Predicting Flood Hazard Indices in Torrential or Flashy River Basins and Catchments

The idealised river basin consisted of a 14 m wide and 1 m deep trapezoidal channel with the side slope of 30° and two 100 m wide floodplains on the each side of the main channel (see Fig. 1a). The idealised river basin was 2000 m long and was divided into two 1000 m sections. The upper section (i.e. the first 1000 m) had a different bed slope for each test case, and with the bed slope ranging between 0.1% to 2.5% (i.e. S = 0.001 and S = 0.025). The upper limit of the bed slope range is consistent with the bed slope of River Valency at Boscastle (O'Connor 2008), the site of one of the most violent flash floods in the history of the UK (Kvočka et al. 2015). The lower section had a constant bed slope which remained nearly horizontal (i.e. S = 0.001) for every test case. At the end of the river basin, there was a 100 m long and 2 m deep reservoir (see Fig. 1b).

The idea behind the design of this test case was to mimic the propagation of a flood wave through a short and steep river basin, as this type of terrain is often associated with occurrence of flash floods. Even though flash flooding is generally a result of a unique combination of meteorological and hydrological conditions the steepness of the terrain is also important, since it can affect the occurrence of flash floods due to its orographic effects, and promote the rapid concentration and propagation of stream flow (Marchi et al. 2010). Examples of such topography (i.e. complex, short and steep river basins and catchments) can be found in many areas prone to flash flooding, such as much of Wales and south-east England in the UK, the Mediterranean region (e.g. Greece, France, Spain), Central Europe (e.g. Slovenia, Slovakia, Austria) etc. (Gaume et al. 2009; Marchi et al. 2010; Foulds et al. 2012).

The idealised river basin computational domain was 2100 m long, 214 m wide and divided into square cells, with each cell having an area of 1 m². An inflow boundary was set as the upstream boundary condition, with a sinusoidal hydrograph being used to represent the flood wave (Lin et al. 2006):

where q_p is the unit-width peak discharge, T is the duration of the flood and t is the time step with the following restriction

In the conducted tests q_p was set to 300 m³/s and T was set to 4 h, which meant that the hydrograph simulated had a relatively high peak discharge (considering the dimensions of the main channel and the valley) and a short time-to-peak (i.e. 2 h). In addition, the overall flood duration was generally also short (i.e. 4 h). Also, the considered peak value of the hydrograph is consistent with the estimated peak value of the 2007 Železniki flash flood, the site of one of the most violent flash floods in the history of the Slovenia (Kvočka et al. 2016). Therefore, the considered hydrograph had all the characteristics of an extreme flash flood (Gaume et al. 2009; Marchi et al. 2010). The downstream boundary was set at the end of the reservoir, with a prescribed water level being specified as the downstream boundary condition. The main channel was assigned a Manning’s roughness coefficient of 0.04, while the floodplains were assigned a Manning’s roughness coefficient of 0.05.

Borth is a coastal village in west Wales, UK. It is a popular holiday seaside resort, with several caravan and camping sites nearby (see Fig. 2a). It is situated at the end of the Leri catchment. The Leri catchment is a relatively small and generally steep catchment, and is susceptible to the occurrence of flash floods (Foulds et al. 2012). The most recent flash flood occurred on 9th of June 2012, which flooded Borth and the nearby villages of Dol-y-bont and Tal-y-bont. Around 60 properties and tens of caravans were flooded around Borth, with a significant number of residents being evacuated from flooded properties in Tal-y-bont and caravan sites near Dol-y-bont as a result of the flood (Ceredigion County Council 2012; BBC 2012). Despite the general perception that the 2012 flash flood was one of the severest in history, the post-flood study revealed that the magnitude of this flood event was more common than first thought. Namely, the estimated return period for this flood event was between 50 and 80 years (Foulds et al. 2012).

