Joe Miemczyk
email: jmiemczyk@audencia.com

Andrew Graves

Managing end-of-life vehicle networks: a longitudinal case of the UK

Keywords: End of life vehicles, strategy, case study, United Kingdom

This research outlines the development of end of life vehicle network management in the UK from the early 1990s to mid-2006. The paper critically appraises the developments over a decade and a half from the development of a voluntary scheme to a legislated regime for recovering end of life vehicles with the goal of reducing landfill waste. The research assesses the management implications (economic implications and industrial responses) of a voluntary versus legislated approach, with regard to legitimacy and competitiveness drivers within vehicle manufacturers and service providers (vehicle dismantlers).

Introduction

This research outlines the development of end of life vehicle network management in the UK from the early 1990s to mid-2006. The paper critically appraises the developments over a decade and a half from the development of a voluntary scheme to a legislated regime for recovering end of life vehicles with the goal of reducing landfill waste. The research assesses the management implications (economic implications and industrial responses) of a voluntary versus legislated approach, with regard to legitimacy and competitiveness drivers within vehicle manufacturers and service providers (vehicle dismantlers).

At the present time the UK industry is in the throes of conforming to the European End of Life Vehicle Directive, one of a raft of producer responsibility directives designed to ensure that the environmental impacts of products are minimised by making their manufacturers responsible for at least some of the costs of recovery, treatment and recycling. Vehicle manufacturers have established contracts with recyclers that conform to the directives stringent requirements on environmental protection, and at the same time government at various levels has put in place enforcement regimes aimed at ensuring the directives can be met. This has meant serious investments in capital and other resources by all the parties involved, as well as a general re-structuring of the vehicle recycling industry overall.

Yet there is a significant history to end of life vehicle management in the UK. Firms in the car producing and recycling (dismantling, shredding) industries are not coming to this issue without an extensive amount of experience, quite the opposite. Furthermore, firms can be a hostage to their own history, and institutional theory tells us that practices and routines of firms can commonly be seen as built up through time as habitual actions more than rationalised decisions based on fact and logic. To understand these interactions it is important to look at strategic changes over time. While, actions such as end of life vehicle management are often viewed at a network level, it is the collective behaviour of individual organisations that determine whether legislated targets set at EU level can be met. Hence this study takes the perspective of individual firm strategy within a network context. To aid the understanding of how firm strategy has developed, the study also takes a longitudinal approach in an attempt to elucidate how firms can affect and be affected by changes in the institutional environment.

Section 1 has given a general introduction to the problem area. The paper continues with section 2 covering some of the key literature in the field of end of life vehicle management, as well as some of the recent management thinking that informs the discussion elements of the paper such as network management, capabilities and legitimacy (through formal regulatory pressures). An outline of the methodological approach is given in section 3. The principal findings are given in section 4 which is divided into 5 sections covering 4.1) the original voluntary approach 4.2) legislated actions, 4.3) structural network changes 4.4) the development of network relationships 4.5) capability development and 4.6) attitudes to changes in the network and process. The paper closes with conclusions and indications for further research.

Background literature

Much of the management literature focussed on the end of life vehicle issue originates from the late 1990s when the ELV Directive (CEC 2000) was first likely to become enshrined in EU law. A number of authors examined the management aspects of reforming ELV organisation [START_REF] Hond | Solving the shredder waste problem: Cooperation among firms in the automotive industry[END_REF], Den Hond 1998). In particular these authors have examined how capabilities for vehicles recycling have been developed by sharing knowledge between vehicle producers, dismantlers and other experts in the field, often in collaborative arrangements. While this research has focussed on the strategic implications, for example adopting a resource based perspective, the dynamic nature of capabilities and the changing legislative landscape call for longitudinal viewpoints.

Some authors examined the issues from an engineering perspective in an attempt to provide an micro-economic analysis of different recovery strategies. This is useful in terms of developing models that could be applied to different contexts accounting for labour cost and technology differentials [START_REF] Johnson | Evaluation policies and automotive recovery options according to the European Union Directive on end-of-life vehicles (ELV)[END_REF]. Some researchers have used less commonly applied theoretical frameworks such as political ecology in order to provide further understanding of the changes occurring in the industry [START_REF] Orsato | The political ecology of automobile recycling in Europe[END_REF]. Importantly, the institutional fields having considerable influence on firm strategies have also been considered through various econo-political viewpoints, for example discussing the role of incentives for innovations in the end of life vehicle field especially were these may be mis-aligned or give undesirable results [START_REF] Mazzanti | Economic Instruments and Induced Innovation: The Case of End-of-Life Vehicles European Policies[END_REF]). Yet literature does not consider the business strategy dynamics of evolving network structures and processes, particularly at the network level. This paper examines the development over time of both the structural elements of vehicle recovery from a network and process perspective, as well as the capability development implications of network members, and potentially the network as a whole. The management literature is replete with research that has covered various aspects of these important domains and the following part of this literature review provides an extremely brief outline of current thinking.

With regard to the network perspective, management researchers have drawn on a wide range of disciplines to enhance understanding. The network view can be stated as a conceptual or analytical tool which provides a complete view of the social environment encompassing relationships, a structure for a set of persons or structural description of a network based on the characteristics of relationships [START_REF] Shulman | Network Analysis: A new addition to an old bag of tricks[END_REF]. General types of networks include the following [START_REF] Lamming | An initial classification of supply networks[END_REF]: social networks such as industrial districts; Bureaucratic networks such as trade associations and consortia with exchange or associational contractual agreements and joint and/or capital ventures with equity and property rights. In particular, sociological approaches to 'network' research has had a particularly influential role. Early researchers have often used concepts such as network centralitystructurally this means the number of direct ties to others in the network and/or independent access and/or control over other actors -to explain how organisations (or individuals) affect or are affected by networks, for example in the ability to acquire knowledge. From the perspective of relationships and firm performance, it is proposed that competitive advantage can be gained by strategically managing the network [START_REF] Harland | Supply chain management: relationships, chains and networks[END_REF]. However the Scandinavian school of supply, in taking a resource dependency perspective, state that networks cannot be managed, only coped with [START_REF] Hakansson | Developing relationships in business networks[END_REF]. Recent research into networks has revealed that firms may be in an enhanced competitive position because their network structures allow them to exploit their internal capabilities more effectively, whereby the innovativeness of network partners indirectly supports the focal firm's performance [START_REF] Zaheer | Benefiting from Network Position: Firm Capabilities, Structural Holes, and Performance[END_REF]. McEvily and Markus (2005) also find that the acquisition of capabilities is facilitated through network embedded ties and alliances and argue that while both trust and information sharing are important, joint problem solving is also key to transferring knowledge. This discussion of the role of networks and the network perspective has quickly turned to how networks can be a source of advantage through capability acquisition or development. Hence it is clear that a brief discussion of recent thinking on capabilities is also required.

Recent definitions of capability include "a high level routine (or collection of routines) that, together with its implementing input flows, confers upon an organization's management a set of decision options for producing significant outputs of a particular type" (Winter 2003: 991) or "a capacity to integrate, combine, and deploy tangible and intangible resources through distinctive organisational processes in order to achieve desirable objectives" (Lavie 2006a: 153). The discussion of capabilities provides an explanation for how firms achieve competitive advantage through the deployment of idiosyncratic resources that competitors do not possess and are not able to imitate. The role of collaboration in providing resources and hence benefits to firms has been a recent development in the capability perspective (For example [START_REF] Das | A Resource-Based Theory of Strategic Alliances[END_REF][START_REF] Dyer | The relational view: cooperative strategy and sources of interorganizational competitive advantage[END_REF]Lavie 2006b).

Although often applied to strategic alliances this previous work is useful in explaining how the alignment of capabilities through similarity and utilisation, and importantly, has identified that the expectations for certain capabilities does not always lead to the expected 'positive' outcomes in terms of performance such as cost reduction, efficiency and so forth.

Method.

A detailed case analysis of the attempt development of vehicle recycling networks in the UK forms the empirical core of the paper. Case studies are particularly useful when exploring new areas of research [START_REF] Voss | Case Research in Operations Management[END_REF][START_REF] Stuart | Effective case research in operations management: a process perspective[END_REF][START_REF] Eisenhardt | Building theories from case study research[END_REF] and equally, the rich qualitative and quantitative data sets generated (Yin, 1994) are particularly important because the measurement of intangible phenomena e.g. roles and relationships [START_REF] Denzin | Handbook of Qualitative Research[END_REF] was a central concern. Studies of organizations or sectors remain popular in management research because they offer the opportunity for deep, longitudinal analysis, exploring the impact of organizational change and often involve the development of conceptual frameworks or interpretative schemes (Mueller et al., 2003).

