Interrelationship between Wall and Beach Erosion in Loc An, Vietnam: Remote Sensing and Numerical Modeling Approaches

Abstract

Beach erosion and coastal protection are complex and interconnected phenomena that have a substantial impact on coastal environments worldwide. Among the various coastal protection measures, seawalls have been widely implemented to mitigate erosion and protect coastal assets. However, the interrelationship between beach erosion and seawalls remains a critical topic for investigation to ensure effective and sustainable coastal management strategies. Seawalls impact the shoreline, particularly through the “end effect”, where the seawall functions similarly to a groin, causing erosion on the downdrift side relative to the direction of wave approach. This study provides a detailed analysis of the interplay between beach erosion and seawall structures in Loc An, Vietnam, employing both remote sensing and numerical approaches. Sentinel-2 images were employed together with an analytical solution to observe the shoreline change at the Loc An sand spit and to determine input values for the numerical model. Based on the shoreline dynamics, a numerical scheme was employed to study the shoreline evolution after the construction of a seawall. Our findings show that the shoreline evolution can be divided into three stages: (1) The first stage corresponds to the elongation of the sand spit without interference from coastal structures. (2) The second stage shows the effect of jetties on the shoreline, as signaled by the buildup of sand updrift of the jetties. (3) The third stage shows the effectiveness of the seawall, where the shoreline reaches its equilibrium condition. The study provides a quick and simple method for estimating shoreline diffusivity (ε) in situations where measured data is scarce.

1. Introduction

Coastal areas are among the most dynamic and vital regions on Earth, serving as the frontier between land and sea [1]. These zones are not only crucial for biodiversity and ecosystem services but also support a significant portion of the global population through tourism, fisheries, and as gateways for trade and transportation. Among these environments, river mouths and inlets hold a special place because of their ecological, economic, and social importance. These areas experience substantial changes in morphological features, driven by a complex interplay of natural factors such as typhoons [2], river flows [3,4], tides [5,6], waves [7,8], and longshore currents [9,10]. The constant reshaping of these landscapes poses challenges for their physical stability and the communities that depend on them.

To mitigate these challenges and stabilize river mouths and inlets, various solutions have been implemented [11,12,13], with the construction of coastal structures like jetties and seawalls being prominent. While the impact of these interventions on shoreline dynamics has been extensively studied globally [14,15,16,17,18], research in Vietnam remains scarce [19]. This gap is primarily due to the lack of measured data, which is crucial for understanding and predicting coastal behavior under the influence of both natural events and human interventions.

With the advancement of remote sensing technology, monitoring and measuring the coastal environment has become easier than ever. This modern technique provides an effective tool for studying the coastal zone, offering coverage at various resolutions and regular temporal frequencies [20]. In recent years, remote sensing has been increasingly used in a wide range of ocean engineering applications, including the monitoring of coastal features such as barrier islands [21], shoreline dynamics [22], changes in mangrove forests [23], and even floating litter on the sea surface [24]. Furthermore, novel approaches have been introduced to enhance coastal applications [25,26,27].

This article aims to explore the effectiveness of combining traditional models (such as shoreline modeling) with modern techniques (like remote sensing) in coastal studies, with a specific focus on the Loc An sand spit at the Loc An estuary in Ba Ria-Vung Tau province (Figure 1). In this study, the data extracted from Sentinel-2 images using remote sensing techniques will be used to calibrate and validate the simple shoreline model. Despite the limited availability of field data, this study combines remote sensing techniques with a simple numerical model to explore the influence of coastal structures on shoreline evolution in regions where direct measurements are scarce. This research adds to the understanding of coastal management and demonstrates the potential of integrating multiple scientific methods to overcome data limitations, offering useful insights for similar studies in other parts of the world.

