**2. Long-term shoreline changes along Yumigahama Peninsula and longshore sand transport**

#### **2.1 Shoreline changes**

breakwaters and cuspate forelands have been stably maintained, fine-grained sand was locally deposited, natural sandy beaches rapidly disappeared, and the coastline was covered by artificial structures in the entire Yumigahama Peninsula [4].

In predicting the beach changes triggered by the imbalance in longshore sand transport on a coast such as the Kaike coast, a long-term prediction in an extensive area is required [5]. In such a case, the time scale changes from years to decades and the calculation domain reaches even up to 10–100 km. The N-line model can be applicable to the prediction of such beach changes [6–12]. We have developed the contour-line-change model considering the changes in grain size of the seabed materials as a type of N-line model to predict long-term beach changes including the prediction of changes in grain size [13, 14]. The model has applied to the longterm prediction of beach changes around a river mouth delta and those along an

Uda et al. [15] have first shown a predictive model of the three-dimensional development and deformation of a river mouth delta. This model enabled the prediction of not only the shoreline change but also the three-dimensional, longterm topographic changes around a river mouth, and offshore sand transport can be taken into account when the sea bottom slope exceeds a critical value given by the angle of repose of sand. Furuike et al. [16] have applied the model to the Enshunada coast. The shoreline changes between the Magome river mouth and Imagireguchi jetty have been investigated where beach changes triggered by the decrease in fluvial sand supply from the Tenryu River occur. It was observed that the shoreline retreated downcoast of the Magome river mouth, and the shoreline advanced in parallel further downcoast, while maintaining the curvature in the shoreline configuration. Miyahara et al. [17, 18] have investigated the long-term evolution of the Tenryu River delta associated with sand bypassing at several dams. When the sediment yield from the river was artificially increased, the supplied sediment was mainly deposited around the river mouth, but it took a longer time for a sandy beach far from the river mouth to recover. Given the annual discharge of sediment with three grain sizes, the recovery of the delta topography and the effect of nourishment on the nearby coast have been predicted. San-nami et al. [19] have analyzed the beach changes of Kujukuri Beach with a 60 km length located in Boso Peninsula. On south Kujukuri Beach, severe beach erosion has occurred since the

arc-shaped shoreline in previous papers.

*Location of Yumigahama Peninsula in Tottori Prefecture, Japan.*

*Sedimentary Processes - Examples from Asia,Turkey and Nigeria*

**Figure 1.**

**64**

The shoreline configurations in the Yumigahama Peninsula were determined from an old geographical map produced in 1899 and the aerial photographs taken in 1947, 1967, and 1973. The shoreline changes in 74 years were investigated between 1899, when rich sediment was supplied from the Hino River during the period of iron sand mining, and 1973, when the construction of detached breakwaters began. In the analysis of the shoreline changes, the correction owing to the changes in tide level was made using the tide level when the aerial photograph was taken and the mean foreshore slope of 1/15. The origin of the coordinate (*x* = 0 km) was set at the left bank of the Hino River and the alongshore distance *x* was taken westward along the coastline (**Figure 1**).

**Figure 2** shows the shoreline changes between 1899 and 1947. The shoreline markedly retreated in an area between *x* = 1 and *x* = 2 km around the Hino River mouth with the decline of mining of iron sand, and the shoreline receded by a maximum of 250 m at the river mouth. On the other hand, in the areas westwards and eastwards of *x* = 3 km, the shoreline advanced of 100 m in parallel at a rate of 2 m/year, except that in the vicinity of Yodoe fishing port. Similarly, **Figure 3** shows the shoreline changes with reference to shoreline in 1947 until 1967 and 1973, after that two detached breakwaters were constructed. Since a seawall and groins have been constructed in an area between *x* = 0 and *x* = 1.5 km westwards of the Hino River until 1967, the shoreline recession was prevented. However, excluding this area, the shoreline retreated westwards of the protected area. The eroded sand was transported westwards of *x* = 5 km, resulting in the shoreline advance. Until 1973, the area where the shoreline markedly receded expanded westwards, and the area where the shoreline advanced until 1967 was eroded. Here, the shoreline advance at two locations near *x* = 1 km was due to the formation of cuspate forelands behind the detached breakwaters constructed by 1973.

**Figure 2.** *Shoreline changes in entire Yumigahama Peninsula (reference year: 1899).*

**Figure 3.** *Shoreline changes until 1967 and 1973 in entire Yumigahama Peninsula (reference year: 1947).*

### **2.2 Analysis of bathymetric survey data**

Using the bathymetric survey data (200 m intervals), the bathymetry in 2000 and the bathymetric changes between 1980 and 2000 have been represented (**Figures 4** and **5**). The eroded and accreted sand volumetric changes relative to those occurring in 1980 are also shown together with the locations of the detached breakwaters (DBs; **Figure 5**). The contours shallower than 8 m markedly protruded near the river mouth delta, and the DBs were constructed immediately westwards of this river delta. Twelve DBs were constructed from 1971 to 1982. With the construction of DBs, sand was deposited behind these breakwaters, whereas erosion occurred in the offshore zone between 4 and 7 m, and down the coast of the accretion area. These beach changes were considered to be due to the blocking of westward alongshore sand transport by the DBs.

sand. The depth distribution of *d*<sup>50</sup> can be separated into three subregions, as shown in **Figure 6**, with longitudinal slopes of 1/6, 1/30, and 1/100, i.e., fluvial sediment supplied from the Hino River is mainly distributed in the depth zone corresponding to the grain size of the material, and coarser material is deposited near the shoreline, and the component finer than the grain size of 0.13 mm is deposited in the zone deeper than 8 m. In addition, fine sand of *d*<sup>50</sup> = 0.3 mm is deposited in the intermediate depth zone between 2 and 8 m. Although westward longshore sand transport prevails on the coast, sand is transported along the depth zones separated in two layers associated with the grain size sorting (**Figure 6**). Thus, the changes in contour lines in the depth zones I and II between *z* = +3 and 2 m and between *z* = 2 and 8 m could be separately calculated, where the average grain

*A Long-Term Prediction of Beach Changes around River Delta using Contour-Line-Change Model*

*Bathymetry in 2000 using expanded coordinates and bathymetric changes with reference to that in 1980.*

**Figure 7** shows the longshore distribution of changes in sand volume in the depth zones I and II between 1980 and 2009. In the depth zone I, the sand volume was almost constant east of *x* = 5 km except for the local deposition of sand behind

sizes are 0.5 and 0.3 mm, respectively.