Primary data was collected using semi-structured interview questions, each interview lasted between 2 and 3 hours, investigating the sub-elements defined by the conceptual model (i.e. operational, processual and contextual factors) and the respondent-driven Critical Incident Technique (CIT) [START_REF] Flanagan | The critical incident technique[END_REF]Bitner et al., 1990;[START_REF] Edvardsson | Service breakdowns, a study of critical incidents in an airline[END_REF]. The former aims at a global case description and understanding of the behaviour of firms over time, whereas the latter is used to map micro-level incidents as individual respondents experience them. For this purpose we defined an event to be 'a retrospective organisation of a set of inter-related incidents into a comprehensive narrative' and for an event to be classified as 'critical' meant that it 'was perceived to have had a positive or negative outcome for a person or the organisation'. A chronology of activities by the companies leading up to and after the implementation of the ELV Directive in the UK was created as a guide to familiarise the investigators with contemporary events in the research context.

The research findings are based on primary interviews with OEMs, dismantlers and policymakers (including Nissan, Honda, PSA, Ford, Volvo, GM Europe, Jaguar, SMMT and the DTI ). The method utilises a case approach that also encompasses site visits and reviews of company documentation and other secondary data. In addition, mapping workshops with dismantlers and vehicle manufacturers were carried out to gain a structural view of the changing nature of the relationships.

Managing end of life vehicle networks (2000-2006)

Between 2000 and 2006 there have been a number of critical events and developments which have shaped today's recovery networks. These events include the development of industry groups, a voluntary agreement, the dissolution of the industry group, the staged and delayed implementation of EU legislation and many meetings between industry bodies, firms and legislators in the UK and EU.

The original voluntary approach

The main action in the UK related to the voluntary approach is the ACORD agreement set up by the Society of Moter Manufactures and Traders on behalf of UK vehicle manufacturers. This agreement put in place measures of vehicle recycling in order to track the level of vehicle recycling in the UK. The main purpose of this agreement was to provide a possible alternative to legislation that was being mooted at European Union levels. If the ACORD agreement could demonstrate to legislators that regulation was not necessary to achieve high levels of recycling (eg 90%), then there would be the possibility that industry could avoid additional regulatory burdens on top of those that are already in place, such as the packaging regulations, IPPC, passive safety requirements and various other laws which have increased the society pressures on the automotive sector (in additional to those that already affect the product in terms of safety, fuel efficiency and exhaust emission). In addition to the ACORD agreement in the UK, the SMMT also facilitated the set up of the CARE group. The CARE group is a membership based organisation established to provide a forum for knowledge sharing on the subject of ELV as well as establishing standards for vehicle recovery. The group holds regular meeting at dismantler sites to discuss current developments in the field and transfer this knowledge across other members of the group, representing both vehicle dismantlers and vehicle manufactures and other associated members. Through the membership funds and coordination activities the group is able to start research projects in certain areas to assess the feasibility of specific technologies or processes that could increase the level of vehicle recycling. For example the group recently completed a feasibility study of recycling plastic bumpers into reusable plastic granules, focussing on how to obtain sufficient volumes from service repair workshops and potentially from dismantlers end of life vehicles.

The CARE group itself ran into some difficulties during the implementation of the ELV regulations in the UK.

Due to the extreme focus on the regulation and the efforts required, efforts in CARE stalled in terms of developing projects and so on. In fact there was a stage in 2002 that many thought that CARE would have to be disbanded as the regulation created a lack of consensus over the future of ELV management in the UK. Since the resolution of the ELV regulation in the UK, the CARE group has a re-asserted itself as a useful mechanism for sharing knowledge in the area, specifically focussing on how to achieve the more challenging aspects of the recycling targets such as the so called 'last 9%' and reducing costs of compliance for all parties. Thus fears of not continuing this apparently important vehicle for knowledge sharing have not been realised to date, and the group continues to hold meetings at dismantler sites and initiate research or feasibility studies.

Legislated action

The UK vehicle producers quickly organised themselves under the umbrella of the SMMT to ensure that a coordinated effort was made to ensure that the transferral of the ELV directive into UK regulation as made with business interests in mind. In fact it was always the publicly stated intention of the UK government to implement the EU ELV directive with a light touch. As the then secretary of state reported "We have consistently made clear our intention to implement the end of life vehicles directive with a light regulatory touch and without disrupting the existing legitimate dismantling, shedding and recycling market".

In developing the UK response to the ELV directive the DTI and Defra coordinated a number of consultations on the implication of the regulation on UK businesses.

Among other EU states the UK was late implementing the ELV directive. A number of bodies threatened to bring legal action against HM Government. This also created considerable uncertainty for the companies actually implicated in the regulation. On the one hand this gave ample time for the industry groups to provide responses to the various consultations, yet on the other hand left less time for companies to actually meet the various implementation deadlines set at EU level.

Network structure

One of the most striking differences between the structure of the recovery networks in 2000 compared with 2006 is the reduction in treatment sites. With the new regulatory requirements on recovery standards, many sites are not able to justify the necessary investments in order to become authorised treatment facilities (ATFs) in the eyes of legislators, notably the Environment Agency (see Table 1). While the current number of sites is roughly half that of six years before, it can be stated that this overall level will continue to increase as dismantlers continue to invest to upgrade their facilities and the Environment Agency, which is severely resource constrained, is able to provide authorisation to existing sites that have made the necessary investments.

Dismantlers 2000 Authorised dismantlers 2006

3375 (2426) 1189 There are a number of implications of this reduction in the overall number of sites available for recovering ELVs. These include, among others, limitations in market choice and oligopolistic practice. In particular the reduction in market choice could theoretically decrease the competition between dismantlers, and therefore notionally raise prices for the 'recovery' service, especially as the geographical spread of dismantlers is partly influenced by the regulation, in that vehicle producers must show evidence that their network meets certain geographical constraints. This could mean that in certain areas where there are few dismantlers which are authorised, the supply of service is restricted, increasing prices for depollution. In reality, the vehicle manufacturers contract through one or two recovery 'brokers' who effectively manage the network, and who can at least have visibility of the 'market' prices for recovery services, and work to ensure limited supply is minimised as much as possible.

The network diagram summarises the linkages between the main players in the ELV recovery network today (figure 1). The main difference from the original voluntary scheme that was in place with ACORD is there are now a number of intermediaries who act as brokers between the OEMs and the actual network of dismantlers across the UK. In order to avoid the situation where OEMs have to manage contracts with multiple dismantlers (in order to ensure that there is a nominated site sufficiently close to last users) the broker manages these relationships on behalf of the OEM. This type of development is common in the automotive sector (where 1 st tier suppliers take on the responsibility of managing many traditional suppliers).

Network relationships

As can be seen from the table (Table 4) below, and figure of the relationship maps, two key end of life vehicle brokers have emerged to provide a compliance solution to the main vehicle producers that sell vehicles in the UK. These two recovery 'brokers' act as a bridge between the vehicle manufacturers and the dismantling network, providing evidence on behalf of the vehicle producers that they have a compliant network in place (for example to the Department of Trade and Industry). Both Autogreen and Cartakeback will ensure that end users of any particular marquee of vehicle will be able to take their end of life vehicle to a 'close by' dismantler at no cost to the end user. The recovery 'broker' will also manage the flow of data on recycling levels across the complete network of dismantlers and report this back to vehicle manufacturers to provide the basis for how the network is contributing to the overall UK recovery targets.

Table 4: Current contracts with the main UK recovery service providers (various public sources)

To an extent OEMs have little strategic interest in integrating product recovery capabilities unless there is a direct effect on their competitive strategy (gaining new markets for example with remanufactured products), and thus contracting to meet regulated norms is the preferred approach. OEMs of this study are intent on meeting legitimacy goals through a least cost approach and are thereby focussed on accessing and developing capabilities that support that aim. Hence, OEMs require collaborative relationships to ensure that service providers can help them meet that aim (through sharing of information and knowledge about product designs and distribution networks), and it is the combination of capabilities that allows the meeting of these goals.

In particular the case of GME and Autogreen showed evidence that the relationship could be developed further to include new area of collaboration, for example in developing a 'green' line of used parts, branded under GM. These were just ideas, but show the potential for existing relationships to be developed further in support of product stewardship. Thus, although a standard set of contracts have been issued between dismantlers and vehicle manufacturers, there is certainly scope to expand the depth of these relationships in order to perhaps capitalise on further synergies (technologies, skills, knowledge) to provide value not just in terms of compliance but also differential costs compared to competing brands and also potentially new revenue streams for both dismantlers and vehicle producers. Hence, the networks could be viewed as providing a basis for building new market opportunities, that could theoretically at least support the targets for recovery (through re-use of parts and components).