The Borth study domain was 9 km long, 7 km wide and covered a wider area around the villages of Borth, Dol-y-bont and Tal-y-bont (see Fig. 2b). The 2 m LiDAR (Laser Imaging Detection and Ranging) data were used to set up a computational hydraulics model.The upstream boundary was set as an inflow boundary for the River Leri and the River Cuelan near the village of Tal-y-bont. A 1:100 year flood event was simulated in this study, with the estimated peak discharge values for River Leri and River Ceulan being 64.5 m³/s and 19.1 m³/s. The downstream boundary was set in the Dyfi estuary, with a prescribed water level being specified as the downstream boundary condition. The selection of the roughness parameters was based on a site-survey conducted by the authors, and on the roughness values proposed in a study conducted for Natural Resources Wales. The floodplains were assigned a roughness value of 0.05, while both the natural river channel and the drainage channel on the Cors were assigned a roughness value of 0.04. It should be noted that there were no relevant data available to optimise these selected roughness values.

The empirical flood hazard assessment methodology proposed by Ramsbottom et al. (2006) was developed for the Department for Environment, Food and Rural Affairs (DEFRA) and the UK Environment Agency. Ramsbottom et al. (2006) tested various empirical formulae by comparing the predictions to experimental datasets obtained from laboratory studies conducted by Abt et al. (1989) and Karvonen et al. (2000). The formula adopted to undertake hazard rating analysis is given as follows:

where HR is the flood hazard rating (m²/s), d is the water depth (m), v is the velocity of the flow (m/s) and DF is a debris factor (m²/s), which can have a value of 0, 0.5 or 1, depending on the place of the flood and on the features of the flow. Four flood hazard classifications were proposed, which are summarised in Table 1.

The empirical expression presented by Ramsbottom et al. (2006) has some shortcomings, such as incorporation of the ability of the test subject to learn how to manoeuvre in the flow with time (i.e. training), inclusion of a debris factor without any prior experimental testing, and exclusion of any upper depth limit, which means that large depth/low velocity flood flows are not necessarily considered as hazardous (i.e. floating is not automatically classified as dangerous) (Cox et al. 2010). Nonetheless, the criterion proposed by Ramsbottom et al. (2006) is well established, both within and outside of the UK (Purwandari et al. 2011; Porter and Demeritt 2012; Foudi et al. 2015), and is therefore regarded as a reliable criterion for assessing and mapping flood hazard risk to people.

The formulae presented by Xia et al. (2014) is based on the physical interpretation of the processes that affect the stability of people in floodwaters. The proposed formulae considers: (i) all forces acting on a human body in floodwaters, i.e. drag, bed frictional, gravity, buoyancy and the normal reaction force, (ii) the effect of a non-uniform upstream vertical velocity profile on the stability of a person standing in floodwaters, and (iii) the impact of the net buoyancy force on the body for rapidly varying water depths.

The formulae proposed by Xia et al. (2014) are based on the mechanisms of toppling and sliding instability, which are the two main instability mechanisms for a human body in floodwaters. The incipient velocity for a human body in floodwater experiencing toppling instability is written as:

where U_c is the incipient velocity, h_f is the water depth (m), h_p is the height of the person (m), m_p is the weight of the person (kg), ρ_f is the density of water (kg/m³), α and β are empirical coefficients and a₁, a₂, b₁ and b₂ are coefficients based on the general characteristics of the human body.

The incipient velocity for a human body in floodwater experiencing sliding instability is written as:

where U_c is the incipient velocity, h_f is the water depth (m), h_p is the height of the person (m), m_p is the weight of the person (kg), ρ_f is the density of water (kg/m³), α and β are empirical coefficients and a₁, a₂, b₁ and b₂ are coefficients based on the characteristics of the human body.

The initially proposed criteria were further refined in order to take into account the effect of different slopes. For example, the formula for the incipient velocity at toppling instability that considers the effect of slope is given as (Xia et al. 2016):

where U_c is the incipient velocity, h_f is the water depth (m), h_p is the height of the person (m), m_p is the weight of the person (kg), ρ_f is the density of water (kg/m³), α and β are empirical coefficients, a₁, a₂, b₁ and b₂ are coefficients based on the characteristics of a human body, θ is the angle of the ground slope and γ is the ratio defined as

where a_gx is the correction coefficient for the distance between the centre of gravity of the body and the bed and a_gy is the correction coefficient for the distance between the position of the centre of gravity of the body and the heel.

