Lahars, defined as gravity-driven flows containing a mixture of volcanic sediment and water (Vallance and Iverson, 2015), have caused severe damage to infrastructure and buildings (e.g. de Bélizal et al., 2013; Pierson et al., 2013; Ettinger et al., 2015; Jenkins et al., 2015) in addition to being responsible for a large proportion of volcanic fatalities (Auker et al., 2013). Assessing the extent of potential lahar damage can be difficult due to the complexity of flow behaviour, varying number of elements (e.g. buildings and bridges) exposed to lahars and a lack of knowledge on the structural capacity of these elements to withstand damage-causing components of the lahar flow. Using the common definitions of Varnes (1984), we define the damaging components of lahar flow (e.g. velocity, depth and pressure) as the hazard, environmental characteristics of exposed elements (e.g. building locations and orientations) as the exposure and the ability of exposed elements to withstand the hazard (e.g. building strength) as vulnerability. Lahar-induced damage is controlled by the interactions of these factors; however, the importance of each component can vary. Here we focus on quantifying and examining the role of hazard, exposure and vulnerability in determining lahar-induced building damage.

Post-event field assessments of building damage can elicit information relating lahar hazard to structural damage. However, these assessments tend to only record information on substantial damage, are affected by terrain changes during the event which alter exposure and often rely on a priori assumptions of building strength and vulnerability (Ettinger et al., 2015). Pre-event assessments are affected by the lack of reliable hazard intensity measures (van Westen et al., 2006; Ettinger et al., 2015), differences in spatial and temporal scales, uncertainty surrounding site-specific lahar triggers (Di Baldassarre and Montanari, 2009) and a lack of structural information on building stock (Ettinger et al., 2015). These issues are reflected in the relative lack of studies on hazard impact in urban areas (Jenkins et al., 2015) and often result in a reliance on expert judgement to develop vulnerability models for lahars and flash floods (Ettinger et al., 2015).

The physical vulnerability of buildings, defined as the susceptibly of a building to damage with respect to the hazard (Künzler et al., 2012), is a function of building characteristics such as size, shape, age, construction materials, structural integrity, maintenance and build quality (Martelli, 2011; Künzler et al., 2012; Ettinger et al., 2015). Information on these building properties is often lacking and hard to collect on a large scale. This commonly leads to the simplification of vulnerability into a measure that can provide a relative indication of vulnerability and consequent damage (Künzler et al., 2012). Studies simplifying vulnerability as a relative index use a combination of qualitative and quantitative metrics obtained through building surveys, interpretation of remote sensing data and GIS techniques to map and analyse vulnerability on a large scale (e.g. Lavigne, 1999; Künzler et al., 2012; Galderisi et al., 2013; Thouret et al., 2013, 2014; Ettinger et al., 2015). These methods can be applied to understand and highlight spatial patterns in vulnerability; however, as a relative measure, they cannot provide guidance on absolute damage for any specific event.

A direct estimation of damage caused by specific events requires quantified relationships describing a building's response to the hazard. Buildings can be damaged through a number of mechanisms including (i) direct damage resulting from static and dynamic forces imposed by the flow, (ii) damage to foundations through erosion and scour, (iii) buoyancy effects of the flow causing structures to float, (iv) direct damage from larger debris (missiles) within the flow and (v) indirect damage caused by chemical and biological actions such as seeping induced weakness of mortar (Kelman and Spence, 2004). All these actions, apart from chemical and biological effects, are related directly to lahar depth, velocity or a combination of depth and velocity. Thus, a common approach in determining building damage thresholds for a particular building type is to relate damage to hazard intensity measures of depth and/or velocity (e.g. Zanchetta et al., 2004; Custer and Nishijima, 2015; Jenkins et al., 2015). However, building typologies are affected by socio-economic, cultural and institutional conditions (Künzler et al., 2012), and hazard intensities (flow depth and velocity) are affected by building environmental factors such as local elevation, distance from main channels and orientation (Thouret et al., 2014). This results in complex interactions between hazard, exposure and vulnerability. These issues cause direct vulnerability relationships to be site-specific and require detailed investigation of the regions at risk to examine the relative effects and role of hazard, exposure and vulnerability on building loss.