Capabilities

One interesting debate occurring between network members in the argument over the capability for recovering through shredding versus capability for recovery through dismantling. The research highlights an interesting example for how different capabilities are potentially in conflict, specifically in the comparison of the shredder or dismantler approach. A recent market experiment was set up between members of the CARE group and a plastic recycler to assess whether significant levels of plastic from car bumpers could be recovered at a low cost. One of the significant results from this study was that while high levels of plastic material could be recycled into re-usable plastic granulate the cost advantage of doing so was nearly equalised by the reduction in revenues from the hulks sold to shredders (which were less heavy).

Specific skills for recovery are still required by many members of the network in the UK. In particular handling items such as airbags and air-conditioning units is complex and fraught with safety and compliance difficulties. In order to cope with this deficiencies in the skills of network companies, the 'knowledge-sharing' groups such as CARE activity provide access to training and current knowledge on this types of issues. Yet at the same time all network members have equal access to these skill acquisition routines and as such form a generic level of capability development applied to the whole sector, separated from areas of commercial sensitivity or competitive advantage. Thus the network structure in place tend to allow for the development of 'industry' or 'sector'-level capability development, providing something akin to a level playing field for compliance. Clearly it is not in the interests of government agencies to allow differentiation based on compliance (non, low levels or high levels of compliance), and certainly firms are not incentivised to go beyond the level of compliance, within the previous or current frameworks of vehicle recovery.

Building on the substantial literature on social capital (Nahapiet and Ghoshal 1998), the ability to network to find expertise by OEMs appears to be a function of their network ties (within trade associations, industry working groups as well as their collaborative relationships with service providers) and allows the identification of suitable partners in the product recovery supply chain. This research also finds that the service providers ability to provide revenue to reduce compliance costs is key to the economics of product recovery (whether through re-use, remanufacturing or recycling) and is built up over time, again through supply chain network ties, but in this case of the service providers. These capabilities tend to be pre-existing within the firms and are accessed by the OEMs planning to establish a product recovery process. The research highlights a couple of instances where capabilities (not identified in other research) are developed through the collaborative relationships and these specifically relate to re-establishing the customer link and building legitimacy. The issue of legitimacy, and perhaps the effect on reputation is difficult to link to competitive advantage. Whether end users are more likely to used dismantlers with higher levels of 'legitimacy' remains to be seen, but seems an unlikely differentiator.

The activities of the OEMs to access and develop new capabilities in response to market (or business environment) changes, could be viewed in terms of dynamic capability [START_REF] Winter | Understanding Dynamic Capabilities[END_REF], a meta-capability that has been defined as "processes to integrate, reconfigure, gain and release resources-to match and even create market change" [START_REF] Eisenhardt | Building theories from case study research[END_REF]Martin 2000: 1107). This type of capability, however, is contingent on firms being able to exercise some degrees of freedom in their response. As shown in the case companies of this research, firms can be constrained in their response to varying degrees. Although collaborative relationships (and their role in providing social capital for example) may form a facilitator in integrating, reconfiguring and gaining resources, and thus capabilities (Blyler and Coff 2003), the constraints of pre-determined pathways and mandated procedures for product recovery limit the usefulness of this explanation. Given that collaborative relationships do, on occasion, lead to the access to and development of capabilities for product recovery, for these capabilities to be of competitive value they must improve performance in some way. The question then arises who actually gains from this improvement. Is it the dismantlers, in terms of lower dismantling costs or is manufacturer who will need to pay less for vehicles with a negative value.

This narrow focus on firm advantage has been challenged on a number of quarters. As Mathews ( 2003) argues benefits also can be at an industry level (drawing on the evolutionary view of the firm), so that shared capability development, or least the development of capabilities that can be of equal access to other industry players is not necessarily in opposition to competitive positioning at an industry level. Thus taking this 'extended resource-based view' implies that capabilities that are not idiosyncratic and can easily be transferred to other firms in an industry could actually benefit the competitiveness of an industry overall. Managers that view product recovery as 'noncompetitive' may support this concept of industry level competitiveness at least at a regional level (perhaps where regions compete against regions -Europe and the Far East as examples).

Attitudes towards changes in ELV recovery

For the main part, this discussion has centred round a structural view of the changes in relationships between actors across industries. An examination of attitudes of industry players provides a deeper understanding of the strategic implication of the changes in the way ELVs are handled in the UK. Overall attitudes towards the changes to network structure over the period indicate a major re-organisation of the competitive environment for ELVs in the UK. "It's a shrinking market". The OEMs primary objective with regard to ELV legislation is to be legal, e.g., "the number one objective is to be legal"(OEM manager), which reinforces the view that this cannot be viewed as a strategic priority but an operational necessity. Alongside this objective is the need to reduce the costs and liabilities of the regulated norms, e.g "..everybody's looked at it and it is incredibly difficult to make this work financially, by working on your own," again highlighting the need to collaborate with 'competitors to gain economies of scale. Furthermore ELV is not seen a competitive priority in general, "materials recycling is one area we don't perceive ourselves as being the best in the industry". OEM liability is restricted to only providing a free take back service to end users, when a vehicle is of negative value "when it comes to the recycling targets unless the legislation changes a great deal I don't think manufacturers see too many individual risks". Thus the implementation of the ELV Directive has moved from a phase of risk minimisation (due to financial accruals), to a realisation that the strategic threat is rather smaller than expected originally.

Conclusions

OEMs still view the ELV issue as a low competitive priority compared to other environmental concerns. Yet at the same time there are high liability risks (due to financial accruals).In the UK, the new network organisation implies a market approach, yet within an atmosphere of limited competition. To an extent the risks of opportunism on the part of contractors are overcome with visibility (legislated and contracted). The case provides lessons for voluntary versus legislated approaches, and the strengths and weaknesses of both. The main challenge on the corporate agenda is balancing competitiveness and legitimacy concerns.

Looking at the development of the ELV issues in the UK over the last decade and a half highlights some interesting insights. The first is that a proactive voluntary approach will not necessarily stave off a legislated action. Although management strategists claim such benefits of proactive environmental strategy (Porter 1991;Hart 1995), this case demonstrates this is not always the case and should not be relied upon. The second main point is that while the voluntary approach has not survived in terms of structure and process, the network links that have been developed remain and have been very important in the response made by industry. In particular, the ability of the industries to act together to influence the exact way the legislation was interpreted is believed by the authors to be one reason why the UK may be seen as having an 'efficient' mechanism for complying to the ELV Directive.

Managerial implications

As this study has primarily taken a strategic capabilities perspective the main managerial implications follow from this. The first point to make is that ELV continues not to be a strategic imperative for vehicle manufacturers (despite a brief period when finance directors at board level were busy calculating the impact of financial accruals on their fragile share valuations). Many other issues compete more successfully to win board level discussion time. However, that is not to say that there could be competitively valuable opportunities. As some vehicle manufacturers have indicated, there may be opportunities outside of the legislative constraints. Some firms are spending time to develop their aftermarket strategies to expand there markets for 'branded' reconditioned parts and perhaps get closer to vehicle last users, as potential new customers.

A further managerial implication is the peculiarity of the UK ELV approach. Under the present many vehicle manufacturers have contracts with a limited number of dismantlers to ensure they have take back network in place, often coordinated through a single broker this is far from ideal for any 'buyer' of service, albeit mandated ones. To ensure that opportunism on the part of dismantlers is minimised, contracts are more akin to supply partnerships, common in the automotive industry. However, without significant investment in the relationship on both sides (information sharing, joint decision making, etc) there are still risks of opportunism. Thus vehicle manufacturer managers must still be vigilant to ensure that the true costs of ELV treatment and recovery are passed on, and still more wary that the true costs associated with their particular brand and models are visible.

The struggle between a dismantling process approach against a shredder based approach is a significant one. It is still to be shown which will be the most effective at achieving the recycling targets at the lowest cost. Shredders claim that the last 9% (made up of plastics etc) is most likely to be recovered with an approach based on post shredder technologies due to the economies of scale which are possible on a material separation level. While prices for complete components sold on the second hand market would give a better return by weight (despite the costs of dismantling), the ability to meet recycling targets is highly dependent on the consumer demand for parts (both in terms of volume and price sensitivity). More studies in this area are required to find the optimal process.