The study area is influenced by prevailing northeast winds, resulting in a net longshore sediment transport (LST) from northeast to southwest. This has led to the elongation of the Loc An sand spit and a tendency for the river mouth to shift southwestward. The waves, predominantly from the east-northeast direction, with significant heights ranging from 1.51 m to 3.0 m, generate longshore currents that drive sediment transport along the coast in the northeast-to-southwest direction [5]. As the Loc An estuary is located along the East Sea of Vietnam, the semi-diurnal tide also plays a role in the morphological changes of the estuary. Information on the tidal conditions is shown in Figure 2. Here, the water level is measured at the Vung Tau Oceanographic Station, which is located approximately 30 km southwest of the Loc An estuary (see Figure 1).

2. Materials and Methods

2.1. Remote Sensing

Launched in 2015 and 2017, respectively, Sentinel 2A and 2B are twin satellites orbiting the poles, established under the European Copernicus initiative [28]. Their purpose is to perform systematic observations of the atmosphere, land, and oceans. These satellites are equipped with a multispectral imager (MSI) that captures imagery in 13 spectral bands ranging from 443 nm to 2190 nm, offering broad coverage with a high spatial resolution. The resolution varies, with most bands in the visible and very near infrared (VNIR) spectrum offering a 10 m resolution, while the short-wave infrared (SWIR) bands have a 60 m resolution. Sentinel 2A and 2B boast a temporal resolution that can reach up to every 5 days, although this varies with latitude, decreasing as one moves further from the equator [29]. This capability enables frequent monitoring and observation of the earth’s land, atmosphere, and oceans.

In this study, Sentinel-2 images covering the period from 2017 to 2023, which depict the sand spit at the Loc An River mouth (Figure 1), were selected for analysis. The details of the Sentinel-2 images are presented in

Table 1. Information about Sentinel-2 images.

No.	Date	Sensor	Resolution (m)	Data Source
1	05 December 2015	MSI	10	Sentinel 2–1LC
2	25 December 2015	MSI	10	Sentinel 2–1LC
3	14 January 2016	MSI	10	Sentinel 2–1LC
4	24 March 2016	MSI	10	Sentinel 2–1LC
5	26 March 2016	MSI	10	Sentinel 2–1LC
6	08 January 2017	MSI	10	Sentinel 2–1LC
7	18 April 2017	MSI	10	Sentinel 2–1LC
8	31 August 2017	MSI	10	Sentinel 2–1LC
9	05 September 2017	MSI	10	Sentinel 2–1LC
10	10 October 2017	MSI	10	Sentinel 2–1LC
11	06 February 2018	MSI	10	Sentinel 2–1LC
12	25 October 2018	MSI	10	Sentinel 2–1LC
13	04 November 2018	MSI	10	Sentinel 2–1LC
14	13 January 2019	MSI	10	Sentinel 2–1LC
15	27 July 2019	MSI	10	Sentinel 2–1LC
16	15 October 2019	MSI	10	Sentinel 2–1LC
17	26 July 2020	MSI	10	Sentinel 2–1LC
19	04 September 2020	MSI	10	Sentinel 2–1LC
20	29 October 2021	MSI	10	Sentinel 2–1LC
21	16 February 2022	MSI	10	Sentinel 2–1LC
22	28 November 2022	MSI	10	Sentinel 2–1LC
23	13 March 2023	MSI	10	Sentinel 2–1LC
24	27 April 2023	MSI	10	Sentinel 2–1LC
25	06 July 2023	MSI	10	Sentinel 2–1LC
26	27 January 2024	MSI	10	Sentinel 2–1LC

.

Shoreline detection was performed based on the Normalized Difference Water Index (NDWI) [30] as follows:

$$NDWI={\displaystyle \frac{X_{green}-X_{nir}}{X_{green}+X_{nir}}}$$

(1)

where X_green and X_nir are the GREEN (B3) and NIR (B8) spectral bands with the central wavelengths of 560 nm and 842 nm, respectively. The extracted shorelines were mapped together to observe the changes in the shoreline. Since the position of the shoreline is influenced by tidal variations, which are particularly important in meso- and macro-tidal regions, tidal correction was applied to the shoreline positions using the method presented in Hoang et al. [31]. Hourly water levels collected at the Vung Tau Oceanographic Station from 2015 to 2023 were utilized. The beach slope was also obtained from the bathymetric map in the topography survey reported by Royal Haskoning DHV for the coastline at Ho Tram Resort, which is 10 km northeast of the Loc An coast [32].

