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Transport and Dispersion of Water-borne

Pollutants and Feasibility of Abatement Measures

Christoffer Carstens

April 2012

TRITA-LWR.LIC. 2064 ISSN 1650-8629

ISRN KTH/LWR/LIC 2064-SE

ISBN 978-91-7501-344-2

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c

Christoffer Carstens 2012 Licenciate Thesis

Division of Water Resources Engineering

Department of Land and Water Resources Engineering Royal Institute of Technology (KTH)

SE-100 44 STOCKHOLM, Sweden

Reference to this thesis should be as follows: Carstens, C (2012) In the Pipe

or End of Pipe? Transport and Dispersion of Water-borne Pollutants and

Feasibility of Abatement Measures TRITA-LWR.LIC. 2064

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Summary in swedish

Overg¨ ¨ odning, eutrofiering, av vattendrag och recipienter ¨ ar ett av da- gens viktigaste milj¨ oproblem, b˚ ade i fr˚ aga om komplexitet och storlek.

F¨ or ¨ Ostersj¨ on ¨ ar ¨ overg¨ odningen ett akut problem, som har lett till syrefria f¨ orh˚ allanden i djupvatten och vid bottnar, en situation som uppr¨ atth˚ alls och f¨ orst¨ arks n¨ ar fosfor frig¨ ors fr˚ an hypoxiska sediment.

Minskad belastning av n¨ arsalter (P och N) har l¨ ange haft h¨ og poli- tisk prioritet, men den nuvarande sv˚ ara situationen antas ¨ aven kr¨ ava aktiva ˚ atg¨ arder inom avrinningsomr˚ aden och recipienter f¨ or att mins- ka b˚ ade belastning och negativa konsekvenser. Ett genomf¨ orande av framg˚ angsrika och kostnadseffektiva metoder f¨ or att minska utsl¨ app och negativa konsekvenser kr¨ aver kunskap om naturliga processer i avrinningsomr˚ aden, vattendrag och recipienter, samt teknisk expertis f¨ or att kunna j¨ amf¨ ora effekterna av olika slags typer av ˚ atg¨ arder och deras lokalisering.

Denna studie f¨ ors¨ oker kombinera processf¨ orst˚ aelse av transportme- kanismer inom avrinningsomr˚ aden, med s¨ arskilt fokus p˚ a kustomr˚ aden, och genomf¨ orbarhet av viss teknik f¨ or att minska n¨ arsaltsbelastningen och negativa effekter av eutrofiering p˚ a plats. Det ¨ overgripande temat

¨ ar ¨ odet f¨ or en individuell f¨ ororening, fr˚ an dess att den introduceras i avrinningsomr˚ adet till dess f¨ orsvinnande fr˚ an recipienten. Studien har delats in i tv˚ a delar d¨ ar den ena behandlar f¨ orst˚ aelse och modellering av dispersion (spridning) av f¨ ororeningar som trasnporteras av vatten genom avrinningsomr˚ aden, grundvatten och vattendrag. Den andra studien utreder potential att nyttja v˚ agkraft f¨ or att genomf¨ ora en storskalig syres¨ attning av ¨ Ostersj¨ ons syrefria vattenmassor och bott- nar.

Transport och spridning i avrinningsomr˚ aden utreds genom en kombinerad metodologi d¨ ar fysiskalsikt baserade, tredimensionella, nu- meriska grundvattenmodeller kopplas till Lagrangiansk Stokastisk Ad- vektiv Reaktiv (LaSAR) transportmodellering. Tillv¨ agag˚ angss¨ attet ¨ ar kraftfullt i den meningen att det tar h¨ ansyn till upptagningsomr˚ adets strukturella och geomorfologiska dispersion i den numeriska model- len och hydrodynamisk spridning p˚ a midere skalor, samt os¨ akerheter i LaSAR-metoden. Studien ger exempel p˚ a de komplicerade transport- tidsf¨ ordelningar som uppst˚ ar n¨ ar man i avrinningsomr˚ aden varierar hydrogeologiska f¨ orh˚ allanden med olika k¨ allors storlek och placering.

Vidare belyses betydelsen av dispersion och retention p˚ a grund av mo- lekyl¨ ar diffusion. Studien visar att geomorfologisk kontroll av sprid- ningen ¨ ar stark ¨ aven f¨ or relativt heterogena system (i form av h¨ og dispersion) och att varken den genomsnittliga uppeh˚ allstiden eller en vanligt anv¨ anda statistiska f¨ ordelningar f¨ or att beskriva uppeh˚ allstider i avrinningsomr˚ aden ger korrekta ˚ atergivningar av hydrologiska sy- stem.