The degree of flood hazard risk according to the criteria proposed by Xia et al. (2014) can be quantified by mimicking the principle of bivalence, and is given as:

where HR is the flood hazard rating, U is the mean velocity of the flow, U_toppling is the toppling incipient velocity and U_sliding is the sliding incipient velocity.

The main difference between the empirical approach and the mechanics based approach is in the way they take into account forces induced by flow conditions. The overturning force on the human body in floodwaters according to the empirical approach (see Eq. (2)) is proportional to the water depth times the velocity (i.e. hv), whereas in the mechanics based approach (see Eqs. (3), (4) and (5)) the overturning force is proportional to the water depth times the velocity squared (i.e. hv²). Therefore, the mechanics based criteria can be much more influenced by higher velocities and momentum, as compared to the empirically based criteria, and thus can quickly adapt to rapidly varying flood events with abrupt changes in the flow regime (e.g. hydraulic jumps). This characteristic enables a more accurate assessment to be made of the flood hazard indices in a short time period, which is a particularly important feature for flood hazard assessment of rapidly varying flood events, such as flash floods (Kvočka et al. 2016).

Figure 3 shows a comparison between the empirically based and the mechanics based flood hazard assessment methods for the idealised river basin test case. It can be seen in Fig. 3 that both criteria generally assessed a similar degree of flood hazard risk when the upper gradient was below 1% (i.e. S = 0.01). This was as expected, since the formulae based on empirical or quasi-theoretical studies are as accurate as mechanics based criteria for flood events that are characterised with generally low velocities (i.e. less than 1 m/s) and a low Froude number (Kvočka et al. 2016). However, the prediction of flood hazard risk indices in the upper part of the valley started to differ more significantly between both methods when the gradient became steeper, i.e. when the gradient was greater than 1%. This again is not surprising, as the mechanics based method can be much more influenced by the forces induced by flow conditions, and where the dynamic force on a body is proportional to the square of the mean velocity.

For example, Figs. 4 and 5 show the predicted maximum velocities and Froude number values for the idealised river basin test case. It can be seen in Figs. 4 and 5 that both velocities and Froude number values in the upper part of the valley are higher with the increase in the bed slope, and with the Froude number being particularly high (i.e. near or above 1) when the slope was greater than 1%. On the other hand, the velocities and Froude number values were similar in the nearly horizontal lower part of the valley for all test cases. When these values are compared to the results presented in Fig. 3, it can be seen that the mechanics based and experimentally calibrated method is much more influenced by the higher velocities and momentum of the flow when compared to the empirically based method. This confirms that the mechanics based approach can better adapt to higher or rapidly changing velocities, and thus can more realistically predict the flood hazard risk indices for flood events characterised by high-velocity and high-Froude number flows, such as flash floods.

Even though this is a simple idealised test case, the results indicate that the mechanics based approach should be used when the river basin (or catchment) gradient is above 1%. This is particularly interesting when one considers how river basins or catchments are generally divided according to the value of the bed slope. In general, rivers are defined as basins having a bed slope that is less than 1%, torrential rivers are defined as basins having a bed slope ranging from 1 to 6%, and finally streams are referred to as torrents when the bed slope is greater than 6% (Ancey 2013). This means that the value of the bed slope, which indicates when it would be more accurate to use a mechanics based flood hazard risk assessment approach, is very similar to the threshold value of the bed slope that is used to distinguish between a river and torrential river.

Moreover, torrential catchments are typically small (i.e. less than 100 km² in size), characterised by a steep terrain and being susceptible to the occurrence of sudden, short and violent flood events (Ancey 2013). However, these characteristics are also characterising features of catchments prone to flash flooding (Marchi et al. 2010), with a mechanics based flood hazard risk assessment approach already being identified as the most appropriate method for predicting flood hazard indices in areas susceptible to the occurrence of flash floods (Kvočka et al. 2016). All in all, the aforementioned considerations and obtained results suggest that for a torrential or flashy river basin or catchment, the flood hazard indices should be predicted using the mechanics based and experimentally calibrated approach.