2.2. One-Line Model

The analytical method to analyze the development of shoreline changes along the Loc An sand spit was based on the concept of the one-line model, as depicted in Figure 3. This approach operates under the premise of the one-line theory, which suggests that the beach maintains a consistent profile, implying that all underwater contours remain parallel to each other. Hence, a single line can represent seaward or landward changes of the shoreline, with this line directly aligning with the shoreline itself, as shown in Figure 3. Figure 3 implies that the shoreline change is determined by the net rate of sand entering and leaving the beach segment over a given time down to a certain depth as

$${\displaystyle \frac{\Delta y}{\Delta t}}={\displaystyle \frac{1}{D_B+D_C}}\left({\displaystyle \frac{\Delta Q}{\Delta x}}\pm q\right)$$

(2)

where Δy/Δt is the shoreline change; D_B is the berm height and D_C is the depth of closure, respectively; ΔQ/Δt is the net rate of sand entering and leaving the beach segment (Δx); and q is the source (q₀) of sink (q_s) of sand along the segment (Figure 3).

The morphological changes of the shoreline’s shape, denoted as y(x,t), are governed by the One-Line Contour Equation (OLCE) [33], a formula first introduced by Pelnard-Considére in 1956 [34]. According to Larson et al. (1987) [35], under conditions of a small breaking wave angle and slight shoreline curvature, the OLCE simplifies into the diffusivity equation (DSE). The DSE is expressed as

$${\displaystyle \frac{\partial y}{\partial t}}=\epsilon {\displaystyle \frac{{\partial }^{2}y}{\partial x^2}}$$

(3)

In this context, ε represents the diffusivity of the shoreline, determining the rate at which the system reacts to external influences. The value of ε is primarily influenced by the height and angle of the breaking waves, as highlighted in various studies within the literature, including works by Ashton and Murray (2006 a,b) [36,37], Falques (2003, 2005) [38,39], and Walton and Dean (2010) [40].

It should be noted that various analytical solutions have been derived for the DSE as presented in books [41,42]. Therefore, by applying appropriate boundary conditions for shoreline evolution, numerous analytical solutions for shoreline change can be derived from Equation (3), which primarily fall into two categories: infinite shorelines and finite shorelines. Those in the first category were studied and introduced by Pelnard Considère for the case of shoreline evolution downdrift of a groin [34] or shoreline evolution of a river delta with infinite shorelines [35]. In the second category, Tanaka and Nadaoka [17] introduced an analytical solution for shoreline evolution bounded by two groins and Duy et al. [19] presented an analytical solution for the formation and deformation of a river delta with finite beach length.

3. Results of Satellite Image Analysis

3.1. Stage 1: Elongation of the Sand Spit

As seen in Figure 4, there are three stages of shoreline evolution. In the first stage, the shoreline of the Loc An sand spit was in its natural condition, as there were no structures on the shoreline (Figure 4a). The shorelines on 5 December 2015 and 14 January 2016 were also added to Figure 4a to highlight the elongation of the sand spit due to LST. This stage of the Loc An sand spit was detailed in the study by Duc Anh et al. [18]. According to Duc Anh et al. [18], the Loc An sand spit was extended westward about 2500 m from 1988 to 2017 by the longshore sediment transport at a rate of 2 × 10⁵ m³/year from northeast to southwest. This direction of LST coincides with the dominant northeast wave direction.

3.2. Stage 2: Intercept of LST by a Jetty

In the second stage, the jetties were constructed (Figure 4b) to stabilize the river mouth. At this stage, the LST was intercepted by the eastward jetty, leading to the accumulation of sand and the seaward advance of the shoreline adjacent to the eastward side of the jetty. According to satellite images, the eastward jetty was constructed in 2017, and the westward jetty was constructed in 2018. The construction of the jetties has caused a change in the direction of the river flow from westward (1988 to 2017) to southward (2018–present). This directional change in river flow can cause sand to bypass to the east side of the jetty and contribute to the buildup of sediment immediately eastward of the jetties.