F¨ or att bek¨ ampa intern belastning av P fr˚ an sediment in situ,

har storskalig luftning av djupvatten s˚ a kallad haloklin ventilering,

f¨ oreslagits. Den grundl¨ aggande tanken ¨ ar att omblanding av de dju-

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pare, syrefria vattnen med syrerika, ytligare vattenmassor, genrerar en syres¨ attande effekt av djupvattnen. Denna syres¨ attning skulle ha tv˚ a stora positiva effekter, dels genom direkt syres¨ attning genom ombland- ningen och minskad belastning av P fr˚ an sediment genom att dessa n¨ ar de syres¨ atts, b¨ orjar fastl¨ agga fosfor. P˚ a s˚ a s¨ att skulle den s˚ a kal- lade onda cirkeln, d¨ ar ¨ overg¨ odningen f¨ orst¨ arker sig sj¨ alv, brytas. Den m¨ angd energi som kr¨ avs f¨ or detta ¨ andam˚ al ¨ ar mycket stor och kr¨ aver en billig och enkel energik¨ alla f¨ or att vara genomf¨ orbar. Denna studie unders¨ oker m¨ ojligheten att m¨ ota en liknande operations energibehov med v˚ agkraft. ¨ Ostersj¨ ons v˚ agklimat utreds och relateras till tv˚ a olika typer av tekniker f¨ or att ¨ oka den vertikala omblandningen av vatten;

en d¨ ar flytande v˚ agbrytare f˚ ar transportera ner ytvatten till ¨ onskat djup och en annan d¨ ar stora kluster av bojar anv¨ ands f¨ or att blan- da om vatten mellan tv˚ a onskade djup. Det visas att den erforderliga m¨ angden syre som beh¨ ovs f¨ or att h˚ alla sedimenten vid oxiska tillst˚ and kan tillhandah˚ allas, billigt och effektivt, med hj¨ alp av v˚ agkraft.

Nyckelord: F¨ ororeningstransportmodellering; Grundvatten;

Dispersion; V˚ agkraft; ¨ Overg¨ odning

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Acknowlegdements

First of all I would like to thank Vladimir for your supervision and endless optimism and encouragement. Gia and Anders for your super- vision. Urban, Patrik and Sven for answering questions and helping me solving all the different problems I’ve encountered with Darcy- Tools. Christian for discussions and your efforts with the WEBAP project.

I would also give a special thanks to the organisations and persons, responsible for making this research happen, the financiers. Nova Re- search and Development and the Swedish Nuclear Waste Company (SKB) made the research on groundwater transport possible and per- sonally I would like to thank Marcus, Bengt, Jan-Olof and Peter for making this possible and for support. For the WEBAP project, the EU Life+ grant LIFE08 ENV/S/000271 supported the research.

For the finalisation of the Licentiate I am grateful to Anders W for a nice and thorough review. Joanne, thanks for most formal things around the PhD studies. Aira and Britt deserves a large thank you for all your administrative help. Bosse, thank you for encouragement and leading me into research as well as for your endless storytelling.

My fellow PhD students in the group: Staffan, Lea, Andrew, thank you. Emma, Anna and Joakim; thanks for help and long discussions.

All other colleagues.

Finally I would like to express my gratitude to my family for sup-

port in general and encouraging my decision to go into research. Made-

lene, thanks for sharing your life with me and for being a strong sup-

port during good and bad times.

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Table of Contents

Summary in swedish iii

Table of Contents vii

List of Figures ix

List of Tables xi

Abbreviations and nomenclature xiii

List of Papers xv

Abstract 1

Introduction 1

Hydrological transport and transport times . . . . 2

Eutrophication . . . . 4

Options for measures through halocline ventilation . . . . 5

Scope and focus of this study . . . . 7

Material and methods 8 Transport times and their distributions . . . . 8

Groundwater flow physics . . . . 9

Transport of solutes . . . 10

Lagrangian description of reactive transport processes . . . 10

Numerical 3D model . . . 13

Forsmark catchment . . . 14

Wave power and floating breakwaters . . . 18

Calculation of linear wave power for buoy clusters . . . 18

Calculation of overtopping flow rates . . . 19

Results 19 Modelling of fluxes and transport times . . . 19

Calibration . . . 19

Basic configuration . . . 21

Effects of source scale and position . . . 21

Effects of non-Fickian dispersion . . . 26

Effects of mass transfer due to molecular diffusion . . . 30

Wave-powered aeration in the Baltic Sea . . . 30

Linear wave power and buoy clusters . . . 30

Floating breakwaters and overtopping flow rates . . . 31

Discussion 33 Transport time modelling . . . 33

Wave-powered halocline ventialtion . . . 35

Costs and feasibility . . . 35

Technological advantages . . . 35

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Open issues . . . 36

Conclusions 36

Transport modelling in near-coastal catchments . . . 36 Wave-powered halocline ventilation . . . 37 Linking the two . . . 38