Political implications

One of the main implication for policy-makers is to clearly legislate so that firms are incentivised to improve their environmental performance, under market forces. Under the current arrangements, car manufacturers are hardly incentivised to increase the level recycling their vehicles through improved designs. While the cost of depollution of vehicles has to be covered by the manufacturers if it is more than the 'value' of the constituent parts, this does not mean the cost of achieving the recovery levels of 85% is met by each vehicle needs to be covered by manufacturers. It may make more sense to link the cost of reaching 85% of each vehicle model rather than brand, so that manufacturers would have to cover differential costs according the recoverability of individual model and hence be influenced to improve the 'design for recover', of less recoverable models. Of course this introduces extra information processing costs to the process, but may be recouped from lower cost model designs.

In order to gain maximum input from industrial partners, it may be prescient to introduce some kind of mix between regulation (where firms mainly aim at compliance only) and a voluntary or more fiscally aligned approach. Under the current regime there is limited incentive for industry to go beyond compliance and perhaps develop innovative solutions. If firms could directly link their environmental performance, e.g. recyclability, with operating profit for example, there would certainly be a greater chance of firms differentiating their responses. However, legislation often limits this differentiation, and so a voluntary code alongside fiscal incentives may in fact be more effective at mobilising companies' considerable resources to solve one of Europe's most pressing issues, the reduction of waste.

Implications for other contexts

Although an obvious point, the UK is not alone in reforming the way ELVs are dealt with, yet this has major implications for the contextual factors affecting this study. Compared to other countries, UK claims to have made a lower regulatory burden on firms. There is not empirical evidence to suggest that other countries have implemented the ELV directive in a less or more rigorous fashion, but ths would certainly be an important perspective to take. This has implications for both the effectiveness and efficiency of the whole ELV recovery process. Legislators may claim that their regulatory design has led to a more efficient process in terms of public and private resources (lower costs to enforce, lower costs to comply, for examples), but it has yet to be seen whether it would be as effective at meeting mandated targets. It should be pointed out that the effect of this is unlikely to affect vehicle manufacturer strategies, but could considerably affect cross-border trade in ELVs and hence the structure of Europe's dismantling and recycling industry.

Further research

Since any longitudinal study is limited firstly by the timeframe, it would be prescient to monitor developments in the UK visa vis the continued development of end of life vehicle and the impacts on both compliance and competitive position of firms within the network. As already mentioned, it could be fruitful to examine the implementation of the ELV Directive across a number of member states, with different routes to compliance. Such as study could include Holland, Germany and Italy which have also had distinctive approaches to ELV recovery in the past, rules on scrappage for example, and duties included on the price of new vehicles.

Figure 1 :

 1 Figure 1: Managing end of life networks: changing business relationships (scope of study) One of the most significant impacts of the new legislative regime is that the business relationships have changed. 2000 (Voluntary approach) 2006 (Legislated approach) CARE group membership ACORD Agreement Selected dismantlers for trials Contracts for factory scrap vehicles Remanufacturing (Engines & transmissions)

Table 3 :

 3 Recycling levels under the ACORD agreement (Source: SMMT)

		1997	1998	1999	2000
	Number of ELVs 1900000	1800000	1800000	2017137
	Recovery	76%	74%	77%	80%
	percentage

The following table show what the ACORD agreement achieved between 1997-2001 in terms of vehicles recycling. Interestingly, due to uncertainty about the ELV directive in terms of recycling targets, certainty over the implementation of the new European law, the efforts to measure recycling levels in the UK ended in 2001.

Table 2 :

 2 Structure of the recovery networks (2000 -2006) 2

Table 3 :

 3 Contractual relationship changes (2000Contractual relationship changes ( -2006) )

Figures based on MVDA, Environment Agency and World partsin brackets) for 2000 and Environment agency for 2006

Andrew Graves is Professor of Technology Management at The University of Bath and directs research into Lean Manufacturing and Agile Production within the IMRC (Innovative Manufacturing Research Centre). He is a mechanical engineer and political scientist and began his industrial career as a trainee manager with the Rootes Group in 1966. At the Science Policy Research Unit at the University of Sussex he undertook research with MIT's IMPV (International Motor Vehicle Programme) on R&D and technology issues and subsequently became European Programme Director to the programme.


Byunghyun Kim

Brett F Sanders
email: bsanders@uci.edu

James S Famiglietti

Vincent Guinot

Urban flood modeling with porous shallow-water equations: a case study of model errors in the presence of anisotropic porosity

Keywords: Porous shallow water equations, Finite volume model, Anisotropic porosity, Dam-break flood, Urban flood

teaching and research institutions in France or abroad, or from public or private research centers.

Urban flood modeling with porous shallow-water equations: a case study of model errors in the presence of anisotropic porosity

Introduction

Urban flood modeling is now possible at centimetric resolution or better with modern laser scanning data and flood models [START_REF] Bates | Integrating remote sensing data with flood inundation models: how far have we got?[END_REF][START_REF] Sampson | Use of terrestrial laser scanning data to derive decimetric resolution urban inundation models[END_REF]), but it is not advisable at this resolution over entire floodplains as the computational costs and memory demands are forbidding except on massively parallel computing architectures. Commonly used models are constrained by the Courant, Friedrichs, Lewy (CFL) condition for both stability and accuracy which dictates nearly an order-of-magnitude increase in computational effort every time the mesh resolution is doubled. For a Cartesian grid with a cell size of ∆x, the computational cost C of integrating a flood over a specified duration will scale as the product of the required number of computational cells n c and time steps n t ,

C ∼ n c n t ∼ 1 ∆x 3 (1)
because n c ∼ ∆x -2 and the CFL requirement to scale ∆t with ∆x. Thus, halving the cell size causes an eight fold increase in computational effort (nearly an order of magnitude) and at least a four-fold increase in memory demands. Previous work has shown that porosity models reduce computational demands by orders of magnitude [START_REF] Yu | Urban fluvial flood modelling using a two dimensional diffusion-wave treatment, part 2: development of a sub-grid-scale treatment[END_REF][START_REF] Mcmillan | Reduced complexity strategies for modelling urban floodplain inundation[END_REF]Soares-Frazão et al., 2008;[START_REF] Sanders | Integral formulation of shallow-water equations with anisotropic porosity for urban flood modeling[END_REF].

Porous shallow-water equations (porosity models) resolve urban flooding at a relatively coarse (and efficient) resolution compared to available geospatial data using additional parameters that account for sub-grid scale topographic features affecting the movement and storage flood water [START_REF] Defina | Two-dimensional shallow flow equations for partially dry areas[END_REF][START_REF] Yu | Urban fluvial flood modelling using a two dimensional diffusion-wave treatment, part 2: development of a sub-grid-scale treatment[END_REF][START_REF] Mcmillan | Reduced complexity strategies for modelling urban floodplain inundation[END_REF][START_REF] Sanders | Integration of a shallow-water model with a local time step[END_REF]Soares-Frazão et al., 2008;[START_REF] Cea | Unstructured finite volume discretization of two-dimensional depth-averaged shallow water equations with porosity[END_REF][START_REF] Chen | A coarse-grid approach to represent building blockage effects in 2D urban flood modelling[END_REF][START_REF] Guinot | Multiple porosity shallow water models for macroscopic modelling of urban floods[END_REF][START_REF] Schubert | Building treatments for urban flood inundation models and implications for predictive skill and modeling efficiency[END_REF]. In practice, the idea is to use a cell size on the order of meters or dekameters instead of a submetric resolution. This gives rise to models that resolve flooding at the pore scale roughly corresponding to the width of roadways and open spaces between buildings, in contrast with classical shallow-water models that resolve flooding at the point scale, as approximated by the grid resolution. [START_REF] Sanders | Integral formulation of shallow-water equations with anisotropic porosity for urban flood modeling[END_REF] and [START_REF] Guinot | Multiple porosity shallow water models for macroscopic modelling of urban floods[END_REF] introduce two alternative formu-lations of porosity models to capture porosity anisotropy, which can be expected in most practical applications. Anisotropy occurs in urban landscapes when there are preferential flow directions such as wide streets and narrow alleys aligned in perpedicular directions. Hypothetical examples of anisotropic flow have been presented in previous studies [START_REF] Sanders | Integral formulation of shallow-water equations with anisotropic porosity for urban flood modeling[END_REF][START_REF] Guinot | Multiple porosity shallow water models for macroscopic modelling of urban floods[END_REF], including numerous cass with angled channel-like flows through urban areas. Additionally, [START_REF] Schubert | Building treatments for urban flood inundation models and implications for predictive skill and modeling efficiency[END_REF] present a field-scale application of an anisotropic porosity model that outperforms models based on the classical shallow-water equations.