**Figure 5.**

**67**

**Figure 4.**

*Bathymetry in 1980 using expanded coordinates.*

*DOI: http://dx.doi.org/10.5772/intechopen.85207*

#### **2.3 Longitudinal profiles and depth change in median diameter**

In July 2002, the sampling of the seabed material was carried out in the depth zone between +3 and 12 m along transect *x* = 1.3 km at 1-m-depth intervals. **Figure 6** shows the longitudinal profile and depth distribution of median diameter (*d*50) along this transect. Coarse sand of *d*<sup>50</sup> ranging from 0.3 to 0.7 mm is deposited on the foreshore, and *d*<sup>50</sup> decreases with increasing depth, and well-sorted fine sand with a grain size of 0.13 mm is deposited in the depth zone deeper than 8 m. Corresponding to this grain size distribution, the seabed slope is as steep as 1/6 on the foreshore, 1/30 between 2 and 8 m, and a very gentle slope of 1/100 in the depth zone deeper than 8 m. From these data, it is found that the depth of closure (*h*c) is approximately 8 m on this coast, and the seabed near *h*<sup>c</sup> is covered by fine

*A Long-Term Prediction of Beach Changes around River Delta using Contour-Line-Change Model DOI: http://dx.doi.org/10.5772/intechopen.85207*

**Figure 4.** *Bathymetry in 1980 using expanded coordinates.*

**Figure 5.** *Bathymetry in 2000 using expanded coordinates and bathymetric changes with reference to that in 1980.*

sand. The depth distribution of *d*<sup>50</sup> can be separated into three subregions, as shown in **Figure 6**, with longitudinal slopes of 1/6, 1/30, and 1/100, i.e., fluvial sediment supplied from the Hino River is mainly distributed in the depth zone corresponding to the grain size of the material, and coarser material is deposited near the shoreline, and the component finer than the grain size of 0.13 mm is deposited in the zone deeper than 8 m. In addition, fine sand of *d*<sup>50</sup> = 0.3 mm is deposited in the intermediate depth zone between 2 and 8 m. Although westward longshore sand transport prevails on the coast, sand is transported along the depth zones separated in two layers associated with the grain size sorting (**Figure 6**). Thus, the changes in contour lines in the depth zones I and II between *z* = +3 and 2 m and between *z* = 2 and 8 m could be separately calculated, where the average grain sizes are 0.5 and 0.3 mm, respectively.

**Figure 7** shows the longshore distribution of changes in sand volume in the depth zones I and II between 1980 and 2009. In the depth zone I, the sand volume was almost constant east of *x* = 5 km except for the local deposition of sand behind

**2.2 Analysis of bathymetric survey data**

*Shoreline changes in entire Yumigahama Peninsula (reference year: 1899).*

*Sedimentary Processes - Examples from Asia,Turkey and Nigeria*

**Figure 2.**

**Figure 3.**

**66**

of westward alongshore sand transport by the DBs.

**2.3 Longitudinal profiles and depth change in median diameter**

In July 2002, the sampling of the seabed material was carried out in the depth zone between +3 and 12 m along transect *x* = 1.3 km at 1-m-depth intervals. **Figure 6** shows the longitudinal profile and depth distribution of median diameter (*d*50) along this transect. Coarse sand of *d*<sup>50</sup> ranging from 0.3 to 0.7 mm is deposited on the foreshore, and *d*<sup>50</sup> decreases with increasing depth, and well-sorted fine sand with a grain size of 0.13 mm is deposited in the depth zone deeper than 8 m. Corresponding to this grain size distribution, the seabed slope is as steep as 1/6 on the foreshore, 1/30 between 2 and 8 m, and a very gentle slope of 1/100 in the depth zone deeper than 8 m. From these data, it is found that the depth of closure (*h*c) is approximately 8 m on this coast, and the seabed near *h*<sup>c</sup> is covered by fine

Using the bathymetric survey data (200 m intervals), the bathymetry in 2000 and the bathymetric changes between 1980 and 2000 have been represented (**Figures 4** and **5**). The eroded and accreted sand volumetric changes relative to those occurring in 1980 are also shown together with the locations of the detached breakwaters (DBs; **Figure 5**). The contours shallower than 8 m markedly protruded near the river mouth delta, and the DBs were constructed immediately westwards of this river delta. Twelve DBs were constructed from 1971 to 1982. With the construction of DBs, sand was deposited behind these breakwaters, whereas erosion occurred in the offshore zone between 4 and 7 m, and down the coast of the accretion area. These beach changes were considered to be due to the blocking

*Shoreline changes until 1967 and 1973 in entire Yumigahama Peninsula (reference year: 1947).*

**Figure 8** shows the distribution of longshore sand transport between 1899 and 1947, and between 1947 and 1967 with a positive sign for westward longshore sand transport. Although westward and eastward longshore sand transports took place on the west and east sides of the river mouth between 1899 and 1947, respectively, the distribution of longshore sand transport changes linearly with the same slope west of *x* = 3 km and east of *x* = 3 km, and they smoothly converge to the distribution between 1947 and 1967 west of *x* = 7 km. Westward and eastward longshore sand transports take maximum rates of 3.2 <sup>10</sup><sup>5</sup> and 4.6 <sup>10</sup><sup>4</sup> <sup>m</sup><sup>3</sup>

*A Long-Term Prediction of Beach Changes around River Delta using Contour-Line-Change Model*

*Distribution of longshore sand transport between 1899 and 1947 and between 1947 and 1967.*

at *x* = 1 and 1.8 km, respectively. On both sides of these points, the longshore sand transport is linearly distributed, so that it is found that the shoreline has advanced in parallel over time except for the erosion area close to the river delta. The longshore

mouth, implying that the same amount of sand was supplied from the river during the period. In contrast, longshore sand transport became westward in the entire area between 1947 and 1967 after the fluvial sand supply was markedly reduced owing to the stoppage of the mining of iron sand with a maximum rate of

a peak value shifted westward compared with that between 1899 and 1967, implying that the erosion area expanded westward. Furthermore, since longshore sand transport on the west and east sides of the river mouth become 1.5 <sup>10</sup><sup>5</sup> and

/year, respectively, sediment inflow from the river results in

increased from the west end to a location of *x* = 7 km in both periods between 1899 and 1947 and between 1947 and 1967, and the effect of the decrease in the longshore sand transport did not reach far from the river mouth, resulting in the parallel advance of the shoreline. Therefore, when this straight line is extrapolated toward the river mouth, a straight line indicated by "before erosion" in **Figure 8** is obtained,

and the longshore sand transport at the river mouth became 3.6 105 m3

is assumed to be the westward longshore sand transport before the initiation of

In the area east of the river mouth, eastward longshore sand transport could have prevailed with the development of the river delta, and the longshore sand transport is assumed to have a linear distribution with the same longshore inclination as that west of the river mouth to satisfy the conditions that the shoreline around the river mouth is continuous and the river delta parallelly advanced in the offshore direction. By extrapolating a straight line with the same inclination as that west of the river mouth from Yodoe fishing port, we obtained a longshore sand

sediment supply from the river in the period when iron sand was mined is assumed

/year at *x* = 5 km. The location where the longshore sand transport takes

/year. In addition, in **Figure 8**, the longshore sand transport linearly

/year on the east side of the river mouth. Finally, the entire

sand transport has a difference of 2.9 <sup>10</sup><sup>5</sup> <sup>m</sup><sup>3</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.85207*

2.3 <sup>10</sup><sup>5</sup> <sup>m</sup><sup>3</sup>

**Figure 8.**

9.0 <sup>10</sup><sup>4</sup> <sup>m</sup><sup>3</sup>

6.0 <sup>10</sup><sup>4</sup> <sup>m</sup><sup>3</sup>

erosion.