References 39

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List of Figures

Wave-energized aeration pumps . . . . 7

Scope of study . . . . 8

TOSS distribution . . . 12

Forsmark domain . . . 14

Hydraulic conductive domain . . . 15

Detail of the grid . . . 16

3D details of grid and permeability . . . 16

Wave stations and hypoxic zones . . . 18

Modeled groundwater levels in Forsmark . . . 20

Trajectories . . . 23

CDFs and CCDFs for mean travel times . . . 24

PDFs for basic configuration . . . 25

PDFs when varying α . . . 28

CDFs and CCDFs when varying α . . . 28

PDFs when varying ζ . . . 29

CDFs and CCDFs when varying ζ . . . 29

PDFs when varying ζ and adding mass transfer process . . . 30

Available wave power and overtopping flow rates . . . 32

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List of Tables

Grid discretisation scheme. . . 17

Source cases, positions and sizes. . . . 17

Calibration results for flow . . . 20

Calibration results for groundwater levels . . . 22

Statistics of the CDF of mean travel times . . . 24

Differences in early and late arrivals due to non-Fickian transport. . 26

Statistical parameters for different wave data . . . 32

Wave energy propagation and overtopping capacities . . . 32

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Abbreviations and nomenclature

c

g

Wave group speed [L T

−1

]

C Concentration of solute in mobile phase [M L

3

] CDF Cumulative Distribution Function

CCDF Complementary Cumulative Distribution Function D

ij

Dispersion coefficient [L

2

T

−1

]

E Energy density [M T

−2

]

f (t) Transport time distribution (TTD) of ideal tracers or water.

g Gravitational acceleration constant, here 9.82 [L T

−2

]

g(t) Memory function describing exchange kinetics between immobile and mobile zones [T

1]

h(t) Discharge of tracer.

H Wave height [L]

H

0

Deep water wave height [L]

H

m0

Significant wave height, obtained from spectral analysis [L]

H

s

Significant wave height, obtained from wave height measurement data [L]

K

ij

Hydraulic conductivity [L T

−1

] L Wave length [L]

L

0

Deep water wave length [L]

L

p

Wave length based on T

p

[L]

m

n

The n’th moment of the wave power spectrum M RT Mean residence time. In this study identical to ¯ ¯ τ . N Concentration of solute in immobile phase. [M L

3

]

P Linear wave power per meter wave crest. Wave energy flux. [M L T

−3

] P DF Probability density function

R

c

Crest freeboard [L]

s Laplace variable.

S

op

Wave steepness factor defined as H/L T Wave period [T]

T T D Transport time distribution. The PDF of transport times.

u Water velocity [L T

−1

]

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δ Dirac delta function η Surface displacement [L]

γ conditional PDF of single particle residence time including retention [T

−1

]

κ Permeability of a porous medium [L

2

] ρ Density [M L

−3

]

σ Standard deviation of mean travel time [T]

τ travel time of a single trajectory (stochastic) [T]

¯

τ mean travel time of a single trajectory [T]

¯ ¯

τ mean travel time of an ensemble of trajectories.[T]

ζ Coefficient of variation of mean travel time, σ/τ [-]

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List of Papers

I. Carstens, C. and Cvetkovic, V (2012): Hydrological dispersion in a coastal catchment. (manuscript) (Carstens responsible for research outline, mod- elling and analysis)

II. Carstens, C., Destouni, G. and Cvetkovic, V. (2011): Wave-power poten- tial for reducing hypoxia in the Baltic Sea. Submitted to Environmental Research Letters (Carstens responsible for research outline, modelling and analysis)

The author has also contributed to the following work (not included in thesis):

III. Cvetkovic, V., Carstens, C., Selroos, J-O. and Destouni, G. (2012): Wa-

ter and solute transport along hydrological pathways. Water Resources

Research (in review)

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Abstract

Eutrophication is one of the key environmental problems of today, both in terms of complexity and magnitude. For the Baltic Sea (BS), eu- trophication is an acute problem, leading to hypoxic conditions at the bottom; a situation that is sustained and amplified, when phosphorus is released from hypoxic sediments. Reducing nutrient loading is a top political priority but the present situation is believed to require active measures within the catchments and recipients to reduce both loading and adverse effects. Implementation of effective and cost-efficient abate- ment methods requires understanding of natural processes in watersheds, streams and recipients as well as technological expertise in order to com- pare the effects of measures of different kinds and locations. This the- sis tries to combine process understanding of catchment transport be- haviour, especially in coastal zones, and feasibility of certain technolo- gies for reducing nutrient loading and effects of eutrophication in-situ.