3.3. Stage 3: Construction of the Seawall and LST in between Two Boundaries

Finally, the shoreline was bounded by both a jetty and a seawall, as seen in Figure 4c. The seawall is clearly visible in Figure 4d, which was taken during June 2022. It should be noted that a groin was constructed eastward of the seawall using geo-tubes to prevent beach erosion along the Loc An sand spit. However, this groin was destroyed within a few months of its construction. As a result, the seawall was built to stabilize the shoreline (Figure 4d).

This seawall was constructed using a prefabricated non-metallic reinforced concrete solution. The revetment is assembled using A-shaped non-metallic reinforced concrete blocks, each 4 m high, 4.1 m wide, and either 2 m or 1.5 m long. These blocks are connected using male and female grooves to form the framework of the seawall. The M300 concrete wave deflector wall is attached to the A-shaped blocks with M300 concrete square piles, measuring 25 cm × 25 cm and 9 m long. Behind the revetment, the earthfill is raised to an elevation of +3.5 m with continuous ART 11D double-layer geotextile. The foot of the revetment is protected against erosion with 60 cm thick rubble stone, 4.2 m wide, placed on two layers of ART 11D geotextile and combined with M300 concrete-coated gabions, 0.8 m wide and 1 m high, which are durable in seawater environments. The A-shaped concrete blocks are connected using vertical sliding locks with a lock depth greater than 2 cm.

Because the seawall at the Loc An sand spit is a semi-infinite seawall where flanking (erosion occurring beside the seawall) is not possible, this seawall can function similarly to a groin or a jetty [35].

The shoreline changes of the Loc An sand spit from 2015–2023 are presented in Figure 5. As can be seen in Figure 5, the shoreline evolution of Loc An can be divided into three stages. The first stage corresponds to the elongation of the sand spit because of longshore sediment transport with no constraint from coastal structures. This stage includes the shorelines on 17 January 2015, 20 January 2016, and 6 January 2017.

In the second stage, the shoreline was affected by the construction of jetties around 31 August 2017. The influence of the jetties became clear from the buildup of the shorelines on 7 February 2018 and 13 January 2019.

The third stage includes the shorelines on 16 February 2022 and 13 March 2023. In this stage, a seawall had been constructed, hence the shorelines were affected by both the jetties and the seawall. Interestingly, there was no significant change in the shoreline position during this stage, showing that the shoreline had approached its equilibrium condition. This phenomenon agrees well with the analytical solution proposed by Tanaka and Nadaoka (1982) [17].

4. Application of One-Line Model

The three stages of shoreline evolution at the Loc An sand spit can be sketched as in Figure 6. In this section, two analytical solutions for shoreline changes, derived from a well-known theoretical one-line model, were used to examine the sandy beach evolution effected by one boundary (stage 2) and two boundaries (stage 3) in Figure 6.

4.1. Solution for Sandy Beach with One Boundary

Beginning with Pelnard-Considere’s initial work in 1956 [34], which presented an analytical solution for shoreline change on the updrift side of a groin, numerous researchers have since developed analytical tools for a variety of applications. A key study of interest for this research was conducted by Larson et al. (1987) [35]. Their study specifically addressed the open (unbounded) coast hypothesis, significantly simplifying the mathematical challenge. Within the range of analytical solutions offered by Larson et al. (1987) [35], they included a scenario addressing shoreline change in the presence of groins and jetties (Figure 7).

As seen in Figure 7, a groin disrupts the alongshore movement of sand, resulting in a build-up on the side facing the incoming waves (updrift side) and a depletion or erosion on the opposite side (downdrift side). The equation characterizing the part where accumulation occurs is

$$y(x,t)=2\tan \theta \left[\sqrt{{\displaystyle \frac{\epsilon t}{\pi }}}{e}^{-{\left(x-x_0\right)}^2/4\epsilon t}-{\displaystyle \frac{x-x_0}{2}}erfc\left({\displaystyle \frac{x-x_0}{2\sqrt{\epsilon t}}}\right)\right]$$

(4)

where θ is the angle of breaking wave crests relative to an axis set parallel to the trend of the shoreline, t is the time, and erfc is the complementary error function.