**69**

transport of 1.1 <sup>10</sup><sup>5</sup> m3

/year

/year, which

/year between both sides of the river

#### **Figure 6.**

*Longitudinal profile along transect x = 1.3 km and depth distribution of d50 measured in July 2002.*

**Figure 7.** *Longshore distribution of changes in sand volume in depth zones I and II between 1980 and 2009.*

detached breakwaters constructed between *x* = 0 and 4 km, and the accreted sand volume was 1.34 <sup>10</sup><sup>6</sup> <sup>m</sup><sup>3</sup> in contrast to the eroded sand volume of 5.8 <sup>10</sup><sup>5</sup> <sup>m</sup><sup>3</sup> . The sand volume in the depth zone II decreased between *x* = 2 and 9 km, whereas it increased west of *<sup>x</sup>* = 9 km; the eroded volume was 2.01 <sup>10</sup><sup>6</sup> <sup>m</sup><sup>3</sup> greater than the accreted volume of 1.06 106 <sup>m</sup><sup>3</sup> . However, the eroded and accreted sand volumes in the entire depth zones I and II were 2.59 <sup>10</sup><sup>6</sup> and 2.40 <sup>10</sup><sup>6</sup> <sup>m</sup><sup>3</sup> , respectively, and the accreted volume accounts for 93% of the eroded volume.

#### **2.4 Longshore sand transport**

The west and east ends of the study area are the solid boundaries of the reclaimed land and Yodoe fishing port; thus, the distribution of longshore sand transport can be calculated by integrating the shoreline change from these boundaries to the Hino River mouth, and multiplying by the characteristic height of beach changes, which is defined by the correlation coefficient between the change in the cross-sectional area of the beach and the shoreline change, and then divided by the elapsed time. Since the relationship of *h* = (1.01.3)*h*<sup>c</sup> holds between the characteristic height of beach changes (*h*) and *h*<sup>c</sup> [4], and *h*<sup>c</sup> on this coast is approximately equal to 8 m, *h* becomes 10 m.

*A Long-Term Prediction of Beach Changes around River Delta using Contour-Line-Change Model DOI: http://dx.doi.org/10.5772/intechopen.85207*

**Figure 8.** *Distribution of longshore sand transport between 1899 and 1947 and between 1947 and 1967.*

**Figure 8** shows the distribution of longshore sand transport between 1899 and 1947, and between 1947 and 1967 with a positive sign for westward longshore sand transport. Although westward and eastward longshore sand transports took place on the west and east sides of the river mouth between 1899 and 1947, respectively, the distribution of longshore sand transport changes linearly with the same slope west of *x* = 3 km and east of *x* = 3 km, and they smoothly converge to the distribution between 1947 and 1967 west of *x* = 7 km. Westward and eastward longshore sand transports take maximum rates of 3.2 <sup>10</sup><sup>5</sup> and 4.6 <sup>10</sup><sup>4</sup> <sup>m</sup><sup>3</sup> /year at *x* = 1 and 1.8 km, respectively. On both sides of these points, the longshore sand transport is linearly distributed, so that it is found that the shoreline has advanced in parallel over time except for the erosion area close to the river delta. The longshore sand transport has a difference of 2.9 <sup>10</sup><sup>5</sup> <sup>m</sup><sup>3</sup> /year between both sides of the river mouth, implying that the same amount of sand was supplied from the river during the period. In contrast, longshore sand transport became westward in the entire area between 1947 and 1967 after the fluvial sand supply was markedly reduced owing to the stoppage of the mining of iron sand with a maximum rate of 2.3 <sup>10</sup><sup>5</sup> <sup>m</sup><sup>3</sup> /year at *x* = 5 km. The location where the longshore sand transport takes a peak value shifted westward compared with that between 1899 and 1967, implying that the erosion area expanded westward. Furthermore, since longshore sand transport on the west and east sides of the river mouth become 1.5 <sup>10</sup><sup>5</sup> and 9.0 <sup>10</sup><sup>4</sup> <sup>m</sup><sup>3</sup> /year, respectively, sediment inflow from the river results in 6.0 <sup>10</sup><sup>4</sup> <sup>m</sup><sup>3</sup> /year. In addition, in **Figure 8**, the longshore sand transport linearly increased from the west end to a location of *x* = 7 km in both periods between 1899 and 1947 and between 1947 and 1967, and the effect of the decrease in the longshore sand transport did not reach far from the river mouth, resulting in the parallel advance of the shoreline. Therefore, when this straight line is extrapolated toward the river mouth, a straight line indicated by "before erosion" in **Figure 8** is obtained, and the longshore sand transport at the river mouth became 3.6 105 m3 /year, which is assumed to be the westward longshore sand transport before the initiation of erosion.

In the area east of the river mouth, eastward longshore sand transport could have prevailed with the development of the river delta, and the longshore sand transport is assumed to have a linear distribution with the same longshore inclination as that west of the river mouth to satisfy the conditions that the shoreline around the river mouth is continuous and the river delta parallelly advanced in the offshore direction. By extrapolating a straight line with the same inclination as that west of the river mouth from Yodoe fishing port, we obtained a longshore sand transport of 1.1 <sup>10</sup><sup>5</sup> m3 /year on the east side of the river mouth. Finally, the entire sediment supply from the river in the period when iron sand was mined is assumed

detached breakwaters constructed between *x* = 0 and 4 km, and the accreted sand volume was 1.34 <sup>10</sup><sup>6</sup> <sup>m</sup><sup>3</sup> in contrast to the eroded sand volume of 5.8 <sup>10</sup><sup>5</sup> <sup>m</sup><sup>3</sup>

*Longshore distribution of changes in sand volume in depth zones I and II between 1980 and 2009.*

*Longitudinal profile along transect x = 1.3 km and depth distribution of d50 measured in July 2002.*

*Sedimentary Processes - Examples from Asia,Turkey and Nigeria*

The sand volume in the depth zone II decreased between *x* = 2 and 9 km, whereas it increased west of *<sup>x</sup>* = 9 km; the eroded volume was 2.01 <sup>10</sup><sup>6</sup> <sup>m</sup><sup>3</sup> greater than the

The west and east ends of the study area are the solid boundaries of the reclaimed land and Yodoe fishing port; thus, the distribution of longshore sand transport can be calculated by integrating the shoreline change from these boundaries to the Hino River mouth, and multiplying by the characteristic height of beach changes, which is defined by the correlation coefficient between the change in the cross-sectional area of the beach and the shoreline change, and then divided by the elapsed time. Since the relationship of *h* = (1.01.3)*h*<sup>c</sup> holds between the characteristic height of beach changes (*h*) and *h*<sup>c</sup> [4], and *h*<sup>c</sup> on this coast is approximately

in the entire depth zones I and II were 2.59 <sup>10</sup><sup>6</sup> and 2.40 <sup>10</sup><sup>6</sup> <sup>m</sup><sup>3</sup>

and the accreted volume accounts for 93% of the eroded volume.

. However, the eroded and accreted sand volumes

accreted volume of 1.06 106 <sup>m</sup><sup>3</sup>

**Figure 7.**

**Figure 6.**

**2.4 Longshore sand transport**

equal to 8 m, *h* becomes 10 m.