The over-arching theme is the fate of the individual contaminant, from injection to removal. Transport and dispersion in catchments are inves- tigated, combining physically-based, distributed, numerical groundwater models with Lagrangian stochastic advective reactive solute (LaSAR) transport modelling. The approach is powerful in the sense that it in- corporates catchment structural, geomorphological dispersion in the nu- merical model with hydrodynamic and sub-scale dispersion as well as un- certainty in the LaSAR framework. The study exemplifies the complex nature of transport time distributions in catchments in general and when varying source size and location, importance of dispersion parameters and retention due to molecular diffusion. It is shown that geomorpho- logical control on dispersion is present even for relatively heterogeneous systems and that neither the mean residence time nor a statistical distri- bution may provide accurate representations of hydrological systems. To combat internal loading of P from sediments in-situ, large-scale aeration of deep waters, halocline ventilation, has been suggested. This study further investigates the feasibility of wave-powered devices to meet the energy demands for such an operation. It is shown that the required amount of oxygen needed to keep the sediments at oxic conditions could be provided, cheaply and efficiently, through the use of wave power.

Key words: Contaminant transport modelling; Groundwater;

Dispersion; Wave power; Eutrophication

Introduction

Anthropogenic pollution of water bodies is a global and long-

time occurring phenomena, where evidence for centennial hu-

man impact on water bodies are available. (Renberg et al.,

2001; Galloway & Cowling, 2002; Bindler et al., 2009) Metal

pollution (mainly airborne) has even been detected in lake

sediments, of several thousands years of age (Branvall et al.,

2001). Excess nutrient loads to water bodies (eutrophication)

has been present at least since medieval times (Renberg et al.,

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2001), even if large increases in loads have been noticed mainly since the industrial revolution and specifically after the World War II and the improvements in artificial fertilizer production, intensification of the agricultural sector in general and indus- trial wastewater handling (Galloway & Cowling, 2002; Gal- loway et al., 2004; Howden et al., 2010). Other pollutants in- clude different organic compounds, e.g. PCBs, that have been released during the last 50 years (Breivik et al., 2002). Com- mon for all these pollutants is their water-borne transport, from sources to recipients, thereby dispersing, accumulating and undergoing physical and biogeochemical transformation, both along transport pathways and within the recipients. De- cisive for efficient pollutant management is a fundamental un- derstanding of the behaviour of these systems; an understand- ing that may form the basis for cost-effective and successful abatement measures. The possibility to compare measures in different systems, in different scales and locations, is desirable from management perspectives.

Hydrological transport and transport times

Understanding of water and material transport through catch- ments; aquifers, streams and lakes, is essential for many types of applications. The time water spends within a particular system is the primary factor governing different kinds of trans- port processes, such as retention, attenuation, biogeochemi- cal transformation. For practical purposes (e.g. when dealing with pollution problems) it is important to appropriately es- timate the water transit time, both for estimates of when a pollutant enters a recipient and for evaluating the importance of different processes along the way.

The time a water particle spends within a hydrological system, such as a catchment and its different subsystems; un- saturated zone, saturated zone, stream network is important for both understanding the functionality of the hydrological system and the behaviour of contaminants within the system.

Water residence times (the time a water particle stays in a system) and water transport (or travel) times, the time it takes for a water particle to travel from a to b in a particular system, are believed to be fundamental descriptors of internal processes, revealing information about flow pathways, storage and sources within the catchment and have long been in the focus of research. The water residence time is defined as as the time (since entry) that water molecules have spent inside a flow system, whereas transit time is defined as the elapsed time when the molecules exits a flow system. (McGuire &

McDonnell, 2006). Often the mean residence time (MRT) is

the focus of studies. The heterogeneity of a hydrological sys-

tem is however characterised by the full spectrum of travel or

transport times, including all possible pathways through a hy-

drological system. The spectrum of possible transport times,

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from an spatially distributed input to a control plane or point, the transport time distribution (TTD), is believed to provide an integrated measure of the large-scale, hydrological disper- sion of a system, representing all possible pathways from the source area to the location of measurement or interest. The TTD gives information about how a catchment stores and releases water, processes that in turn control important re- tention and transformation processes of geochemical and bio- geochemical cycling and contamination persistence. Stream water is an integrated mixture of water sources with an age that reflects all precipitation events (and their individual ages) that has contributed to the runoff. It also provides limits for exposure and reaction times for different biogeochemical processes. There are strong reasons for considering TTDs in- stead of MRT in studies of catchments and transport issues.