As shown in Figure 4b, the shoreline change at this stage (stage 2) is markedly influenced by the jetties functioning as a groin. Hence, Equation (4) can be applied to model the shoreline’s evolution during this phase. The shoreline data extracted from Sentinel-2 images at this stage were compared with the modeled shoreline to ascertain the values of ε and θ.

To facilitate a comparison between the shorelines extracted from Sentinel-2 images and the modeled shorelines (Equation (4)), the images were converted into local coordinates as depicted in Figure 8.

The idea for determining ε and θ is to make a comparison between the modeled shoreline and the measured shoreline (extracted from Sentinel-2 images). The modeled shoreline is obtained using Equation (4). However, Equation (4) requires the values of ε and θ for the calculation. Therefore, iteration was performed to determine ε and θ. To evaluate the agreement between the modeled shoreline and the extracted one, the root mean square error (RMSE) was used as in the following equation:

$$RMSE=\sqrt{{\displaystyle \frac{{\displaystyle \sum {\left(y_{pi}-y_{oi}\right)}^2}}{n}}}$$

(5)

where y_pi is the predicted shoreline position at point i and y_oi is the observed shoreline position at point i along the shoreline, and n is the number of observation points along the shoreline.

The iteration process will stop when the minimum RMSE is obtained. The initial values of the parameters required in Equation (4) are shown in

Table 2. Initial values of variables used for modeling the shoreline using Equation (4).

Diffusion coefficient, ε (m2/day)	Unknown
Breaking wave angle, θ (degrees)	Unknown
Modeling time, t (years)	1.9

.

At this stage, the shoreline extracted from the Sentinel-2 image dated 27 July 2019 was selected as the measured shoreline. The reason for choosing the shoreline on this day, was that the seawall had not yet been constructed, allowing Equation (4) to apply. It is clear in Figure 9a that the construction of the jetties began around 31 August 2017. Therefore, the shoreline as of 31 August 2017 is considered the initial shoreline. The modeling time can be determined as the duration between 31 August 2017 and 27 July 2019, or t = 1.9 years.

Figure 10 shows the comparison between the measured shoreline and the computed shorelines, whereby the measured shoreline is extracted from the Sentinel-2 image captured on 27 July 2019, and the computed shorelines were obtained using Equation (4). As illustrated in Figure 10, several values of ε were used in the calculations, and a value of ε = 230 m²/day demonstrates good agreement with the measured shoreline. Furthermore, the RMSE values presented in Figure 11 confirm that the best value of ε is 230 m²/day.

At the same time during the iteration to determine the value of ε, various values of θ were also input into the model for calculation. The final values of ε and θ are shown in

Table 3. Best fit values of ε and θ.

Diffusion coefficient, ε (m2/day)	230
Breaking wave angle, θ (degrees)	18
Modeling time, t (years)	1.9

.

In order to clarify the suitability of Equation (4) for solving the problem at the Loc An sand spit, the model was applied with another period while still keeping the values of ε = 230 m²/day and θ = 18 degrees. At this stage, the shoreline position on 26 July 2020 was used. Hence, the modeling time is determined as the duration between 31 August 2017 and 26 July 2020, or t = 2.9 years. The validation results are shown in Figure 12. It can be seen that the RMSE in the validation step is 8.9 m which is quite close to the RMSE in the calibration step. This indicates the good performance of the model.

4.2. Solution for Sandy Beach with Two Boundaries

As illustrated in Figure 4c (Stage 3), the shoreline evolution at this stage is influenced by two structures: the jetties and the seawall. Equation (4) was no longer valid and a new analytical solution had to be employed as in the study of [19]. Tanaka and Nadaoka (1982) [17] introduced an analytical solution for shoreline evolution between the two jetties as sketched in Figure 13.