**68**

.

, respectively,

**Figure 9.** *Distribution of longshore sand transport calculated from beach changes between 1980 and 2009.*

to be 4.7 � <sup>10</sup><sup>5</sup> <sup>m</sup><sup>3</sup> /year as the sum of longshore sand transport on both sides of the river mouth. In addition, since the longshore inclination of longshore sand transport in **Figure 8** gives 21 m<sup>3</sup> /m/year as the rate of sand accretion, the rate of the advance of the river delta becomes 2.1 m/year, when dividing this rate by the characteristic height of beach changes of *h* = 10 m.

mentioned above. The decrease in the breaker angle west of the river mouth corre-

*A Long-Term Prediction of Beach Changes around River Delta using Contour-Line-Change Model*

A numerical model described in [13, 14] was used. Let the *x*- and *z*-axes be the longshore distance and seabed elevation relative to the still water level, respectively, and *Y* (*x, z, t*) is the offshore distance to a specific contour line to be solved. To consider the sorting of sand of different grain sizes by cross-shore sand transport, the depth distribution of cross-shore sand transport considering the grain size effect in mixing was considered. The sorting of grain size populations was modeled by

(*k*)

mobility of sand of each grain size population by cross-shore movement is the same as that of longshore sand transport, the coefficient of the sediment transport rate

Furthermore, assuming that the ratio of the exposed area of each grain size population to the entire sea bottom area is equal to the content of each size population in the exchange layer *<sup>μ</sup>*ð Þ*<sup>k</sup>* ð Þ *<sup>k</sup>* <sup>¼</sup> <sup>1</sup>*;* <sup>2</sup>*;* <sup>⋯</sup>*; <sup>N</sup>* , the cross-shore sand transport of each grain

bcos2

ffiffiffiffiffiffiffi

population *k*. In this case, a grain size population was assumed to have a single

, which corresponds to each grain size

, which was given in [21, 22], was introduced.

*<sup>α</sup>*<sup>b</sup> sin *<sup>β</sup>* � cot *<sup>β</sup><sup>=</sup>* cot *<sup>β</sup>*ð Þ*<sup>k</sup>*

*<sup>d</sup>*ð Þ*<sup>k</sup>* <sup>p</sup> *; k* <sup>¼</sup> <sup>1</sup>*,* <sup>2</sup>*,* <sup>⋯</sup>*, N* (2)

. By assuming that the

<sup>c</sup> � 1

(1)

� �*;*

sponds to the decrease in the westward longshore sand transport.

introducing the equilibrium slope angle *β*<sup>c</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.85207*

(*k*) is derived.

according to the grain size *d*(*k*)

Cross-shore sand transport:

*<sup>z</sup>* <sup>¼</sup> *<sup>μ</sup>*ð Þ*<sup>k</sup>* � *<sup>ε</sup>z*ð Þ� *<sup>z</sup> <sup>γ</sup>* � *<sup>K</sup>*ð Þ*<sup>k</sup>*

*k* ¼ 1*,* 2*,* ⋯*, N*

size population *q*<sup>z</sup>

**Figure 10.**

*q*ð Þ*<sup>k</sup>*

**71**

equilibrium beach slope with a characteristic grain size *d*(*k*)

<sup>1</sup> � *EC*<sup>g</sup> � �

*K*ð Þ*<sup>k</sup>* <sup>1</sup> <sup>¼</sup> *<sup>A</sup>*

**4. Contour-line-change model considering change in grain size**

*Change in apex angle of Hino River delta and predominant wave direction. (a) 1899. (b) 1947.*

Similarly, the distribution of longshore sand transport after 1980 was evaluated by integrating the topographic changes between 1980 and 2009 in the east from the west end, and dividing by the elapsed time. **Figure 9** shows the results. The longshore sand transport took a maximum of 1.0 � <sup>10</sup><sup>5</sup> <sup>m</sup><sup>3</sup> /year at *x* = 7.5 km in 1990 with a secondary peak at *x* = 0 km. After 2000, it has decreased in the eastern part, and it took a maximum of 6.5 � <sup>10</sup><sup>4</sup> <sup>m</sup><sup>3</sup> /year in 2009.

## **3. Change in apex angle of Hino River delta and predominant wave direction**

**Figure 10** shows the geographical map in 1899 and aerial photograph in 1947. The apex angle of the river delta (angle AOB) in 1899 was 170°. Assuming that the wave crest line of the predominant wave incident to the river delta is given by a straight line of A<sup>0</sup> OB<sup>0</sup> in **Figure 10(a)**, the breaker angles between the shoreline on both sides of the river delta and the wave crest line are given by AOA<sup>0</sup> and BOB<sup>0</sup> . These breaker angles can be evaluated as follows. First, the sum of these breaker angles becomes 10° by subtracting the apex angle of the river delta of 170° from 180°. In the distribution of the longshore sand transport shown in **Figure 8**, the longshore sand transport on the west and east sides of the Hino River mouth are 3.6 � <sup>10</sup><sup>5</sup> and 1.1 � <sup>10</sup><sup>5</sup> <sup>m</sup><sup>3</sup> /year, respectively, resulting in the ratio of longshore sand transport on both sides of 3:1, because the longshore sand transport is proportional to the breaker angle. Taking these conditions into account, the breaker angles on the left and right banks of the river mouth decrease to 7.5° and 2.5°, respectively, when separating 10° by the ratio of 3:1. Finally, the predominant wave direction normal to A<sup>0</sup> B<sup>0</sup> is assumed to be N13°E.

In 1947, the shoreline of the river delta receded because of the decrease in sand supply from the river, as shown in **Figure 10(b)**, and the apex angle increased to up to 175°, and the breaker angles on the left and right banks of the river became 5° and 0°, respectively, taking the predominant wave direction of N13°E calculated as

*A Long-Term Prediction of Beach Changes around River Delta using Contour-Line-Change Model DOI: http://dx.doi.org/10.5772/intechopen.85207*

#### **Figure 10.**

to be 4.7 � <sup>10</sup><sup>5</sup> <sup>m</sup><sup>3</sup>

**Figure 9.**

**direction**

straight line of A<sup>0</sup>

normal to A<sup>0</sup>

**70**

3.6 � <sup>10</sup><sup>5</sup> and 1.1 � <sup>10</sup><sup>5</sup> <sup>m</sup><sup>3</sup>

in **Figure 8** gives 21 m<sup>3</sup>

height of beach changes of *h* = 10 m.

part, and it took a maximum of 6.5 � <sup>10</sup><sup>4</sup> <sup>m</sup><sup>3</sup>

B<sup>0</sup> is assumed to be N13°E.

/year as the sum of longshore sand transport on both sides of the

/m/year as the rate of sand accretion, the rate of the advance

/year in 2009.