(McGuire & McDonnell, 2006; Godsey et al., 2010; McDonnell et al., 2010) The literature provides several other notations, e.g. travel time, weighting function, exit time, etc. (Mal- oszewski & Zuber, 1982; Lindgren et al., 2004; Darracq et al., 2010; Botter et al., 2010), and even though commonly used for describing the identical feature, it is worth to note that in many cases and by strict definitions, they are not the same (e.g. Rinaldo et al., 2011).

Modelling of transport times in complex real-life appli- cations in natural systems poses great challenges. Material transport is three-dimensional, time-variant and taking place in complex, mainly unknown geological structures. Several different approaches have been employed all with their respec- tive advantages and disadvantages. Simply put; distributed physically based models provide an intuitive and attractive advantage by the possibility to represent geological structures and local velocities but are problematic in their demand for (often) uncertain, unrepresentative, distributed data, mainly of the underground and the tendency for over-parametrisation.

Lumped models of different kinds, on the other hand, can pro- vide simplicity in implementation and require less data (but some data is still required) but gives little or no information on internal or spatial processes. Both approaches face problems of data scarcity and model equifinality, i.e. the possibility that several different model set-ups can produce equally good (or bad) results (Beven, 1989, 2001).

Through the aid of tracers, residence times and their sta- tistical distributions can be modelled or fitted to data (Din¸cer et al., 1970; Maloszewski & Zuber, 1982; McGuire & Mc- Donnell, 2006). Still, the proper choice of statistical TTD for a particular problem is a difficult but important task.

The choice of distribution affects both the result of the mean

transport time estimate and, perhaps more importantly, the

assumptions on early and late arrivals of tracers, due to the

asymmetrical form of most commonly used distributions. The

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exponential distribution (fully mixed reactor) is commonly used but studies show that this might be appropriate only in large systems with lakes, where mixed conditions should be expected. The gamma distribution has been proposed to be more appropriate in many cases. (Kirchner et al., 2000; God- sey et al., 2010) Other models (advection-disperson, piston etc.) have also been proposed but in the review by McGuire

& McDonnell (2006) as many as 66% of the studies used an exponential description (Godsey et al., 2010). Recently Cvetkovic (2011) proposed that the TOSS family of distri- butions might be useful as it includes basically all earlier used models and many in between, spanning non-fickian dis- persion from anomalous behaviour to plug flow. However, the commonly used, formal statistical descriptions of TTDs requires several assumptions and imposes numerous mathe- matical constrains in terms of modality and shape. Whereas the actual travel times are virtually impossible to measure in real-world cases, modelling studies suggest that many of the distributions, commonly used, are to different degrees of accuracy reproduced by models. (e.g. McGuire et al., 2007;

Fiori & Russo, 2008; Dunn et al., 2010; Darracq et al., 2010) The inherently time-variant nature of TTDs has recently been highlighted (e.g. Rinaldo et al., 2011); though important for time-variant, short-term problems, the implications for long- term transport, as in the present study is still not clear.

The discussion on transport time distributions have com- monly only focussed on catchment transport times, i.e. TTDs of water molecules, uniformly distributed introduced over the whole catchment and measured in a single point, at the outlet of a stream. Influence of source size and position on the TTD has not very often been studied, even though highlighted as important for transport characteristics (for some general in- vestigations, see Ibaraki, 2001; Tonina & Bellin, 2008). Effects of different control planes, where pollutant discharge is also expected to alter the TTDs. This in turn relates to the impact of diffuse loadings and submarine groundwater discharge from near-coastal catchments, which has been highlighted in sev- eral studies (e.g. Moore, 1996; Li et al., 1999; Windom et al., 2006; Moore et al., 2008; Destouni et al., 2008; Moore, 2010).