Figure 6 shows a comparison between the empirically based and the mechanics based flood hazard assessment methods in the wider area around Tal-y-bont. In Fig. 6, it can be seen that the mechanics based method generally assessed a higher degree of flood hazard risk when compared to the empirically based method. This is not surprising as the mechanics based approach was much more influenced by the flows conditions, i.e. particularly higher velocities and changes in the flow regime. The flow characteristics can be seen in Fig. 7, which shows the predicted maximum velocities and Froude numbers in the wider area of Tal-y-bont. It can be seen in Fig. 7 that both velocity and Froude number values were generally higher in this area, with the maximum velocities generally being well above 1 m/s and maximum Froude number values generally being near or above 1.

Based on the results presented in Figs. 6 and 7, it again appears that the mechanics based method is more accurate in representing the higher degree of risk associated with the high-velocity and high-Froude number flows when compared to the empirically based method. Consequently, the mechanics based approach generally predicts higher flood hazard indices for higher Froude number flood events, and therefore provides a more accurate flood hazard risk assessment in areas prone to the occurrence of flash floods. In addition, the average local bed slope in the wider area around Tal-y-bont is typically 2%, which means that the results obtained agree well with the previous results obtained for the idealised river basin test case. All this suggests that the mechanics based approach should be used for flood hazard risk assessment in areas characterised with a steep terrain (i.e. with a bed slope in excess of 1%), which are prone to the occurrence of sudden, short and extreme flood events (e.g. flash floods).

Figure 8 shows a comparison between the empirically based and the mechanics based flood hazard assessment methods in the wider region around Dol-y-bont. In Fig. 8, it can be seen that the mechanics based method generally predicted higher flood hazard indices in the wider region around Dol-y-bont. The terrain around Dol-y-bont is generally less steep than around Tal-y-bont, with the average local bed slope being around 0.6%. Nonetheless, the flow conditions in this area were still characterised with relatively high-velocity flows (i.e. above 1 m/s) and Froude number values reaching up to 1 (see Fig. 9). This indicates that the mechanics based method is more suitable for assessing flood hazard indices in torrential or flash flood response river basins and catchments. These areas are often characterised with a complex terrain and steep bed slopes (generally greater than 1%), which can significantly change in relatively small areas (e.g. from 2% around Tal-y-bont to around 0.6% in Dol-y-bont). This further suggests that in steeper torrential or flashy river basins and catchments the flood hazard indices should be predicted using flood hazard criteria based on the hydrodynamic interpretation of the processes, such as the mechanics based and experimentally calibrated method considered in this study.

Figure 10 shows a comparison between the empirically based and the mechanics based flood hazard assessment method in the wider region around Borth. In Fig. 10, it can be again seen that the mechanics based method generally assessed higher degrees of flood hazard risk in the areas characterised with higher velocities and Froude number values (see Fig. 11). Figure 10 also shows that in the areas where the velocities were smaller (i.e. 0.5 m/s or less), both methods predicted similar degrees of flood hazard risk indices. This is not surprising since the area around Borth is nearly horizontal and therefore the effect of the terrain on the flow characterises is relatively small, i.e. the rapid concentration and propagation of flood flow is minimised due to the flat ground. For example, similar results were observed for the idealised test case hazard risk when the upper gradient was below 1% (see Section 5.1).However, it can be also seen in Fig. 10 that the empirically based method predicted a higher degree of flood hazard index in areas where the velocities are smaller (see Fig. 11), but the floodwater is generally deeper (see Fig. 12).

This later finding is due to the different nature of both methods. The empirically based method considered in this study defines the flood hazard risk as being a function of the product of the depth and velocity. Consequently, the empirically based approach in this example defines a large depth/low velocity flow as hazardous (although this is not always the case, see Section 3.1). In contrast, the mechanics based method is based on the laws of fluid mechanics (or physics), and takes into account the characteristics of a human body (e.g. height and weight). Thus, the mechanics based approach defines the flood hazard index as being primarily related to a critical velocity of the flow that leads to the loss of a person’s stability in floodwaters. However, the critical velocity is calculated for a specific water depth, which means that a person will lose stability at a specific depth only if the velocity of the flood flow is higher than the critical velocity. Therefore, the mechanics based method will not automatically classify a large depth/low velocity flow as being hazardous, and particularly if the velocity of the flow is below the critical velocity.