Figure 13 shows that the analytical solution proposed by Tanaka and Nadaoka (1982) [17] agrees well with the case of Loc An shoreline in stage 3, bounded by the jetties and the seawall. Therefore, the analytical solution of Tanaka and Nadaoka (1982) [17] can be used to study the shoreline evolution of the Loc An sand spit in stage 3, and the solution is presented as follows:

$$\begin{array}{l}y=L.\tan \theta .{\displaystyle \sum _{n=1}^{\infty }{\left(-1\right)}^n.\left[\sqrt{\frac{4\epsilon t}{\pi L^2}}.\exp \left\{-\frac{L^2}{4\epsilon t}{\left(\frac{x’}{L}+\frac{1}{2}-n\right)}^2\right\}+\left(\frac{x’}{L}+\frac{1}{2}-n\right).\mathrm{erfc}\left\{-\frac{L}{\sqrt{4\epsilon t}}\left(\frac{x’}{L}+\frac{1}{2}-n\right)\right\}\right]}\\ +L.\tan \theta .{\displaystyle \sum _{n=-\infty }^0{\left(-1\right)}^n.\left[\sqrt{\frac{4\epsilon t}{\pi L^2}}.\exp \left\{-\frac{L^2}{4\epsilon t}{\left(\frac{x’}{L}+\frac{1}{2}-n\right)}^2\right\}-\left(\frac{x’}{L}+\frac{1}{2}-n\right).\mathrm{erfc}\left\{\frac{L}{\sqrt{4\epsilon t}}\left(\frac{x’}{L}+\frac{1}{2}-n\right)\right\}\right]}\end{array}$$

(6)

where L is the length of the shoreline bounded by the jetties and the seawall, L = 960 m (Figure 8), and x′ is defined as

$$x’=x-x_1$$

(7)

The temporal variation of the shoreline upstream of the jetties, denoted as y₁ in Figure 6, will be examined by plotting it under two scenarios: one with a single boundary (the jetties) and the other with two boundaries (the jetties and the seawall). Furthermore, we utilized the dimensionless form of the shorelines to establish the criteria for when the shoreline approaches its equilibrium. The dimensionless shoreline position and the dimensionless time are defined as follows [17]:

$${\eta }_1={\displaystyle \frac{2y_1}{L\tan \theta }}$$

(8)

where η₁ represents the dimensionless form of the shoreline, y₁ is the position of the shoreline upstream of the jetties, and L is the distance between the jetties and the seawall (as shown in Figure 8).

$$\tau ={\displaystyle \frac{4\epsilon }{L^2}}t$$

(9)

where τ is the dimensionless time.

The dimensionless shoreline positions are plotted in Figure 13 with the normal axis in Figure 14a and the log-log plot in Figure 14b. As can be seen in Figure 14, in the initial stage of the shoreline evolution (τ < 0.5), there is good agreement between the shoreline positions with one boundary and two boundaries. This indicates there is no effect of the second boundary (the seawall in the case of Loc An beach). However, for τ > 0.5, there is a significant gap between the shoreline positions of one boundary and two boundaries, indicating that the longshore sediment transport has reached the second boundary and the shoreline changes were affected by it. It is also observed from Figure 14 that after τ = 2.0, the shoreline position affected by two boundaries will reach the equilibrium condition, as indicated by a horizontal line. This phenomenon is similar to the study of Duy et al. [19].

4.3. Comparison with Field Measured Data

In this section, the shoreline positions in stage 3, modeled by Equation (6), will be compared with the shoreline positions extracted from the satellite images to determine whether the shoreline position at Loc An beach has reached its equilibrium condition.

Equation (6) requires a straight shoreline as the initial condition for the calculation (Figure 13). This condition cannot be met with the Loc An sand spit because the shoreline at Stage 3 had already been influenced by the jetties and was no longer straight. Hence, a numerical approach was employed to model the shoreline evolution with an arbitrary initial shoreline.