Porosity heterogeneity exists when the size of flow paths is spatially variable, and different porosity models resolve heterogeneity over different scales.

Isotropic porosity models are restricted to scales larger than the length scale of the Representative Elemental Volume (REV). This is typically an order of magnitude larger than the scale of flow obstructions in urban flood applications, nominally a kilometer or more [START_REF] Guinot | Multiple porosity shallow water models for macroscopic modelling of urban floods[END_REF]. On the other hand, the anisotropic porosity model developed by [START_REF] Sanders | Integral formulation of shallow-water equations with anisotropic porosity for urban flood modeling[END_REF] does not require the existence of an REV and can resolve heterogeneity at the grid scale.

Since porosity anisotropy is a critical consideration for practical applications, this study presents modeling of a unique experimental test case involving dam-break flow through an anistropic array of obstructions, which builds on earlier experimental work and modeling studies focused on isotropic arrays of obstructions [START_REF] Testa | Flash flood flow experiment in a simplified urban district[END_REF]Soares-Frazão and Zech, 2008). A classical shallow-water model and both isotropic and anisotropic porosity models are applied and calibrated. The objective is to measure and report the magnitude of porosity model errors in an absolute sense and also relative to other errors which collectively limit the overall accuracy of the model. A Results point to significant differences in porosity model errors across alternative porosity model formulatoins.

Methods and Materials

Porosity Definition

Porosity can be defined in more than one way, namely as a volume average fraction of pore space in a porous media or as an areal average fraction of pore space, as in a slice through the porous medium [START_REF] Bear | Dynamics of Fluids in Porous Media[END_REF]. Both volumetric and areal porosity can be expected to vary spatially in the case of a heterogeneous porous medium, and areal porosity can also vary with the orientation of the plane over which the areal average is taken, and thus exhibit anisotropy. If an urban land surface filled with solid features is taken as a porous medium, then the pore space represents the gaps between the solid features, the volumetric porosity represents the fraction of the land surface able to store water, and the areal porosity represents the fraction of space available for flood conveyance which is directionally dependent.

Porous Shallow-Water Equations

The anisotropic porosity model of [START_REF] Sanders | Integral formulation of shallow-water equations with anisotropic porosity for urban flood modeling[END_REF] is written as integral statements of mass and momentum conservation for an arbitrary 2D domain Ω with boundary Γ and unit outward normal vector n as follows,

∂ ∂t Ω iU dΩ + Γ iE • n dΓ = Γ iH • n dΓ + Ω iS dΩ (2)
where

U =      h uh vh      E =      uh vh u 2 h + 1 2 gh 2 uvh uvh v 2 h + 1 2 gh 2      (3) S =      0 -(c f D + c b D )uV -(c f D + c b D )vV      H =      0 0 1 2 gh| 2 ηo 0 0 1 2 gh| 2 ηo      (4)
where u=x-component of velocity, v=y-component of velocity, g=gravitational

constant, V = (u 2 + v 2 ) 1/2 , c f D is a ground friction drag coefficient, c b D is a
drag coefficient for sub-grid scale flow obstructions, and h| ηo is the depth corresponding to a piecewise constant water surface elevation η o and piecewise linear ground elevation z within Ω. The H term is introduced to transform the classical ground slope source term to a boundary integral that preserves stationary solutions. Based on the limits of this transformation, the momentum equations appearing in Eq. 2 are restricted to numerical schemes that are first-or second order accurate in space [START_REF] Sanders | Integral formulation of shallow-water equations with anisotropic porosity for urban flood modeling[END_REF].

The variable i(x, y) appearing in Eq. 2 is defined for the spatial domain D ∈ R 2 and represents a binary density function that takes on a value of zero or unity depending on the presence or absence of a solid flow barrier as follows [START_REF] Sanders | Integral formulation of shallow-water equations with anisotropic porosity for urban flood modeling[END_REF],

i(x, y) =    0 if (x, y) ∈ D b 1 otherwise (5)
where D b is a subdomain of D that corresponds to solid obstacles. Two gridbased porosity parameters are dependent on the density function (Eq. 5) as follows,

φ j = 1 Ω j Ω j i dΩ ψ k = 1 Γ k Γ k i dΓ (6)
where Ω j corresponds to the two-dimensional (2D) spatial domain of the j th computational cell and Γ k corresponds to the k th computational edge of a mesh. Note that φ j represents the fraction of a cell area occupied by voids, and ψ k represents the fraction of a cell edge occupied by voids. Consequently, these parameters affect the relative storage of cells and conveyance between cells, respectively. Importantly, anisotropic blockage effects are explicitly resolved by the distribution of ψ k values across the computational mesh. It is noted that isotropic porous shallow-water equations can be recovered from Eq. 2 under the assumption that φ j =ψ k ∀k. Additionally, Eq. 2 revert to the classical shallow-water equations in the limit that i(x, y) = 1.

Presently it is not clear how well isotropic and anisotropic porosity models resolve flow at the pore scale where information is needed to assess the risks facing individual land parcels in an urban area, especially when the obstructions exhibit anisotropy. Eqs. 2 resolve flow properties on a grid-cell by grid-cell basis which corresponds to the pore scale since the model requires a grid that aligns cells with pore spaces [START_REF] Sanders | Integral formulation of shallow-water equations with anisotropic porosity for urban flood modeling[END_REF]. In contrast, isotropic models require the existence of an REV where the porosity is scale-independent and where areal and volumetric porosities converge to a single scalar value [START_REF] Bear | Dynamics of Fluids in Porous Media[END_REF]. The length scale of the REV is roughly an order of magnitude larger than the length scale of obstructions in urban landscapes [START_REF] Guinot | Multiple porosity shallow water models for macroscopic modelling of urban floods[END_REF], so assuming that pore sizes and obstructions are similarly sized, the isotropic models theoretically resolve flow at roughly an order of magnitude larger scale than the anisotropic model presented here.

On the other hand, [START_REF] Guinot | Multiple porosity shallow water models for macroscopic modelling of urban floods[END_REF] suggests that isotropic models can yield representative results at scales 2-3 times smaller than the REV scale.

The ground friction drag coefficient is parameterized by a Darcy-Weisbach f as follows, c f D = f /8 which is in turn computed using a modified form of the Haaland equation [START_REF] Haaland | Simple and explicit formulas for the friction factor in turbulent pipe flow[END_REF] presented by [START_REF] Arega | Dispersion model for tidal wetlands[END_REF] which considers the Nikuradse sand-grain roughness height k s and the depth-based Reynolds number Re h = V h/ν, where ν represents the kinematic viscosity. The building drag coefficient is scaled by the projected area of solid barriers as follows, c b D = 1 2 c o D a f h where a f represents frontal area [START_REF] Nepf | Drag, turbulence and diffusion in flow through emergent vegetation[END_REF]. The units of a f are length -1 , corresponding to the frontal width of obstructions in Ω normalized by Ω. c o D is classical drag coefficient that accounts for shape and Reynolds number effects on drag [START_REF] Sanders | Integral formulation of shallow-water equations with anisotropic porosity for urban flood modeling[END_REF].

Numerical Methods

The integral porosity model is solved using a Godunov-based finite volume scheme that allows for triangular, quadrilateral, or mixed meshes [START_REF] Kim | Mesh type tradeoffs in 2D hydrodynamic modeling of flooding with a Godunov-based flow solver[END_REF]. The scheme uses Roe's approximate Riemann solver with a critical flow fix, an adaptive method of variable reconstruction for uneven topography that minimizes numerical dissipation [START_REF] Begnudelli | An adaptive Godunovbased model for flood simulation[END_REF], a local time stepping scheme [START_REF] Sanders | Integration of a shallow-water model with a local time step[END_REF], a improved Volume-Free Surface-Reconstruction (VFR) technique for wetting and drying, and inclusion of grid based porosity parameters [START_REF] Sanders | Integral formulation of shallow-water equations with anisotropic porosity for urban flood modeling[END_REF] which is of particular interest here. The scheme is explicit and conditionally stable in accordance with a CFL condition [START_REF] Kim | Mesh type tradeoffs in 2D hydrodynamic modeling of flooding with a Godunov-based flow solver[END_REF].

Laboratory Experiment

Laboratory-scale modeling of anisotropic blockage effects was carried out in a physical model constructed at the Korea Institute of Construction Technology (KICT).

