: The Authors

Figure 6 Source: The Authors

The comparison of the velocity data showed, as did the water levels, that the model reproduces the measurements appropriately, both in calibration and validation, as can be seen in Fig. 7. The statistical values used in the model evaluation revealed that the values obtained (see Table 3) are in the acceptability ranges suggested by [2].

Figure 7 Source: The Authors

According to the values of the parameters RMS, Correlation Coefficient and Skill presented in Table 3 for both water levels and current velocities, the authors considered the model calibrated and validated.

The results of the model showed a good approach with the field measurements, however, in the specific case of the tide a very good fit was achieved during the ebb tide reflux, while during the flow the peaks were not captured perfectly. This situation may suggest that the flow dominates the hydrodynamic regime of Buenaventura Bay, however there are few previous studies that allow us to conclude on the matter. The results of the current velocities showed an equal tendency to reproduce the peaks of the water levels with a smaller approach. Some simplifications made at the time of implementation of the model, among them, attributing constant flows to the main tributaries and neglecting the smaller tributaries can influence the peaks of water levels and transport during ebb and flow, this simplification is due to the scarcity of data measuring the flows of these streams. Despite this, the statistics used to evaluate the accuracy of the model yielded values with acceptable error ranges.

From the results of the calibrated and validated model, the spatial distribution of the mean velocity (m/s) in Buenaventura Bay is presented in Fig. 8 for the scenario with bathymetry before the dredging intervention. The highest proportion of the domain has average speeds of less than 0.30 m/s. In the sectors close to the access channel, the model showed average speeds between 0.3 and 0.6 m/s. Few areas in the bay have speeds close to 1 m/s or higher, see box in the discontinuous red line in Fig. 8, enlarged in Fig. 9; according to the results of the extended period run (simulation of a year) the only area where speeds of the order of 1 m/s are recorded corresponds to a strait formed between the island and the mainland of Buenaventura (see Fig. 9).

Figure 8 Source: The Authors

Figure 9 Source: The Authors

The application of the particle trajectory model in the velocity fields of the initial bathymetry scenario without dredging shows that the spatial distribution of residence time varies from 1.3 to 89 days (Fig. 10). In the outer part of the bay the lowest residence times are presented, generally less than 30 days, these zones correspond to a more direct communication with the Pacific Ocean and in general show higher average speeds compared to the internal part of the bay. In the middle part of the bay, residence times of between 30 and 60 days are obtained.

Figure 10 Source: The Authors

By varying the bathymetry of the access channel in the way projected by the dredging works (scenario 2), lower velocities are obtained than those of scenario 1. In Fig. 11 it can be appreciated how the spatial variation of the velocity fields decreases significantly in comparison with scenario 1 shown in Fig. 8. This local decrease in the velocity field implies an increase in residence time which was confirmed by the results of the particle trajectory model applied to scenario 2 (see Fig. 12).

Figure 11 Source: The Authors

Figure 12 Source: The Authors

When making the difference with the results of the particle trajectory model, it was found that the deepening of the access channel of Buenaventura Bay can increase the residence time in the same by up to 12 days depending on the area of the same, as shown in Fig. 13.

Figure 13 Source: The Authors.

A complete and rigorous simulation of residence time requires a greater availability of freshwater input data, which in turn can have a marked seasonality in the amount of flow, given the periods of winter and summer: in the rainy season the rivers drastically increase their flows and therefore their discharge to the bay. For the development of this work it was not possible to count with the time series of the tributary rivers flows, for this reason their discharges were considered constant throughout the simulations. The flows of the rivers during the rainy season can influence the decrease of the time of residence in the bay in relation to the increase of flows.

These aspects were not considered in this work and therefore corresponds to one of the simplifications made in the implementation of the model.

Later on, it is necessary to consider the influence of the of tributaries discharges in order to make a more precise estimation of the time of residence of the bay. Since the hydrodynamic model reproduces the velocity fields very well, the results obtained in estimating the residence times are considered close to the behavior of this parameter in the bay. New considerations must be made including tributary discharges and thermohaline conditions. The estimation of the time of residence was considered with the average and constant discharge of the tributaries, the results obtained are indicators of average conditions in the bay of Buenaventura. The extreme values due to seasonal changes generated by the maximum and minimum flows and by the time series of the tributaries discharges were not considered in this study.

The Residence time identifies the time it takes for a particle to leave the study domain. The results of the simulations show that the time of residence in Buenaventura Bay has a high spatial variation, from a few days at the exit of the bay, where it communicates with the Pacific Ocean, to several months in the innermost part. This variation is related to currents, given that in the area where there is a shorter residence time, in addition to coinciding with a more direct communication with external waters, there are current velocities of greater magnitude. The higher currents force the plots of water to leave the bay faster.

The high residence times of the bay of Buenaventura suggest that it functions as a trap for the sedimentary solids discharged by the tributary rivers. The high load of solids thrown by the tributary rivers (Fig. 14) and the high residence time combined with weak currents cause the solids to sediment and as they accumulate in the bed of the bay, they generate greater requirements for dredging over time.

Figure 14 Source: Google Earth, 2019

Buenaventura bays has high residence times which negatively affects the possibility of self-purification of the sheet of water. The pollutants and nutrients discharged into this body of water tend to be concentrated because of the low dilution potential shown by the high residence times found.

Higher residence times in Buenaventura Bay are related to the magnitudes of weak currents in most of it, in general there are magnitudes lower than 0.4 m/s in a large portion of it, as a consequence the water remains a longer time.

The dredging processes produce a local decrease in the velocity fields inside the Buenaventura Bay, generating a drastic increase in residence time. This increase can be up to 12 days in some places.