In order to apply the numerical model, an initial shoreline is required. This initial shoreline must be established after the construction of the seawall to ensure that the shoreline is affected by both the jetties and the seawall. Based on a photo taken by a contractor, the seawall was constructed around 25 September 2021 (Figure 15b). Additionally, a Sentinel-2 image captured on 29 October 2021 also showed the existence of the seawall, indicated by a straight shoreline (Figure 15b). Therefore, the shoreline position on 29 October 2021 was chosen as the initial shoreline for the numerical model.

The comparison between the modeled shorelines and the measured shorelines is shown in Figure 16. In this figure, the modeled shorelines were obtained from the numerical model by discretizing Equation (6). The measured shorelines were extracted from Sentinel-2 images. It should be noted that the shoreline positions presented in Figure 16 were transformed to local coordinates (Figure 8). The locations of the seawall and the jetties, as well as the direction of prevailing offshore waves, are also presented in Figure 16. Interestingly, the shoreline position appears to be stable after 2 years of the seawall’s construction, especially at the location close to the jetties. This shows the effectiveness of the seawall in stabilizing the shoreline along the Loc An sand spit. At the location down-drift of the seawall, a disagreement between the modeled and measured shorelines can be observed, which is attributed to the buildup of the shoreline. This discrepancy is caused by wave refraction, which was not considered in the numerical model.

The evolution of the shoreline at the jetties (x = 960 m) is plotted in Figure 17 to highlight the equilibrium of the shoreline achieved some time after the construction of the seawall (see Figure 6 for a graphical illustration of y₁). In this Figure, the numerical shorelines are represented by a blue line, and the measured shoreline positions are denoted by circles. Although there are fluctuations in the measured shorelines, they follow the trend of the numerical calculations. The numerical results predict that, after 4 years of the seawall’s construction, the shoreline at the Loc An sand spit will stabilize. This underscores the effectiveness of the seawall.

It should be noted that the equilibrium shoreline position, denoted by the dashed red line, is also presented in the figure. Based on the equilibrium shoreline position and the numerical shorelines, it can be concluded that the shoreline bounded by the jetties and the seawall at the Loc An sand spit will be stable by mid-2027. This equilibrium condition of the shoreline at the Loc An sand spit is based on the assumption that sand bypass is neglected. However, it is clear that sand bypass at the eastern jetty and sand transport along the seawall are inevitable. This phenomenon is disregarded in the simple model of this study to quickly analyze basic beach behavior with an acceptable level of accuracy.

5. Discussion

5.1. Tidal Effect to the Shoreline Positions

As shown in Figure 2, the tidal range is approximately 3 m, which can significantly affect the shoreline position as determined from satellite images. Consequently, a detailed discussion of the impact of tides on shoreline shifting would be useful. Figure 18 illustrates the shoreline positions at the jetties (y₁) before and after tidal corrections are applied. It is evident that the maximum shoreline shift due to tidal effects at the Loc An river mouth is approximately 20 m (on 12 April 2022).

5.2. Determination of ε Value

In this section, the value of ε is calculated based on measured data to discuss the value of ε determined by the iteration shown in the previous sections. According to Rosati et al. (2002) [43], the value of ε is calculated using the following equation:

$$\epsilon ={\displaystyle \frac{K{\left(H^2{C}_g\right)}_b}{8}}\left({\displaystyle \frac{\rho }{{\rho }_s-\rho }}\right)\left({\displaystyle \frac{1}{1-n}}\right)\left({\displaystyle \frac{1}{D_B+D_C}}\right)$$

(10)

Here, K represents the dimensionless coefficient for the sediment transport rate formula, H is the wave height, and Cg is the group velocity. The subscript “b” denotes the quantity at the wave-breaking point. ρ and ρ_s represent the density of sea water and sand, ρ = 1025 kg/m³ and ρ_s = 2650 kg/m³, respectively, n denotes the sediment porosity (n = 0.4), D_B is the berm height, and D_C is the depth of closure.