Summary of Models

A classical shallow-water model (CSW), the anisotropic porosity model (PSW-A), and four isotropic porosity models (PSW-I) were applied. Additionally, results of the classical shallow-water model were averaged over each porosity-model grid cell to yield a pore scale classical shallow-water model result (CSW-P). Table 1 presents a summary of the seven models, and Fig. 2 presents the computational meshes used. Note that Fig. 2b corresponds to the gap-conforming mesh required of the anisotropic model [START_REF] Sanders | Integral formulation of shallow-water equations with anisotropic porosity for urban flood modeling[END_REF], where vertices are placed at the centroid of obstructions, cells are aligned with pore spaces, and edges intersect constrictions in the pore space.

Additionally, Fig. 2c corresponds to a region conforming mesh that precisely circumscribes the subdomain filled with flow barriers (Soares-Frazão et al., 2008;[START_REF] Guinot | Multiple porosity shallow water models for macroscopic modelling of urban floods[END_REF]. Four variants of the isotropic porosity model are used to account for both mesh designs and two alternative porosity values corresponding to the region-average volumetric porosity Soares-Frazão et al.

( 2008) and the areal porosity [START_REF] Guinot | Multiple porosity shallow water models for macroscopic modelling of urban floods[END_REF], as shown in Table 1. It is noted that an REV cannot be rigorously established in this test case due to the anisotropy, heterogeneity and limited spatial extent of the flow barriers, so the assumptions required to apply the isotropic model are not satisfied.

However, isotropic models have yielded credible predictions in other applications where these requirements were not satisified [START_REF] Guinot | Multiple porosity shallow water models for macroscopic modelling of urban floods[END_REF], motivating further study here.

Definition of Errors

Three

Model Parameterization and Calibration

In all seven models, mesh vertex heights were assigned based on reservoir or floodplain bed elevations, and mesh cells were assigned a Nikuradse sandgrain roughness height k s to model bottom shear. Further, a no-normal-flux boundary condition was enforced along the reservoir boundaries and concrete wall separating the reservoir and floodplain, and a free-outflow boundary condition was enforced along the remaining three sides of the floodplain.

The gate opening was modeled as an instantaneous breach since the time scale of opening (<3 s) is short compared with the time-scale of the breach flow (>100 s).

To apply the anisotropic porosity model, the cell-based porosity φ j and edge-based porosity ψ k were computed based on the intersection of the mesh with the footprint of the solid blocks following previously described methods [START_REF] Sanders | Integral formulation of shallow-water equations with anisotropic porosity for urban flood modeling[END_REF][START_REF] Schubert | Building treatments for urban flood inundation models and implications for predictive skill and modeling efficiency[END_REF]. Additionally, the frontal area parameter a f required to parameterize drag was computed on a cell-bycell basis in accordance with the projected area facing the dam as described previously [START_REF] Sanders | Integral formulation of shallow-water equations with anisotropic porosity for urban flood modeling[END_REF].

To apply the isotropic porosity models, φ j and ψ k were assigned a uniform value inside the block zone as shown in Table 1. Volumetric porosity values used in PSW-I-1A and PSW-I-2A are based on the spatial extent of cells that contact the obstructions, and the porosity values differ slightly based on the mesh. Areal porosity values used in PSW-I-1B and PSW-I-2B are based on the transect E-E' in Fig. 1d. A uniform frontal area parameter was also specified inside the block zone equal to the total frontal area facing the dam, normalized by the size of the block zone. This corresponds to 0.83 and 1.29 m -1 (Table 1) for the meshes shown Fig. 2b and2c, respectively.

Outside the block zone, a porosity value of unity was assigned in all porosity models. Also, the frontal area was set to zero.

The roughness parameter, k s , was manually calibrated by applying CSW to the first KICT flow scenario (h 0 =0.30 m) with k s values ranging from 0.03 to 0.3 cm, which is an established range for concrete [START_REF] Munson | Fundamentals of fluid mechanics[END_REF]).

The k s value achieving the best agreement between predicted depths and gage measurements (minimum L 1 norm) was subsequently used in all other models and in the second KICT flow scenario (h 0 =0.45 m).

To calibrate c o D , each of the porosity models was applied to the first KICT flow scenario with c o D values ranging from 1.0 to 3.0. This range corresponds to rectangular shaped blocks in an idealized two-dimensional flow [START_REF] Munson | Fundamentals of fluid mechanics[END_REF], and it is recognized that c o D may also vary depending on sheltering effects from the clustering of solid barriers and three-dimensional flow effects [START_REF] Sanders | Integral formulation of shallow-water equations with anisotropic porosity for urban flood modeling[END_REF]. Several options deserve consideration as the reference solution for the L 1 error norm. Calibration to gage measurements is the first option and is motivated by the goal of minimizing the overall error in the porosity model prediction, whereas another option is calibration to CSW-P predictions which is motivated by the goal of minimizing porosity model errors. Further, calbration to CSW-P depth and/or velocity predictions is possible. Here, all three options are pursued: calibration to depth measurements, CSW-P predictions of depth at gage locations, and CSW-P predictions of velocity at gage locations.

Results

Convergence of the CSW model

A resolution of 0.05 m was selected for CSW after a convergence check with a 0.025 m mesh of approximately 1.3 million computational cells. This showed that the average convergence error (measured over the simulation period at each gage) of the CSW depth prediction was less than 2 mm at all stations except Gage 2, where the convergence error was found to be 6 mm. Over all stations, the average convergence error was approximately 1 mm. Gage 2 is located in front of the leading row of obstructions (see Fig. 1). Here, super-critical flow through the breach strikes the first row of blocks, and a bow shock (hydraulic jump) forms across the width of the blocks as shown in Fig. 3. Based on the curvature of the shock wave, Gage 2 is on the windward side of the shock and Gages 11 and 18 are on the leeward side. Further, the width of the shock wave (measured in y direction on Fig. 3) is minimal at Gage 2: over a distance of 30 cm in the y direction, the water depth rises up from 5 cm to 16 cm, and then down again to 10 cm, approximately, based on results shown in Fig. 3(b). As the mesh is coarsened from 0.025 to 0.05 m resolution, this narrow band of super-elevated water is diffused slightly and its windward edge moves closer to Gage 2, leading to higher water depth predictions. Hence, the relatively large convergence error at Gage 2 is explained by its position at the leading edge of a shock wave.

It is noted that porosity models use a 30 cm mesh resolution (Fig. 2(c) and (d), and Table 1), which is too coarse to sharply resolve the narrow band of super-elevated water at Gage 2. This shows that pore scale and point scale values of flood predictions may differ substantially as a result of localized wakes and wave reflections from flow obstructions.

Calibration of k s

Fig. 4 shows CSW model predictions of depth using k s values from 0.03 to 0.3 cm, compared with measurements for a selection of gages. Additionally, In the analysis of model errors which follows, results of all three calibrations are considered and referenced as Calib1 (measured depth), Calib2

(CSW-P depth prediction), and Calib3 (CSW-P velocity prediction).

Model Predictions and Errors

Table 4 provides a summary of all model configurations and run times,

Structural Model Errors

The CSW prediction is shown to yield a good approximation of flood depths across the spatial domain (Fig. 5), with an average error of only 0.63 cm (Table 5), which represents just 2% of the initial depth in the reservoir.

The main limitations of CSW are noted at Sta. 18 where a spurious wave is measured in the experiment that is not explained by the model, and at Sta.

5 where the model overpredicts flood depths roughly by a factor of two. In a second test case involving h 0 =0.45 m (Fig. 6), the average error is 0.89 cm (Table 5) which is again just 2% of the initial depth in the reservoir. Hence, after calibration of the model to the first test case, the model performs with the same relative error in a second test case.

Scale Errors

Differences between point scale (CSW) predictions and pore-scale (CSW-P) predictions of flood depth constitute the scale error which is at least 65% smaller than the structural model error according to L 1 norms shown in Table 5. In particular, the scale error in depth is 0.18 cm in the first test case where the structural model error is 0.63 cm. In the second test case, the scale error is 0.30 cm while the structural model error is 0.89 cm.

Table 5 also shows that the scale error in velocity is 7.45 and 9.12 cm/s, which corresponds to about 2% of the theoretical peak velocity of a dry-bed dam break flood wave, (gh 0 ) 1/2 . Fig. 5 and 6 illuminate the origin of the scale error. In the first test case (Fig. 5), CSW-P notably departs from CSW at Sta. 2 which is explained by the shock waves shown in Fig. 3. This occurs because at the point scale, the prediction corresponds to one side of the shock or the other, while at the pore scale, the prediction corresponds to a spatial average around the shock.

Noticeable differences also occur at two other stations outside perimeter of the obstructions (e.g., Sta. 17 and 18), while differences away from the obstructions (Sta. 5, 6, and 7) and at stations off center from the main flow path (Sta. 19 and 20) are minimal.