We attempt to quantify and examine the components that determine building damage in a small area within the city of Arequipa, Peru. A relative index of vulnerability on a city-block scale was developed for Arequipa in Thouret et al. (2013, 2014). The studies by Thouret et al. (2013, 2014) highlighted two groups of vulnerability indicators (building characteristics and the physical setting) that play an important role in determining vulnerability within Arequipa. In this study, we separate the building characteristics (i.e. vulnerability) from the physical setting (i.e. exposure) to examine the effect of hazard (flow rate and rheology), exposure (building orientation and location) and vulnerability (building type) components on building loss within Arequipa. Physical vulnerability of buildings is explicitly separated from exposure through the development of a building damage model that is dependent on flow velocity, depth and sediment concentration. Simulations of lahar flow using smoothed particle hydrodynamics are used to examine how flow characteristics and the physical setting of city blocks affects forces on buildings and the consequent damage. While, for reasons explained earlier, damage functions presented here are necessarily specific to Arequipa, the hazard modelling approach and vulnerability model development are described in detail to support risk assessment in other regions affected by lahars.

The central business district of Arequipa, the second largest city in Peru, is situated 17 km south-west of the summit of El Misti (Fig. 1), a steep stratovolcano with a history of explosive eruptions. Rapid population growth since 1960 has resulted in an expansion of the city towards the ring plain and steep slopes of El Misti (Thouret et al., 2013). Arequipa is drained by several ravines (locally called quebradas or torrenteras), shown in Fig. 1, that have been shaped by lahars and floods originating from the volcano on volcanoclastic fans north-east of the city. These quebradas are normally dry but carry water sporadically during the December to March rainy season (Vargas Franco et al., 2010; Martelli, 2011; Thouret et al., 2013; Sandri et al., 2014). Flash floods and hyperconcentrated flows occur relatively frequently in the quebradas, with return periods between 2 and 10 years (Vargas Franco et al., 2010; Thouret et al., 2013). Previous studies of lahar hazard and vulnerability for Arequipa identified seven alluvial terraces (T0, T1, T1′, T2, T2′, T3 and T4) based on stratigraphy and local elevation above the quebrada and the Rio Chili valley (Martelli, 2011; Thouret et al., 2013, 2014). The likelihood of inundation by a lahar or flash flood decreases with each terrace. Terrace levels T0 and T1 (up to 3 m above the quebrada) are frequently flooded (approx. every 2 to 10 years). The higher terraces (T1′ to T2′, 3 to 10 m above the quebrada) are rarely flooded (estimated 20 to 100 years) and the highest terraces (T3 and T4) are only likely to be inundated by lahars linked to large eruptions (Thouret et al., 2013, 2014). A city-wide vulnerability study by Thouret et al. (2014) identified that the city blocks most vulnerable to flash floods and lahars were on the lower terraces and typically within 100 m of a quebrada.

To build on this study and investigate the vulnerability of the quebrada channel and banks in detail, simultaneous photogrammetry and building surveys were undertaken along short sections (approximately 200 m) of several quebradas during September 2013. Here we focus on one 150 m-long section of Quebrada Dahlia to examine lahar hazard and building damage. Quebrada Dahlia is a small tributary of Quebrada Mariano Melgar-Huarangal (Fig. 1), which is situated in the Mariano Melgar District on the north-easternmost fan of Arequipa, shown in detail in Fig. 2. The case study area was chosen for the following reasons:

The quebrada channel is reasonably straight, reducing the effect of bends in the watercourse on lahar dynamics.

A three-dimensional reconstruction of the terrain and buildings along Quebrada Dahlia was created using the photogrammetry method described in Mead et al. (2015). The surface reconstruction, shown in Fig. 2b, contained 1.4 million points with a surface density of between 150 and 750 points per m2. A GNSS-D survey undertaken in October 2014 enabled georeferencing of the terrain reconstruction for possible inclusion in future GIS applications. The surface reconstruction was smoothed and reduced to create the lower-resolution terrain model shown in Fig. 2c to be used in the lahar simulations. The terrain model in Fig. 2c contains 22 buildings identified during building surveys in 2013. Streets and the quebrada (shown in Fig. 2a) separate these buildings into five city blocks (labelled in Fig. 2c), referred to hereafter as “blocks”. The typology of each building was characterised through surveys undertaken in 2013 following the approach of Thouret et al. (2014). In this approach, buildings are classified as one of eight structural types (1A–8C) based on a visual inspection to determine construction material, roof type and structural support (see Table 1). These types were then grouped into larger simplified structural classes. Using this building classification system, the study area contains eight class A0 buildings, seven class A buildings and seven class B buildings (see Table 2 for a description of building types and structural classes).