According to Del Valle et al. (1993) [44], K varies with the median grain size (D₅₀) of the sediment and can be calculated using the empirical relationship

$$K=1.4e^{\left(-2.5D_{50}\right)}$$

(11)

According to a survey map from Duc Anh (2021) [5], the median grain size of sand in the study area (Loc An sand spit) ranges from 0.56 to 0.85 mm (Figure 19). By taking the interval of 0.05 mm between 0.56 mm and 0.85 mm, the values of K can be determined as in

Table 4. Variation of K based on the values of D₅₀.

D50 (mm)	K
0.56	0.35
0.60	0.31
0.65	0.28
0.70	0.24
0.75	0.21
0.80	0.19
0.85	0.17

.

The breaking wave height (H_b) was calculated based on the measured waves from 18 September 2016 to 9 October 2016 at the location as shown by the wave buoy in Figure 15. The automatic method for calculating breaking waves proposed by Sana [45] was employed and this resulted in H_b = 0.54 m.

The group velocity C_gb is calculated based on the following equation:

$$C_{gb}=\sqrt{g{\displaystyle \frac{H_b}{\kappa }}}$$

(12)

where the value of the breaker index, κ, is often taken as 0.78 [46].

The depth of closure, D_C, was determined from the bathymetric map in the topography survey reported by Royal Haskoning DHV for the coastline at Ho Tram resort which is 10 km North-East of the Loc An coast. According to the report of Royal Haskoning DHV, the value of D_C is 6.5 m [32].

From the value of D_C, the berm height (D_B) can be determined as follows:

$$D_B=0.32D_C=2.1 (m)$$

(13)

By substituting all the values into Equation (10), we obtain the values of ε as in

Table 5. Variation of ε based on the values of D₅₀.

D50 (mm)	K	ε (m2/day)
0.56	0.35	347
0.60	0.31	314
0.65	0.28	277
0.70	0.24	245
0.75	0.21	216
0.80	0.19	191
0.85	0.17	168

.

Figure 20 shows the comparison between the values of ε calculated by two methods. The blue bars represent the variation of ε calculated by Equation (5) based on measured data, while the red line shows the value of ε determined by the simple method (see Figure 11). The result from the simple method falls within the upper and lower bounds of the ε value range. Hence, the simple method can determine the value of ε in cases of data scarcity.

6. Conclusions

Shoreline evolution of a sand spit affected by coastal structures at Loc An, Ba Ria–Vung Tau province, Vietnam was investigated by combining remote sensing techniques and a numerical model. In this approach, a numerical method was applied based on a simple model that can be quickly used to determine the diffusion coefficient (ε) for LST, which accounts for wave energy and sediment grain size on the beach. Although the analytical solutions are based on several approximations, they have been proven effective in numerous study areas worldwide, particularly where field measurement data are limited. The results of this study can support the sustainable development of the coastal area using placed structures as erosion control measure. The main findings of this study can be summarized as follows:

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The evolution of the Loc An sand spit was divided into three stages in which the first stage showed the elongation of the sand spit because of LST without human intervention. The second stage showed the shoreline buildup upstream of the jetties. In the third stage, the shoreline at the Loc An sand spit was affected by both the jetties and the seawall and the shoreline approached equilibrium conditions.

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The value of shoreline diffusivity and the breaking wave angle at the Loc An coastline were determined as 230 m²/day and 18 degrees, respectively.

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Although erosion occurred after the construction of the seawall, this erosion was limited and the shoreline still progressed towards equilibrium conditions.

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A comparison between the measured and calculated data of shoreline evolution at the jetty was conducted. Although there were some fluctuations in the measured data, they generally followed the trend of the calculated shoreline evolution. This indicates the reliability of the simple shoreline model used in this study.

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There is a good agreement between the simple method proposed in this study and the measured data for calculating ε. Hence, the method can determine the value of ε for data-scarce localities.

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If the sand bypass at the eastern jetty and sediment transport along the seawall are neglected, the shoreline at the Loc An sand spit is expected to reach its equilibrium condition by mid-2027.