Eutrophication

Eutrophication of aquatic ecosystems by excess anthropogenic

nutrient discharges create problems worldwide (Diaz & Rosen-

berg, 2008; Galloway et al., 2008; Conley et al., 2009c; Rock-

strom et al., 2009), requiring effective solutions to reduce the

proliferation of harmful algal blooms (Huisman et al., 2005)

and the formation of “dead zones” in coastal marine ecosys-

tems (Diaz & Rosenberg, 2008). The semi-enclosed geogra-

phy of the BS, restricting large in- and outflows of oxygen-

rich saltwater from the Atlantic ocean, combined with large

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freshwater inputs produces a stable halocline (stratification) at approximately 60 m depth, inhibiting exchange of the deep waters. In all, this makes the BS sensitive to excess nutrient inputs and hypoxia has naturally been intermittently present in deeper parts (Conley et al., 2009a). Anthropogenic eu- trophication has increased the natural oxygen-depleted occur- rences and other effects, that are well described (Elmgren &

Larsson, 2001; Diaz & Rosenberg, 2008; Conley et al., 2009a, 2011) and have led to the world’s largest dead zone due to hypoxia (Diaz & Rosenberg, 2008). Reducing the specific BS eutrophication has been on the agenda for decades, through the Convention on the Protection of the Marine Environment of the Baltic Sea (Helsinki Commission, HELCOM) in the late 1980s. Until recently, HELCOM has worked to imple- ment an agreed 50 percent reduction target for anthropogenic nitrogen discharges (Backer & Lepp¨ anen, 2008). BS policies have so far mainly considered measures aimed directly at re- ducing nutrient inputs at their inland sources, including the international initiative, the Baltic Sea Action Plan (BSAP), with specific phosphorus and nitrogen reduction allocations for each BS country (HELCOM, 2007). Reducing nutrient inputs is a long-term effort and effects of progress might not be noticed in a long time. In spite of decades-long attempts to reduce eutrophication in this way, however, much progress still remains to be achieved.

Options for measures through halocline ventilation

Frustration over the lack of success in reducing the BS eu-

trophication so far has also led to increasing calls for addi-

tional, rapid and radical in-situ engineering measures to re-

duce the BS hypoxia. Consideration of such measures would

bring BS policies more in line with policies for other large-

scale international environmental problems, such as climate

change, for which the necessity of combining different mitiga-

tion and adaptation measures has been recognized. A com-

bination of load reductions on land and in-situ measures to

reduce internal loading would increase the speed of recovery,

compared to load reductions only (Stigebrandt & Gustafsson,

2007). Due to the long recovery time of BS from eutrophi-

cation all means to speed up the process could be considered

favourable from the public and stakeholder perspective. Con-

ley et al. (2009b) recently reviewed the theoretical potential

of different possible engineering measures to reduce the BS

hypoxia. Based on the results of oceanographic simulations

(Gustafsson et al., 2008), they concluded that only increased

oxygenation of the BS bottom waters, through halocline ven-

tilation, has the potential to reduce hypoxia and internal load-

ing, without important negative impacts. However, the envi-

sioned practical challenges posed by the large-scale engineer-

ing projects are considered daunting and stressed the impor-

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tance of reducing loading from land. (Conley et al., 2009b) The main idea of engineered halocline ventilation is to mimic, enhance and sustain the natural erosion of the halo- cline by circulating oxygen-rich surface water around the halo- cline (Stigebrandt & Gustafsson, 2007). In this way, oxygen could be supplied to the deepwater both by enhanced verti- cal mixing and directly through pumping. By improving the oxygen conditions in the deepwater, the bottom sediments would act as a phosphorus sink, with the phosphorus binding to iron oxides.(Stigebrandt & Gustafsson, 2007; Gustafsson et al., 2008; Conley et al., 2009b) Stigebrandt & Gustafsson (2007) have estimated the total amount of oxygen required for effectively ventilating the entire BS to be in the order of 100 kg s

−1

, which roughly corresponds to a requirement of pumping 10 000 m

3

s

−1

of oxygen-saturated water from 50 to 120 meters depth. The estimated annual power needed for this mixing, including extra mixing power and pumping, is 60 MW (Stigebrandt & Gustafsson, 2007). It is rather obvious that this kind of undertaking has to be carried out, using sim- ple technology and renewable, ”free” energy. Stigebrandt &

Gustafsson (2007) propose floating wind turbines to facilitate the pumping. In a recent publication Stigebrandt & Lilje- bladh (2011) describes the on-going prototype project BOX, which aims at investigating some of the crucial biogeochemi- cal, ecological and technological questions and risks, related to such large-scale oxygenation projects, by pumping in the bay Byfjorden on the Swedish west coast. Similar questions are addressed on the east coast of Sweden, viz. in the bay Kan- holmsfj¨ arden. The related ongoing project “Wave-Energized Baltic Aeration Pump” (WEBAP, www.webap.ivl.se) aims at investigating the potential of using wave power to facilitate the pumping, through overflow column devices.