Buildings and infrastructure can be damaged through a variety of mechanisms brought upon by the actions of a lahar. Here, as in most other studies of lahar damage (Zanchetta et al., 2004; Toyos et al., 2008; Ettinger et al., 2015; Jenkins et al., 2015), we focus on the direct damage resulting from hydrostatic and hydrodynamic forces applied to buildings. We regard these actions as the most important, although scour and large debris missiles within the flow can also cause significant damage (Jenkins et al., 2015). Scour and debris actions are neglected here as they are currently too difficult to predict and incorporate into large-scale loss analyses (Kelman and Spence, 2004), particularly in regions with limited hazard and exposure information.

The building stock within Arequipa is characterised mostly by masonry structures of varying quality, with some reinforced concrete structures (Thouret et al., 2014). Therefore, we develop vulnerability relationships that are primarily focused on masonry buildings. A structural failure model similar to those employed by Roos (2003), Custer and Nishijima (2015) and Zeng et al. (2015) is implemented. In these models, masonry walls are presumed to fail when the lateral pressure imposed on the wall results in a bending moment or shear force greater than the wall's calculated ultimate bending moment or ultimate shear force.

The ultimate bending moment (Mu) is calculated using the following equation (Roos, 2003): Mu=(ft+fd)wb26, where ft is the tensile strength of the masonry wall, fd is the design compressive stress acting on the wall, w is the width of the wall facing the flow and b is the thickness of the wall, which is assumed equal to the brick width. Tensile strength and design compressive stresses for buildings in Arequipa are calculated using the approach specified in Australian Standard (AS) 3700–2011 and summarised in Appendix A in addition to an approach calculating ultimate shear force. Preliminary investigations using these two approaches suggested that the force required to overcome the ultimate moment was consistently lower than the force required to overcome the ultimate shear force. Therefore, we chose to focus the remainder of this study on the ultimate bending moment only.

The use of a foreign standard to calculate the ultimate moment should still be valid for the study area if construction material properties from Arequipa are used as inputs. However, some specifications and assumptions of the standard may not be relevant. Notably, observations during the building survey suggest that construction methods and conformity to specifications within the standard differs substantially to those specified in AS 3700–2011. This difference will influence ultimate bending moments, particularly those for low-quality unreinforced building types (i.e. building types 1A–2B) due to the makeshift nature of construction. For these classes, calculated bending moments will represent a best case scenario in which masonry unit strength and quality have not been compromised by construction methods.

The range of design compressive stress for each building typology is shown in Fig. 3. The range was obtained by calculating the design compressive stress for every configuration of masonry compressive strength (fc), wall thickness (b) and thickness coefficient (kt) in Appendix A. Buildings with reinforced frames (types 3, 4 and 6) can withstand much greater compressive stresses than non-reinforced buildings (types 1, 2, 5). The wall thickness has a large effect on building strength, which is consistent with observations by Jenkins et al. (2015). Notably, the design compressive stresses are similar for building types that share the same simplified structural class identified in Thouret et al. (2014), based on the structural classes of Zuccaro et al. (2008). Given these similarities, we also use the simplified structural classes (A0, types 1A–2B; A, types 3 and 5; and B, types 4 and 6A–6C, see Table 2) from Thouret et al. (2014).

The critical depth (i.e. hydrostatic pressure) and dynamic pressure required to overcome the ultimate bending moment (Eq. 1) for each structural class is shown in Fig. 4. These curves assume that both hydrostatic and dynamic pressure act on walls. Other studies (e.g. Jenkins et al., 2015) assume only dynamic pressure acts on walls due to an equalisation of lahar depths on the inside and outside of buildings. This equalisation can take a reasonable amount of time, which is likely to be much longer than the simulation duration studied here (see following sections). The curves in Fig. 4 indicate the structural limit of each class; combinations of depth and pressure that fall above the curves indicate an applied moment greater than the building can withstand. Conversely, combinations of depth and pressure that fall below the curves indicate an applied moment less than the maximum the building can withstand. Figure 4 shows that the critical depth decreases with the density of flows as the hydrostatic pressure gradient is much larger for sediment-rich lahars. The critical depths and pressures are also affected by the structural class, with A0 structures being much less resilient than A and B structures. However, wall thickness has the most dominant effect on determining the strength of buildings. Wider walls increase the section modulus (wb2/6 in Eq. 1), resulting in stiffer walls that also have a higher compressive stress capacity.