OB<sup>0</sup> in **Figure 10(a)**, the breaker angles between the shoreline on

/year, respectively, resulting in the ratio of longshore

/year at *x* = 7.5 km in

.

river mouth. In addition, since the longshore inclination of longshore sand transport

of the river delta becomes 2.1 m/year, when dividing this rate by the characteristic

west end, and dividing by the elapsed time. **Figure 9** shows the results. The

*Distribution of longshore sand transport calculated from beach changes between 1980 and 2009.*

**3. Change in apex angle of Hino River delta and predominant wave**

longshore sand transport took a maximum of 1.0 � <sup>10</sup><sup>5</sup> <sup>m</sup><sup>3</sup>

*Sedimentary Processes - Examples from Asia,Turkey and Nigeria*

Similarly, the distribution of longshore sand transport after 1980 was evaluated by integrating the topographic changes between 1980 and 2009 in the east from the

1990 with a secondary peak at *x* = 0 km. After 2000, it has decreased in the eastern

**Figure 10** shows the geographical map in 1899 and aerial photograph in 1947. The apex angle of the river delta (angle AOB) in 1899 was 170°. Assuming that the wave crest line of the predominant wave incident to the river delta is given by a

both sides of the river delta and the wave crest line are given by AOA<sup>0</sup> and BOB<sup>0</sup>

These breaker angles can be evaluated as follows. First, the sum of these breaker angles becomes 10° by subtracting the apex angle of the river delta of 170° from 180°. In the distribution of the longshore sand transport shown in **Figure 8**, the longshore sand transport on the west and east sides of the Hino River mouth are

sand transport on both sides of 3:1, because the longshore sand transport is proportional to the breaker angle. Taking these conditions into account, the breaker angles on the left and right banks of the river mouth decrease to 7.5° and 2.5°, respectively, when separating 10° by the ratio of 3:1. Finally, the predominant wave direction

In 1947, the shoreline of the river delta receded because of the decrease in sand supply from the river, as shown in **Figure 10(b)**, and the apex angle increased to up to 175°, and the breaker angles on the left and right banks of the river became 5° and 0°, respectively, taking the predominant wave direction of N13°E calculated as

*Change in apex angle of Hino River delta and predominant wave direction. (a) 1899. (b) 1947.*

mentioned above. The decrease in the breaker angle west of the river mouth corresponds to the decrease in the westward longshore sand transport.

#### **4. Contour-line-change model considering change in grain size**

A numerical model described in [13, 14] was used. Let the *x*- and *z*-axes be the longshore distance and seabed elevation relative to the still water level, respectively, and *Y* (*x, z, t*) is the offshore distance to a specific contour line to be solved. To consider the sorting of sand of different grain sizes by cross-shore sand transport, the depth distribution of cross-shore sand transport considering the grain size effect in mixing was considered. The sorting of grain size populations was modeled by introducing the equilibrium slope angle *β*<sup>c</sup> (*k*) , which corresponds to each grain size population *k*. In this case, a grain size population was assumed to have a single equilibrium beach slope with a characteristic grain size *d*(*k*) . By assuming that the mobility of sand of each grain size population by cross-shore movement is the same as that of longshore sand transport, the coefficient of the sediment transport rate according to the grain size *d*(*k*) , which was given in [21, 22], was introduced. Furthermore, assuming that the ratio of the exposed area of each grain size population to the entire sea bottom area is equal to the content of each size population in the exchange layer *<sup>μ</sup>*ð Þ*<sup>k</sup>* ð Þ *<sup>k</sup>* <sup>¼</sup> <sup>1</sup>*;* <sup>2</sup>*;* <sup>⋯</sup>*; <sup>N</sup>* , the cross-shore sand transport of each grain size population *q*<sup>z</sup> (*k*) is derived.

Cross-shore sand transport:

$$\begin{aligned} q\_x^{(k)} &= \mu^{(k)} \cdot \varepsilon\_{\overline{x}}(\mathbf{z}) \cdot \boldsymbol{\chi} \cdot \boldsymbol{K}\_1^{(k)} \cdot \left(\mathbf{E} \mathbf{C}\_{\overline{\mathfrak{g}}}\right)\_{\mathbf{b}} \cos^2 \boldsymbol{a}\_{\mathbf{b}} \sin \overline{\boldsymbol{\beta}} \cdot \left(\mathbf{cot } \boldsymbol{\beta}/\cot \boldsymbol{\rho}\_c^{(k)} - \mathbf{1}\right); \\\ k &= \mathbf{1, 2, \cdots, N} \end{aligned} \tag{1}$$

$$K\_1^{(k)} = \frac{A}{\sqrt{d^{(k)}}}; k = 1, 2, \cdots, N \tag{2}$$

$$\mathbf{\dot{x} \cot \beta = -\partial Y/\partial \mathbf{z}} \tag{3}$$

Change in content of each grain size population:

*∂y*ð Þ*<sup>k</sup> <sup>∂</sup><sup>t</sup>* � *<sup>∂</sup><sup>Y</sup>*

*∂y*ð Þ*<sup>k</sup> <sup>∂</sup><sup>t</sup>* � *<sup>∂</sup><sup>Y</sup>*

landward of the initial exchange layer. The width *B* of the exchange layer is determined with reference to the mixing depth reported by Kraus [25]. The abovementioned equations were solved simultaneously. In the numerical simulation, the calculation points of the contour-line position and sand transport rate are set in staggered meshes with a difference of 1/2 mesh; the cross-shore and longshore sand transport are calculated using Eqs. (1) and (5), respectively. In this case, the upcoast value depending on the direction of sand transport is employed as *μ*(*k*) in Eqs. (1) and (5). The bathymetric changes can be calculated using Eqs. (6) and (7). *μ*(*k*) is calculated using Eqs. (8) and (9), and the calculation point of *μ*(*k*) is set at the

*k* ¼ 1*,* 2*,* ⋯*, N*

*k* ¼ 1*,* 2*,* ⋯*, N*

*<sup>∂</sup><sup>t</sup>* � *<sup>μ</sup>*ð Þ*<sup>k</sup> ,*

> *<sup>∂</sup><sup>t</sup>* � *<sup>μ</sup>*ð Þ*<sup>k</sup> B*

*<sup>B</sup>* is the content of each grain size population on the sandy beach

The numerical simulation of topographic changes relative to the initial topography in 1899 was carried out until 1967, when a large amount of sand was supplied from the river as a result of the extensive mining of iron sand, and then sand supply (*Q*in) markedly decreased after the stoppage of the mining. The calculation period was separated into four periods corresponding to the sand discharge and the structural conditions of the coast. In the first period between 1899 and 1920, *Q*in of

river delta. In this period, the shoreline in the entire area advanced at a rate of

In the second period between 1920 and 1947, *Q*in began to decrease from

was calculated by subtracting the sand volume in the first period (4.7 � 105 m3

which was determined from the shoreline changes between 1899 and 1947, and divided by the elapsed time of 27 years. In the third period between 1947 and 1962,

west of the Hino River mouth, although *<sup>Q</sup>*in remained constant at 6 � 104 m3

which was calculated from the measured shoreline changes between 1947 and 1967. In the fourth period between 1962 and 1967, the seawall and groins were constructed

By setting the coastline lengths in the areas west and east of the river mouth to be *L*<sup>1</sup> and *L*2, we found that the wave directions �*θ*<sup>1</sup> and *θ*<sup>2</sup> immediately west and east of the river mouth, respectively, satisfy a relationship of �*θ*1/*θ*<sup>2</sup> = *L*1/*L*<sup>2</sup> = 3. In

2 m/year, and the entire shoreline advance reached 42 m by 1920.

year � 21 year) from that between 1899 and 1947 (2.9 � 105 m3

/year before 1920 to 1.5 � 105 m3

/year was supplied from the river, resulting in the development of the

/year after 1920. *<sup>Q</sup>*in of 1.5 � 105 m3

/year owing to the stoppage of mining of iron sand,

The content of each grain size population in the new exchange layer formed

*A Long-Term Prediction of Beach Changes around River Delta using Contour-Line-Change Model*

*,*

*∂Y ∂t* ≥ 0*;*

*∂Y ∂t* ≤ 0*;* (8)

(9)

/year

/

/year.