The reasons for investigating wave power are its relative

abundance, renewable nature and the possibilities to con-

struct robust, simple devices for pumping; we consider these

as important prerequisites for successfully implementing a

large-scale off-shore project such as oxygenation of the Baltic

Sea. Even though the BS is a relatively sheltered sea, it has

been shown to have potential for wave power production. Ear-

lier investigations in the BS indicate an annual mean power

of 5-10 kW m

−1

wave crest (Henfridsson et al., 2007; Cruz,

2008). Furthermore, wave power as a general means to cre-

ate artificial pumping or upwelling has been discussed for a

couple of decades (Isaacs et al., 1976; Liu & Jin, 1995; Liu

et al., 1999). The wave-power potential in the BS can then

be exploited for halocline ventilation by use of at least two

different types of pumping devices: i) buoy-driven upwelling

devices (Figure 1a) and ii) floating breakwaters (Figure 1b).

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Figure 1: Principles and conceptual design of the two investigated types of aeration pumps. (a) Buoy- driven upwelling devices. The flap valve could be placed for pumping in both directions. (b) Floating breakwaters with overflow columns.

Buoy-driven devices

The buoy-driven pumping device (Figure 1a) uses a construc- tion proposed by Isaacs et al. (1976) and Liu & Jin (1995).

The suggestion is to connect a buoy to a pipe, which extends from approximately 60 meters depth down to approximately 125 meters depth. The functional mechanism is a flap valve, located either at the bottom or at the top of the pipe, gener- ating water flow in the desired direction. Placed in clusters, it is assumed that the buoys would be able to capture the linear wave power, transported in each wave front.

Breakwaters and wave-driven overflow columns

The second suggested alternative (Figure 1b) is to use wave- driven overflow columns in floating breakwaters, where the higher hydraulic head within a reservoir in the column will drive the transport of oxygen-rich surface water to the sub- halocline outlet depth. This method has been used and tested for power production, using low-head turbines and short pipes, in the project Wave Dragon (Frigaard et al., 2004a,b; Ko- foed, 2002). WEBAP is currently testing the feasibility of the method for halocline ventilation by both wave-powered systems and electrical devices at local scale.

Scope and focus of this study

The full problems of pollutant transport, recipient processes

and different measure options are diverse, complex and far-

(24)

System understanding Measure efficiency

Land

Catchment transport processes

• Advection - dispersion

• Retention

• Attenuation

• Biogeochemistry

• Modelling framework

Efficiency of measures in catchments

• Load reductions

• Wetlands

• Drainage systems

R ecipien t

Recipient processes

• Eutrophication reasons

• Biogeochemistry

Effects of in-situ measures

• Aeration

• Sequestration

Figure 2: Schematical picture of the scope of the study and possible directions for future studies.

reaching and not possible to even briefly cover within the scope of the present work. In general the thesis has focussed on system understanding and possibilities and effects of mea- sures in the two systems; catchments and recipients. This could be illustrated as a matrix, (Figure 2), with different topics. This particular licentiate study, focus on i) catchment system understanding (Paper I), through the investigation of influence of deterministic and hydrodynamic hydrological fea- tures on catchment transport behaviour by the use of com- bined modelling methods and on ii) scrutinising the practical challenges and possibilities to aerate one specific recipient (the BS), through the means of wave power (Paper II).

Material and methods

Transport times and their distributions

For groundwater systems the simple, steady state, turnover time, T , can be defined as the ratio of storage capacity, S [L

3

] of the catchment and the volumetric flow rate, Q [L

3

T

−1

] through the system as:

T = S

Q (1)

For a tracer the mean transit time at an outlet is defined as:

¯ τ =

R

0

tC

I

(t)dt R

0

C

I

(t)dt (2)

(25)

where C

I

(t) [M L

−3

] is the tracer concentration, observed at the outlet, as a result of a instantaneous injection at time, t = 0. The TTD, can be interpreted as the response break- through (h(t)) of a conservative tracer, applied uniformly and instantaneously to the system (catchment surface, groundwa- ter surface, or any other system boundary) at time t = 0 (Maloszewski & Zuber, 1982):

f (t) = C

I

(t) R

0

C

I

(t)dt = C

I

(t)Q/M (3)

where M is the total mass of injected tracer and Q still de- scribes a steady state flow rate.

The mean transit time and the TTD of a tracer is equal to the mean transit time and the TTD of the system only if the injected tracer is ideal and it is both injected and measured in the flux (Kreft & Zuber, 1978; Maloszewski & Zuber, 1982).