Depth at the maximum value of pressure acting on block walls for each scenario is used to determine whether individual buildings in the study area can withstand the bending moment applied by hydrostatic and dynamic pressure. Figures 9 to 11 plot the peak pressure and “surge depth” (depth at the time of peak pressure) for Newtonian, hyperconcentrated and debris flows alongside critical depth–pressure curves for vulnerability classes A0, A and B with a wall thickness of 150 mm (results for 250 mm wall thicknesses are provided in the Supplement). The hazard variables of flow rate and lahar rheology appear to influence building damage, although the size of the effect is difficult to determine since most scenarios place depth and pressure combinations well above the critical curves for each block. The flow depth, which affects hydrostatic pressure and bending moment location, generally increases with the flow rate while the dynamic pressure appears to be mostly controlled by the rheology in combination with flow rate. The forces applied to the West 2 block, containing one class A and 4 class B buildings, are lower than the other blocks. This is possibly due to the relative elevation and orientation of each block to the quebrada (i.e. exposure) affecting dynamic pressure and lahar depth. Debris flow scenarios at flow rates of 25, 50 and 75 m3 s-1 indicate depths and pressures below the critical limit for this block's building classes.

The orientation of walls to the flow direction is another element of exposure that affects the normal pressure exerted on walls. In several scenarios, perpendicular walls are subjected to higher dynamic pressures and lower depths than parallel walls. However, this effect appears to be conditional to the rheology of the flow as the opposite is true for debris flow scenarios. These two effects demonstrate the importance of considering exposure elements separately to vulnerability.

The proportion of buildings with depths and pressures above the critical curve for each scenario is shown in Fig. 12 for 150 mm brick widths and Fig. 13 for 250 mm wall thicknesses. Assuming a binary damage state model where damage is complete for depths–pressure combinations above the curve, these proportions can be used to directly represent building loss. For the thinner walls, all class A0 buildings are above the curve for all scenarios apart from the 25 m3 s-1 debris flow. The East 1 block is not inundated in this scenario, resulting in two undamaged class A0 buildings. Class A and B buildings are also mostly destroyed, with the exception of lower flow rate hyperconcentrated and debris flow scenarios where some blocks are on the edge of inundation and therefore subjected to much lower depth–pressure combinations. Slightly fewer building losses occur with larger brick widths (Fig. 13) as the larger section modulus results in a greater resistance to bending moments. However, most buildings are still destroyed in Newtonian and hyperconcentrated flow scenarios. An exception to this is the 75 m3 s-1 Newtonian flow where the highest pressure on the East 1 block occurs early in the simulation when the surge depth is low, reducing the magnitude of hydrostatic pressure and lowering the size of the applied moment.

The building loss results indicate that class A0 buildings are most vulnerable, with class A buildings marginally stronger due to the roof support. Losses for type B buildings in this area are much lower; however, this appears to be more closely related to building exposure than structural strength as most type B buildings are in two blocks subjected to lower depth–pressure combinations for all scenarios. Overall, similarly to the observations in Jenkins et al. (2015), the data presented here suggest that building strength (i.e. the vulnerability component) has a minimal effect on losses, and building location (i.e. exposure) relative to flow rate and type (i.e. hazard) plays a much greater role.

The losses shown in Figs. 12 and 13 are estimates based on several assumptions that, while necessary for the estimation of building loss, could limit the accuracy of results. Firstly, the depth–pressure curves are created using ultimate bending moments derived from a foreign standard and do not consider proportional losses, only assuming damage is complete for depths and pressures above the critical curve. Secondly, the flow scenarios modelled here are a subset of likely scenarios and do not replicate all damage-causing actions of lahar flow. Finally, the maximum total pressure was assumed to be the sum of hydrostatic and dynamic pressure and to occur when dynamic pressure was at its peak. These limitations are discussed and justified in this section to highlight areas of improvement necessary for robust, quantitative estimation of lahar damage and vulnerability.

Development of fragility functions in the form of critical depth–pressure curves for building classes within Arequipa has helped to provide insight into possible building losses and their cause. Building vulnerability is largely controlled by social, cultural and institutional factors (Künzler et al., 2012), so the depth–pressure curves are necessarily specific to Arequipa building typologies. However, given sufficient data on building strength, depth–pressure curves can be generated through the same approach as in Appendix A and used to quantify masonry building loss in terms of flow depth and pressure in other regions.

Estimated building losses (Figs. 12 and 13) are caused by the intersection of lahar hazard (flow rate, flow type) with building exposure (location, proportion of building types and orientation within blocks) and vulnerability (building type and strength). The almost total simulated building loss for all scenarios indicates that substantial losses can be expected in the event of inundation. Furthermore, lahar depths and pressures obtained from simulations were much greater than most of the buildings in the study area could withstand, even if retrofitting to improve structural strength was undertaken. This suggests that, in this study area at least, exposure and lahar hazard have a larger role in determining building loss than vulnerability.