/year � 48 year),

*∂μ*ð Þ*<sup>k</sup> <sup>∂</sup><sup>t</sup>* <sup>¼</sup> <sup>1</sup> *B*

*DOI: http://dx.doi.org/10.5772/intechopen.85207*

*∂μ*ð Þ*<sup>k</sup> <sup>∂</sup><sup>t</sup>* <sup>¼</sup> <sup>1</sup> *B*

same point for calculating the contour-line position.

**5. Reproduction of reduction in size of Hino River delta**

during erosion is expressed as:

**5.1 Calculation conditions**

*<sup>Q</sup>*in decreased until 6 � 104 m3

4.7 � <sup>10</sup><sup>5</sup> <sup>m</sup><sup>3</sup>

4.7 � 105 m3

**73**

where *μ*ð Þ*<sup>k</sup>*

$$\varepsilon\_{\mathbf{x}}(\mathbf{z}) = \begin{cases} \left(2/h\_{\mathbf{c}}^3\right) \left(h\_{\mathbf{c}}/2 - \mathbf{z}\right) \left(\mathbf{z} + h\_{\mathbf{c}}\right)^2, \ -h\_{\mathbf{c}} \le \mathbf{z} \le h\_{\mathbf{R}} \\\\ \mathbf{0}, \mathbf{z} \le -h\_{\mathbf{c}}, \mathbf{z} \ge h\_{\mathbf{R}} \end{cases} \tag{4}$$

Here, *q*ð Þ*<sup>k</sup> <sup>z</sup>* ð Þ *k* ¼ 1*;* 2*;* ⋯*; N* is the cross-shore sand transport per unit length in the longshore direction for each grain size population, *μ*(*k*) is the content of each grain size population (*k*) in the exchange layer of sand, (*EC*g)b is the wave energy flux at the breaking point, *K*<sup>1</sup> (*k*) is the coefficient of longshore sand transport, *εz*(*z*) is assumed to be equivalent to the depth distribution of the longshore sand transport *εx*(*z*) given by Uda and Kawano [23], and *d*(*k*) is a typical grain size of the grain size population. *A* is a coefficient that depends on the physical conditions of the beach. *d*(*k*) in Eq. (2) has a unit of mm. *γ* is the ratio of the coefficient of cross-shore sand transport to that of longshore sand transport (*γ* = *K*<sup>z</sup> (*k*) /*K*<sup>1</sup> (*k*) ) and expresses the mobility of cross-shore sand transport relative to that of longshore sand transport, *α*<sup>b</sup> is the angle between the wave crest line at the breaking point and each contour line, and *β* is the beach slope angle at each contour line. *β* is the average beach slope angle between the berm height *h*<sup>R</sup> and the depth of closure *h*c, and *β*<sup>c</sup> (*k*) is the equilibrium beach slope angle. When the beach slope becomes steeper than the angle of repose of sand, sand is transported offshore by gravity. By this procedure, we can calculate the formation of a scarp in a zone larger than *h*<sup>R</sup> and the sinking of sand in a zone larger than *h*c.

Longshore sand transport:

$$\begin{aligned} q\_x^{(k)} &= \mu^{(k)} \cdot \varepsilon\_x(\mathbf{z}) \cdot K\_1^{(k)} \cdot \left( \mathbf{E} C\_\mathbf{g} \right)\_\mathbf{b} \cdot \left( \cos \alpha\_\mathbf{b} \sin a\_\mathbf{b} - \xi \frac{\mathbf{1}}{\tan \beta\_\mathbf{b}} \cdot \cos a\_\mathbf{b} \cdot \frac{\partial H\_\mathbf{b}}{\partial \mathbf{x}} \right); \\ k &= \mathbf{1}, \mathbf{2}, \cdots, N \end{aligned} \tag{5}$$

Here, *q*ð Þ*<sup>k</sup> <sup>x</sup>* ð Þ *k* ¼ 1*;* 2*;* ⋯*; N* is the longshore sand transport per unit depth for each grain size population, □*εx*(*z*) is the depth distribution of longshore sand transport, and *ξ* is the constant given by *K*<sup>2</sup> (*k*) /*K*<sup>1</sup> (*k*) , which depends on the physical conditions of the beach, where *K*<sup>2</sup> (*k*) is a function of *K*<sup>1</sup> (*k*) and is equivalent to the coefficient of Ozasa and Brampton [24]. tan*β*<sup>b</sup> is the beach slope in the surf zone and *H*<sup>b</sup> is the breaker height.

Mass conservation for each grain size:

$$\begin{aligned} \frac{\partial \mathbf{y}^{(k)}}{\partial t} &= -\frac{\partial q\_x^{(k)}}{\partial x} - \frac{\partial q\_x^{(k)}}{\partial x} + \mathcal{S}^{(k)};\\ k &= 1, 2, \cdots, N \end{aligned} \tag{6}$$

Here, *<sup>S</sup>*ð Þ*<sup>k</sup>* ð Þ *<sup>k</sup>* <sup>¼</sup> <sup>1</sup>*;* <sup>2</sup>*;* <sup>⋯</sup>*; <sup>N</sup>* is the additional term for calculating the sand source for each grain size population, by which sediment supply from a river or the sand supply by beach nourishment is given. The total contour-line change at a certain position is determined by the summation of the contour-line changes of all grain size populations at that position.

$$\frac{\partial Y}{\partial t} = \sum\_{k=1}^{N} \frac{\partial \mathbf{y}^{(k)}}{\partial t} \tag{7}$$

*A Long-Term Prediction of Beach Changes around River Delta using Contour-Line-Change Model DOI: http://dx.doi.org/10.5772/intechopen.85207*

Change in content of each grain size population:

cot *<sup>β</sup>* ¼ �*∂Y=∂<sup>z</sup>* (3)

*,* � *h*<sup>c</sup> ≤*z*≤ *h*<sup>R</sup>

(*k*) /*K*<sup>1</sup> (*k*)

tan *β*<sup>b</sup>

� �

� cos *α*<sup>b</sup> �

, which depends on the physical conditions

(*k*) and is equivalent to the coefficient of

*<sup>∂</sup><sup>t</sup>* (7)

*∂H*<sup>b</sup> *∂x*

*;*

(5)

(6)

(4)

) and expresses the

(*k*) is the

*εx*ð Þ¼ *z*

Here, *q*ð Þ*<sup>k</sup>*

the breaking point, *K*<sup>1</sup>

sand in a zone larger than *h*c. Longshore sand transport:

ð Þ*<sup>k</sup>* <sup>¼</sup> *<sup>μ</sup>*ð Þ*<sup>k</sup>* � *<sup>ε</sup>x*ð Þ� *<sup>z</sup> <sup>K</sup>*ð Þ*<sup>k</sup>*

and *ξ* is the constant given by *K*<sup>2</sup>

size populations at that position.