Differences between the point scale and pore-scale velocities in Fig. 7 and 8 are noted at Sta. 2, 15 and 16 where relatively high velocities occur due to the alignment of this channel with the dam-break flood wave. Here, faster velocities occur along the centerline and slower velocities occur near the blocks as a result of wakes, and the monitoring stations sample the fastest moving water. Relatively large scale effects are also noted at Sta. 18 and 21.

Porosity Model Errors

Attention is now focused on porosity model errors in flood depth and velocity, which are measured by a comparison of porosity model predictions and CSW-P. Table 5 shows that the anistropic porosity model introduces a significantly smaller error in depth and velocity than all of the isotropic porosity models. For example, in the first and second test cases, isotropic model errors in depth were 65-210% and 77-240% greater than the anistropic model, respectively, based on Calib2. Additionally, isotropic model errors in velocity were 83-97% and 80-86% greater than the anistropic model for the first and second test cases, respectively, based on Calib3. Data in Table 5 also shows that the magnitude of the porosity model errors is mostly greater than or equal to the scale error, but less than the structural model errors, for both depth and velocity. The exception is the second test case where the anisotropic porosity model errors in depth are actually smaller than the scale error.

The total error of the porosity models relative to point-scale predictive skill is also shown in Table 5, with L 1 norms based on gage depth measurements. The total errors of the anisotropic porosity model are nearly identical to CSW and CSW-P based on Calib1, while all of the isotropic models yield larger total errors. Errors in the isotropic models range from 16 to 59% higher than CSW errors in the first test case, and 2 to 29% higher in the second test case, based on Calib1.

Spatial Variability

Previously shown results reveal at-a-station dynamics, but it is also worthwhile to examine the spatial structure of flood predictions. For the h 0 =0.30 m case, Fig. 9 shows contours of pore-scale flood depth and vectors representing the pore scale velocity magnitude and direction 50 s after the dam-break as depicted by: (Fig. 9a) CSW-P model, (Fig. 9b) PSW-A model, and (Fig. 9c-f) the four isotropic porosity models. CSW-P model predicts a zone of elevated water (region colored green, yellow and red) that approximates a triangular shape, and this shape is retained fairly well by PSW-A model, but not as well by the isotropic models. The isotropic models predict a more rounded shape which reflects a lack of directionality. Focusing on the bow shock in front of the obstructions, CSW-P model and PSW-A model predict a laterally distorted shape, while the isotropic models predict a more rounded shape, again reflecting a lack of directionality. block from the dam. On the other hand, the isotropic porosity models fail to capture this depth variability and instead predict a relatively smooth variation of the flood depth through the block zone. This is a result of using a uniform porosity value through the region of obstacles, and consistent with the design of isotropic models to predict flow properties at the REV scale which is considerably larger than the pore scale. Fig. 9 and 10 also reveal insight into the sensitivity of isotropic porosity models to the porosity value.

Generally, with a decrease in the porosity value, the height of the bow shock increases and it shifts forwards towards the dam.

Discussion

The preceding results show that porosity model errors may be significantly larger than scale errors which poses an opportunity for improved porosity models. The margin for improvement of the anisotropic model relative to flood heights is small, but the potential for improvement of the velocity predictions is greater and motivates improved models of flow resistance, possibly allowing for more spatial variability in parameters, or even fundamentally new approaches or more advanced calibration procedures. However, research directed at improving porosity model formulations should be mindful of structural model errors. Based on the data presented here, the anisotropic model is equally accurate as the point-scale classical shallow-water model relative to flood depth prediction, so further reduction in porosity model errors cannot be expected to reduce total errors. Broadly, porosity models cannot be expected to predict flood heights any more accurately than the pore-scale average of the foundational flow model, in this case the classical shallow-water equations.

There is critical need for urban flood inundation models that can be efficiently applied over practical scales such as a city or regional flood plain, and these results and previous studies [START_REF] Yu | Urban fluvial flood modelling using a two dimensional diffusion-wave treatment, part 2: development of a sub-grid-scale treatment[END_REF][START_REF] Mcmillan | Reduced complexity strategies for modelling urban floodplain inundation[END_REF]Soares-Frazão and Zech, 2008;[START_REF] Sanders | Integral formulation of shallow-water equations with anisotropic porosity for urban flood modeling[END_REF][START_REF] Guinot | Multiple porosity shallow water models for macroscopic modelling of urban floods[END_REF] reveal great potential to address this need. But aside from accuracy, another critical question to address is whether any of the porosity models can be more easily parameterized and validated in practical applications. High quality site data is often available for flood modeling studies but calibration data is rare, so there is a need for flood models with parameters that can be estimated deterministically and relied upon to make accurate predictions.

This further supports use of the anisotropic model presented here because porosity parameters are a deterministic function of the flow obstructions [START_REF] Sanders | Integral formulation of shallow-water equations with anisotropic porosity for urban flood modeling[END_REF][START_REF] Schubert | Building treatments for urban flood inundation models and implications for predictive skill and modeling efficiency[END_REF], in contrast with the isotropic model where it is unclear how to define a porosity given that a range of values could be used corresponding to volumetric and aerial porosities defined at different spatial scales. However, calibration data may still needed to estimate porosity model drag parameters (e.g., [START_REF] Schubert | Building treatments for urban flood inundation models and implications for predictive skill and modeling efficiency[END_REF]). In the less common scenario where high quality site data are not available to guide the porosity specification, but calibration data exists, the isotropic model may be preferred as the porosity value itself can be used as a calibration parameter.

Conclusions

Urban also show that porosity model errors in depth and velocity are significantly higher using an isotropic porosity model compared with an anisotropic model, and that the anisotropic porosity model is no less accurate than a fine grid shallow-water model, based on the total error. Recognizing that all porosity models reduced run times by a factor of nearly a thousand compared with the classical shallow-water models, the anistropic porosity model stands out as the most efficient approach for pore-scale modeling based on its low level of error, among models considered here. Additionally, the anistropic porosity model used here is more successful at resolving pore-scale flow variability than isotropic models because the latter are constrained to scales larger than the REV.

Results show that significant differences may exist between pore-scale and point-scale flood conditions in close proximity to flow obstructions, for example due to wave reflections and wakes, so porosity model flood predictions should be used cautiously to inform point-scale flood risk decision-making, such as whether flood heights will rise above the threshold of a door along a roadway. However, results validate the utility of porosity models for mapping flood heights at the pore-scale, i.e., the average flood height across a roadway.

Further research into porosity models should be directed at reducing porosity model errors in velocity, for example with improved drag parameterizations, but should be mindful of limitations posed by structural model errors. Finally, the cell averaging of fine-scale classical shallow-water model predictions is found to be an effective approach for gaging the merits of alternative porosity model formulations, as this enables a direct measure of the porosity model error.

• Fig. 10. Profile of flood depth after dam-break for h 0 =0.30 m at B-B'in 1 Fig. 1(c).

2

Captions of Tables

• Table 1. Shallow-water model formulations and corresponding meshes shown in Fig. 2.

• Table 2. L 1 norms of flood depth for calibration of roughness height (k s ) on CSW (unit: cm).

• Table 3. L 1 norms of flood depth for calibration of drag coefficient (c o D )

on PSW-A and PSW-I.

• Table 4. Model parameters and run time.

• Table 5. L 1 norms of flood depth and velocity based on calibration and reference solution.

29 x (m) Sta. 17 Sta. 17 Sta. 17 .360 20.532 20.343 20.204 20.152 20.122

y (m) -3 -2 -1 0 1 2 3 0 1 2 3 4 5 (a) x (m) y (m) -3 -2 -1 0 1 2 3 0 1 2 3 4 5 (c) x (m) y (m) -3 -2 -1 0 1 2 3 0 1 2 3 4 5 (b)

  better understanding of errors is needed to effectively use porosity models in flood hazard mapping. Three types of errors are reported: (a) structural model errors associated with the shallow-water equations which constitute the foundation of the porosity models, (b) scale errors arising from a grid size that matches the pore scale instead of the point scale, and (c) porosity model errors associated the parameterization of sub-grid scale obstructions.