Mass conservation for each grain size:

*k* ¼ 1*,* 2*,* ⋯*, N*

of the beach, where *K*<sup>2</sup>

Here, *q*ð Þ*<sup>k</sup>*

breaker height.

**72**

*qx*

2*=h*<sup>3</sup> c

sand transport to that of longshore sand transport (*γ* = *K*<sup>z</sup>

<sup>1</sup> � *EC*<sup>g</sup> � �

(

*Sedimentary Processes - Examples from Asia,Turkey and Nigeria*

� �ð Þ *<sup>h</sup>*c*=*<sup>2</sup> � *<sup>z</sup>* ð Þ *<sup>z</sup>* <sup>þ</sup> *<sup>h</sup>*<sup>c</sup> <sup>2</sup>

longshore direction for each grain size population, *μ*(*k*) is the content of each grain size population (*k*) in the exchange layer of sand, (*EC*g)b is the wave energy flux at

assumed to be equivalent to the depth distribution of the longshore sand transport *εx*(*z*) given by Uda and Kawano [23], and *d*(*k*) is a typical grain size of the grain size population. *A* is a coefficient that depends on the physical conditions of the beach. *d*(*k*) in Eq. (2) has a unit of mm. *γ* is the ratio of the coefficient of cross-shore

mobility of cross-shore sand transport relative to that of longshore sand transport, *α*<sup>b</sup> is the angle between the wave crest line at the breaking point and each contour line, and *β* is the beach slope angle at each contour line. *β* is the average beach slope angle between the berm height *h*<sup>R</sup> and the depth of closure *h*c, and *β*<sup>c</sup>

equilibrium beach slope angle. When the beach slope becomes steeper than the angle of repose of sand, sand is transported offshore by gravity. By this procedure, we can calculate the formation of a scarp in a zone larger than *h*<sup>R</sup> and the sinking of

<sup>b</sup> � cos *<sup>α</sup>*<sup>b</sup> sin *<sup>α</sup>*<sup>b</sup> � *<sup>ξ</sup>* <sup>1</sup>

*<sup>x</sup>* ð Þ *k* ¼ 1*;* 2*;* ⋯*; N* is the longshore sand transport per unit depth for each

<sup>þ</sup> *<sup>S</sup>*ð Þ*<sup>k</sup> ;*

grain size population, □*εx*(*z*) is the depth distribution of longshore sand transport,

Ozasa and Brampton [24]. tan*β*<sup>b</sup> is the beach slope in the surf zone and *H*<sup>b</sup> is the

*x <sup>∂</sup><sup>x</sup>* � *<sup>∂</sup>q*ð Þ*<sup>k</sup> z ∂z*

Here, *<sup>S</sup>*ð Þ*<sup>k</sup>* ð Þ *<sup>k</sup>* <sup>¼</sup> <sup>1</sup>*;* <sup>2</sup>*;* <sup>⋯</sup>*; <sup>N</sup>* is the additional term for calculating the sand source for each grain size population, by which sediment supply from a river or the sand supply by beach nourishment is given. The total contour-line change at a certain position is determined by the summation of the contour-line changes of all grain

*∂y*ð Þ*<sup>k</sup>*

(*k*) /*K*<sup>1</sup> (*k*)

(*k*) is a function of *K*<sup>1</sup>

*<sup>∂</sup><sup>t</sup>* ¼ � *<sup>∂</sup>q*ð Þ*<sup>k</sup>*

*k* ¼ 1*,* 2*,* ⋯*, N*

*∂Y <sup>∂</sup><sup>t</sup>* <sup>¼</sup> <sup>∑</sup> *N k*¼1

*∂y*ð Þ*<sup>k</sup>*

*<sup>z</sup>* ð Þ *k* ¼ 1*;* 2*;* ⋯*; N* is the cross-shore sand transport per unit length in the

(*k*) is the coefficient of longshore sand transport, *εz*(*z*) is

0*, z*≤ � *h*c*, z*≥*h*<sup>R</sup>

$$\frac{\partial \mu^{(k)}}{\partial t} = \frac{1}{B} \left\{ \frac{\partial \boldsymbol{\eta}^{(k)}}{\partial t} - \frac{\partial \boldsymbol{Y}}{\partial t} \cdot \boldsymbol{\mu}^{(k)} \right\}, \frac{\partial \boldsymbol{Y}}{\partial t} \ge \mathbf{0};\tag{8}$$
 
$$k = \mathbf{1}, \mathbf{2}, \cdots, N$$

The content of each grain size population in the new exchange layer formed during erosion is expressed as:

$$\frac{\partial \mu^{(k)}}{\partial t} = \frac{1}{B} \left\{ \frac{\partial \mathbf{y}^{(k)}}{\partial t} - \frac{\partial Y}{\partial t} \cdot \mu\_B^{(k)} \right\}, \frac{\partial Y}{\partial t} \le 0;\tag{9}$$
 
$$k = 1, 2, \dots, N$$

where *μ*ð Þ*<sup>k</sup> <sup>B</sup>* is the content of each grain size population on the sandy beach landward of the initial exchange layer. The width *B* of the exchange layer is determined with reference to the mixing depth reported by Kraus [25]. The abovementioned equations were solved simultaneously. In the numerical simulation, the calculation points of the contour-line position and sand transport rate are set in staggered meshes with a difference of 1/2 mesh; the cross-shore and longshore sand transport are calculated using Eqs. (1) and (5), respectively. In this case, the upcoast value depending on the direction of sand transport is employed as *μ*(*k*) in Eqs. (1) and (5). The bathymetric changes can be calculated using Eqs. (6) and (7). *μ*(*k*) is calculated using Eqs. (8) and (9), and the calculation point of *μ*(*k*) is set at the same point for calculating the contour-line position.

## **5. Reproduction of reduction in size of Hino River delta**

#### **5.1 Calculation conditions**

The numerical simulation of topographic changes relative to the initial topography in 1899 was carried out until 1967, when a large amount of sand was supplied from the river as a result of the extensive mining of iron sand, and then sand supply (*Q*in) markedly decreased after the stoppage of the mining. The calculation period was separated into four periods corresponding to the sand discharge and the structural conditions of the coast. In the first period between 1899 and 1920, *Q*in of 4.7 � <sup>10</sup><sup>5</sup> <sup>m</sup><sup>3</sup> /year was supplied from the river, resulting in the development of the river delta. In this period, the shoreline in the entire area advanced at a rate of 2 m/year, and the entire shoreline advance reached 42 m by 1920.

In the second period between 1920 and 1947, *Q*in began to decrease from 4.7 � 105 m3 /year before 1920 to 1.5 � 105 m3 /year after 1920. *<sup>Q</sup>*in of 1.5 � 105 m3 /year was calculated by subtracting the sand volume in the first period (4.7 � 105 m3 / year � 21 year) from that between 1899 and 1947 (2.9 � 105 m3 /year � 48 year), which was determined from the shoreline changes between 1899 and 1947, and divided by the elapsed time of 27 years. In the third period between 1947 and 1962, *<sup>Q</sup>*in decreased until 6 � 104 m3 /year owing to the stoppage of mining of iron sand, which was calculated from the measured shoreline changes between 1947 and 1967. In the fourth period between 1962 and 1967, the seawall and groins were constructed west of the Hino River mouth, although *<sup>Q</sup>*in remained constant at 6 � 104 m3 /year.

By setting the coastline lengths in the areas west and east of the river mouth to be *L*<sup>1</sup> and *L*2, we found that the wave directions �*θ*<sup>1</sup> and *θ*<sup>2</sup> immediately west and east of the river mouth, respectively, satisfy a relationship of �*θ*1/*θ*<sup>2</sup> = *L*1/*L*<sup>2</sup> = 3. In

**Figure 11.** *Wave direction and distribution of longshore sand transport used for calculation. (a) Wave direction. (b) Longshore sand transport.*

addition, *θ*<sup>1</sup> and *θ*<sup>2</sup> are approximately given by 7.5° and 2.5°, respectively, from the geographical map in 1899, as shown in **Figure 10**. Thus, after the trial-and-error calculation of the shoreline changes, while satisfying the relation of *θ*1/*θ*<sup>2</sup> = 3, we obtained *θ*<sup>1</sup> = 12° and *θ*<sup>2</sup> = 4° for the best fit results. Finally, a linear distribution shown in **Figure 11(a)** was assumed with angles of 12° and 4° immediately west and east of the Hino River mouth, respectively. **Figure 11(b)** shows the distribution of longshore sand transport at the initial stage. For the wave condition, the breaker height was determined as *H*<sup>b</sup> = 1.1 m on the basis of the energy mean significant wave height measured between 1995 and 2008 at Hiezu wave observatory.

The initial seabed slope was assumed to be 1/6 between *z* = +3 and 2 m, and 1/30 between *z* = 2 and 8 m. The number of grain sizes (*N*) was set to 2 with characteristic grain sizes of *d*(1) = 0.3 mm for fine sand and *d*(2) = 0.5 mm for coarse sand. The initial contents *μ*<sup>1</sup> and *μ*<sup>2</sup> for fine and coarse sand were, respectively, assumed to be 0.0 and 1.0 in the cell between *z* = +3 and 1 m, 0.5 and 0.5 in the cell at *z* = 2 m, and 1.0 and 0.0 in the cell between *z* = 3 and 8 m. The equilibrium slopes tan*β*<sup>c</sup> (1) and tan*β*<sup>c</sup> (2) of fine and coarse sand were, respectively, assumed to be 1/30 and 1/6, on the basis of the measured longitudinal profile. As for the sand back pass on the coast, sand was extracted from the foreshore with an elevation between 0 and + 3 m at *x* = 13 km, and the same amount of sand was supplied from the foreshore at *x* = 8 km. **Table 1** summarizes the other calculation conditions.

> bathymetry in 1920 (**Figure A1**). During the first period, the parallel contours in 1899 simply advanced by 42 m until 1920, so that a new coordinate was taken for the shoreline to be shifted by 42 m seaward. During the second period, a V-shape shoreline recession began around the river mouth because of the decrease in *Q*in from

**Calculation method Contour-line-change model considering grain size changes**

*A Long-Term Prediction of Beach Changes around River Delta using Contour-Line-Change Model*

1 (1899–1920): 4.7 <sup>10</sup><sup>5</sup> <sup>m</sup><sup>3</sup>

2 (1920–1947): 1.5 <sup>10</sup><sup>5</sup> <sup>m</sup><sup>3</sup>

3 (1947–1962): 6 <sup>10</sup><sup>4</sup> <sup>m</sup><sup>3</sup>

4 (1962–1967): 6 <sup>10</sup><sup>4</sup> <sup>m</sup><sup>3</sup>

Initial bathymetry Straight parallel contours between *z* = +3 m and 2 m with 1/6

fine sand of *d*(1) = 0.3 mm coarse sand of *d*(2) = 0.5 mm

• Initial contents

(1) = 1/30 and tan*β*<sup>c</sup>

and 2008 at Hiezu observatory

Berm height *h*<sup>R</sup> =3m Coefficient of sand transport Coefficient of longshore sand transport *A* = 0.12 for initial

Uniform

Mesh sizes *Δx* = 100 m (longshore direction) and *Δz* = 1 m (vertical direction)

Width of exchange layer *B* = 6 m (thickness of exchange layer = 1 m)

Incident wave condition • Breaker height: *Hb* = 1.1 m,

Depth of closure and berm height Depth of closure *h*<sup>c</sup> =8m

Critical slope of falling sand On land: 1/2 and on seabed: 1/2

Boundary conditions Left and right boundaries: *qx* = 0

Calculation range of depth *z* = +3.5 to 8.5 m

Time intervals *Δt* =1h

/year

/year

/year + seawall

*z* = 3 m to 1 m: *μ*<sup>1</sup> = 0.0 (fine sand) and *μ*<sup>2</sup> = 1.0 (coarse sand) *z* = 2 m: *μ*<sup>1</sup> = 0.5 (fine sand) and *μ*<sup>2</sup> = 0.5 (coarse sand) *z* = 3 m to 8 m: *μ*<sup>1</sup> = 1.0 (fine sand) and *μ*<sup>2</sup> = 0.0 (coarse sand)

(2) = 1/6

energy-mean significant wave height measured between 1995

• Initial breaker angle: linear distribution between 0° (Sakai channel) and 12° (Hino River mouth), and between 4° (Hino River mouth) and 0° (Yodoe fishing port)

longshore sand transport to be *<sup>Q</sup>* = 3.6 105 <sup>m</sup><sup>3</sup>

Landward and offshore boundaries: *qz* = 0

Ratio of cross-shore and longshore sand transports *γ* = 0.1 Coefficient of Ozasa and Brampton's [18] term *ξ* = 3.24

/year

slope, between *z* = 2 m and 8 m, 1/30 slope

• For characteristic grain sizes,

Calculation domain Sakai channel to Yodoe fishing port

Grain size • *N* = 2

*DOI: http://dx.doi.org/10.5772/intechopen.85207*

Equilibrium slope tan*β<sup>c</sup>*

Tide condition MSL

Depth distribution of longshore and cross-shore sand transport

*Q*in in each calculation period Reproduction calculation between 1899 and 1967

/year. However, the contour lines far

/year

/year before 1920 to 1.5 105 m3

4.7 105 m3

**75**

**Table 1.**

*Calculation conditions.*

#### **5.2 Results of numerical simulation**

**Figure 12** shows the bathymetric changes from the first to the fourth periods together with an additional expression of bathymetric changes relative to the


*A Long-Term Prediction of Beach Changes around River Delta using Contour-Line-Change Model DOI: http://dx.doi.org/10.5772/intechopen.85207*