  Fig.1(c). Two different flow scenarios are considered corresponding to an initial reservoir water depth (h 0 ) of 0.30 m and 0.45 m, measured relative to the floodplain elevation[START_REF] Yoon | Experimental study on flood inundation considering urban characteristics (FFC06-05)[END_REF].Within each 3x3 cluster, the gap between buildings is 0.1 m facing the dam (section E-E' in Fig.1(d)) and 0.4 m perpendicular to the dam (section G-G' in Fig.1(d)). This introduces a strong degree of anisotropy in the porosity field, a 1 to 4 ratio in the cross-sectional area available for flow between blocks. The KICT problem also introduces pore scale heterogeneity in the porosity distribution. For example, considering again Fig.1(d), the areal porosity ψ varies significantly between Sections D-D' and E-E' in the y direction, with ψ E < ψ D , and between Sections G-G' and F-F' in the x direction, with ψ G < ψ F . Similarly, the volumetric porosity φ varies significantly between domain a and b shown in Fig. 1(d), with φ b < φ a .

  types of errors are reported: (a) structural model errors, (b) scale errors and (c) porosity model errors. Structural model errors are defined by the difference, as measured by L 1 = N j=1 |(w 1 ) j -(w 2 ) j |/N, between the converged CSW prediction and gage measurements of flood depths. Scale errors are defined by the difference between the CSW (point scale) and CSW-P (pore scale) predictions at gage locations, and are computed for both depth and velocity. Porosity model errors are defined by the difference between porosity model predictions and CSW-P at gage locations (pore scale com-parison), and are evaluated for both depth and velocity.

3. 3 .

 3 Calibration of c o D Table 3 presents L 1 norms in porosity model predictions as a function of c o D and different reference solutions. This shows that optimal c o D depends on the porosity model and also depends on whether the goal is to minimize total errors or porosity model errors. In four of the five models, minimizing porosity model errors calls for a drag coefficient on the low end of the range (1.0) while minimizing total errors calls for a drag coefficient at the high end of the range (3.0). We conjecture that the goal of a porosity model should be to reproduce as accurately as possible the pore-scale averaged solution of the shallow-water equations, and not necessary match measurements. However, the results here clearly indicate that c o D can be tuned to improve the agreement with measurements. The calibration also shows that over a range of physically realistic drag coefficient values, the anisotropic model consistently produces smaller total errors and porosity model errors in flood depths. Further, the anisotropic model performs particularly well with respect to velocity predictions, as the porosity model errors are nearly twice as large for isotropic models versus the anisotropic model.

  Figs. 5 and 6 present predictions and gage measurements of flood depth for the first (h 0 =0.30 m) and second (h 0 =0.45 m) test cases based on Calib1, and Figs. 7 and 8 present model predictions of velocity for the first and second test cases based on Calib1. Results from Calib2 and 3 are not shown graphically, but Table 5 shows L 1 norms according to the porosity model, the calibration, and the reference solution. L 1 norms based on flood depth measurements are used to measure the structural model error in the CSW model and the total error in the porosity models, while L 1 norms based on the CSW-P prediction are used to measure porosity model errors. The scale

Fig. 10

 10 Fig.10shows the flood depth distribution for the h 0 =0.30 m case at four

  flood models based on porous shallow-water equations predict flood depths and velocities with three types of errors: (a) structural model errors associated with the limitations of the 2D shallow-water equations (e.g., hydrostatic pressure, vertical uniform velocity distributions), (b) scale errors associated with use of a relatively coarse, pore scale grid comparable to the spacing between buildings, and (c) porosity model errors related to the treatment of sub-grid scale obstructions. Results show that in this unique test case with anisotropy in the porosity distribution as in practical applications, porosity model errors are mostly greater than scale errors but less than structural model errors, although in one test case the porosity model error of the anisotropic model was slightly less than the scale error. Results

Fig. 1

 1 Fig.1

Fig. 3 .

 3 Fig. 3. Computational mesh for (a) CSW and CSW-P, (b) PSW-A, PSW-I-2A and PSW-I-2B, and (c) PSW-I-1A and PSW-I-1B.

Fig. 3 .

 3 Fig.2

Fig. 4 .

 4 Fig.3

Fig. 5 .Fig. 6 .

 56 Fig. 5. Comparison of predicted flood depth and measurement for h 0 =0.30 m.

Fig. 7 .

 7 Fig. 7. Comparison of predicted velocity for h 0 =0.30 m.

Fig. 8 .Fig. 10 .Fig. 9 .

 8109 Fig. 8. Comparison of predicted velocity for h 0 =0.45 m.

Table 2 shows

 2 L 1 norms for CSW model. These results demonstrate that the influence of roughness depends on the gage location, but overall roughness does not exhibit a strong influence on the average error. The implication is that momentum losses are dominated by the geometric constriction and form drag associated with the solid blocks, not skin friction from the bottom boundary. All subsequent modeling uses k s =0.03 cm since this leads to the most accurate prediction based on the values considered.

Table 1 .

 1 Shallow-water model formulations and corresponding meshes shown in Fig.2

	Case	Description	Mesh in Fig. 2	Num. of nodes	Num. of cells	Resolution (m) Avg. Max. Min.
	CSW	Classical shallow-water	(a)	330464 328612 0.05	0.05	0.05
	CSW-P	Pore-scale average of CSW	(a)	330464 328612 0.05	0.05	0.05
	PSW-A	Anisotropic porosity model	(b)	9216		0.30	0.33	0.25
	PSW-I-1A	Isotropic porosity model ( 𝜙= 𝜓=0.74)	(c)	9412		0.30	0.33	0.25
	PSW-I-1B	Isotropic porosity model ( 𝜙= 𝜓=0.40)	(c)	9412		0.30	0.33	0.25
	PSW-I-2A	Isotropic porosity model ( 𝜙= 𝜓=0.83)	(b)	9216		0.30	0.33	0.25
	PSW-I-2B	Isotropic porosity model ( 𝜙= 𝜓=0.50)	(b)	9216		0.30	0.33	0.25

Table 2 .

 2 L 1 norms of flood depth for calibration of roughness height (k s ) on CSW (unit: cm).

	Case	k s (cm)	Gages inside block zone 2 11&18 12&19 13&20 14&21 15	16	Gages outside block zone Entire Avg. 17 Avg. 3&7 4&6 5 Avg.
		0.03 1.08 1.22 0.66 0.44 0.58 1.49 0.23 0.29 0.75 0.33 0.39 0.81 0.51 0.63
		0.05 0.94 1.23 0.66 0.44 0.62 1.49 0.23 0.29 0.74 0.35 0.44 0.82 0.54 0.64
	CSW	0.10 0.71 1.24 0.66 0.44 0.70 1.48 0.23 0.29 0.72 0.40 0.53 0.89 0.61 0.66
		0.20 0.56 1.25 0.66 0.45 0.83 1.46 0.23 0.30 0.72 0.49 0.66 1.03 0.72 0.72
		0.30 0.57 1.26 0.67 0.46 0.91 1.44 0.23 0.30 0.73 0.56 0.74 1.12 0.81 0.77

Table 3 .

 3 L 1 norms of flood depth and velocity for calibration of drag coefficient (𝑐 𝐷 𝑜 ) on PSW-A and PSW-I. Case L 1 of flood depth (unit: cm) L 1 of flood depth (unit: cm) L 1 of flood velocity (unit: cm/s) Calib1: Ref.-Measured ℎ Calib2: Ref.-Predicted ℎ on CSW-P Calib3: Ref.-Predicted 𝑉 on CSW-P PSW-A 0.705 0.681 0.669 0.663 0.660 0.165 0.186 0.21 0.231 0.248 10.958 11.816 12.513 13.111 13.695 PSW-I-1A 1.068 1.021 1.015 1.012 1.003 0.507 0.545 0.578 0.592 0.590 21.730 21.893 22.132 22.136 21.960 PSW-I-1B 0.751 0.732 0.726 0.728 0.732 0.337 0.387 0.422 0.446 0.464 21.581 21.812 21.956 22.123 22.171 PSW-I-2A 1.152 1.088 1.04 1.003 0.974 0.601 0.529 0.533 0.543 0.533 22.084 21.05 20.798 20.505 20.142 PSW-I-2B 0.815 0.78 0.761 0.752 0.749 0.278 0.284 0.317 0.341 0

	𝑐 𝐷 𝑜 =1.0 1.5	2.0	2.5	3.0 𝑐 𝐷 𝑜 =1.0 1.5	2.0	2.5	3.0 𝑐 𝐷 𝑜 =1.0 1.5	2.0	2.5	3.0

Acknowledgements

This work was supported by the MRPI program of the University of California Office of the President and the Infrastructure Management and Extreme Events program of the National Science Foundation (CMMI-1129730). The authors wish to thank K. Yoon and his research team for their efforts to conduct the laboratory experiments presented here.

Captions of Figures

• Fig. 1. Experiment set-up of Yoon [START_REF] Yoon | Experimental study on flood inundation considering urban characteristics (FFC06-05)[END_REF]