The response at the catchment outlet, or the discharge of an ideal tracer can, for uniform injections, be expressed as the convolution of an instantaneous concentration input at time t − t

0

, δ

in

(t − t

0

) and the TTD of water, f (t

0

) as:

h(t) = Z

0

f (t

0

in

(t − t

0

)dt

0

= g(t) ∗ δ

in

(t) (4) A more general expression, allowing for time-variant TTDs but still spatially uniform inputs can be written (McGuire &

McDonnell, 2006):

h(t) = Z

0

f (t, t

0

in

(t − t

0

)dt

0

(5) Groundwater flow physics

Groundwater flow is controlled by mass balance equations.

The Darcy’s law states that groundwater flows from a higher potential towards a lower

q

i

= −K

ij

∂h

∂x

j

(6)

where q

i

is the Darcy flux or Darcy velocity, which can be in- terpreted as the volumetric flowrate over a unit cross-sectional area. h represents the groundwater head and K

ij

the hydraulic conductivity (in tensor notation), which is a combined rela- tion of both fluid properties, as viscosity, µ and density, ρ, and the permeability, κ, of the porous medium and gravity, g:

K

ij

= gρκ

ij

µ (7)

(26)

Conservation of mass leads to the continuity equation:

∂(nρ)

∂t = − ∂(ρq

i

)

∂x

i

+ q

s

(8)

where q

s

is a source/sink term and n represents the poros- ity. Assuming time-invariant porosity (incompressible matrix) and introducing the Darcy law (6) into (8), the continuity equation can be written:

n ∂ρ

∂t = − ∂

∂x

i

 ρ

2

g κ

ij

µ

∂h

∂x

i



+ q

s

(9)

A mass balance over a infinitesimal element in Cartesian co- ordinates yields the groundwater flow equation

S

s

∂h

∂t = ∂

∂x

 K

x

∂h

∂x

 + ∂

∂y

 K

y

∂h

∂y

 + ∂

∂z

 K

z

∂h

∂z



(10) Equation (10) in some form is usually the governing equation in common groundwater codes. From the solution for the pressure head, the flow pattern and local velocities can be solved through the use of Darcy’s law (6).

Transport of solutes

Transport of a contaminant C in groundwater is usually mod- elled by the Advection-Dispersion Equation (ADE)

∂C

∂t + u

i

∂C

∂x

i

= ∂

∂x

i



D

ij

∂C

∂x

j



(11) where D is a lumped dispersion coefficient. The equation (11) is coupled to the modelled flow field through equation (10).

Often these equations have to be solved simultaneously.

Lagrangian description of reactive transport processes Transport along an ensemble of individual pathways is de- scribed using the LaSAR framework (e.g. Dagan, 1984; Da- gan et al., 1992; Cvetkovic & Dagan, 1994; Cvetkovic et al., 2012). Considering only advective transport of a solute but with reactions between mobile (C) and immobile phases (N ), the governing system for concentrations is written (in vector notation)

∂C

∂t + V · ∇C = − ∂N

∂t (12a)

F (∂N/∂t, N, C) = 0 (12b)

By following solute particles along each individual trajectory,

replacing the Cartesian coordinate system (x

1

, x

2

, x

3

) by a

streamline coordinate system (ξ

1

, ξ

2

, ξ

3

), following an individ-

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ual streamline, equation (12) reduces to

∂C

∂t + ∂C

∂τ = − ∂N

∂t (13)

In the Laplace domain we have (from Villermaux, 1974) d ˆ C

dτ = −s  ˆ N + ˆ C 

; N / ˆ ˆ C = ˆ g(s) (14)

where ˆ g, the ”memory function”, represents the exchange ki- netics between the mobile and immobile zones. For an instan- taneous injection of unit mass, the solution in the Laplace domain is

ˆ

γ(s|τ ) = exp [−sτ (1 + ˆ g)] (15)

which describes the PDF of mass, conditional on the mean travel time. To get the unconditional tracer residence time γ is ensemble averaged over all possible residence times

h(s) = ˆ ˆ f [−s (1 + ˆ g)] (16)

where ˆ f is the Laplace transform of the TTD.

In the real domain, the distribution of unit mass input of a non-reactive, non-interacting solute (i.e. a water molecule) along several individual trajectories (each with the mean travel time ¯ τ ) is formulated as

f (τ ) = Z

0

f (τ |¯ τ )p(¯ τ )d¯ τ (17)

where p(¯ τ ) is the distribution of mean travel times in the domain. Unconditional breakthrough of a several trajectories of a reactive solute is then represented by

h(t) = Z

0

Z

∞ 0

γ(t|τ )f (τ |¯ τ )p(¯ τ )dτ d¯ τ (18) where

Z

∞ 0

γ(t|τ )f (τ |¯ τ )dτ = L

−1

hˆ γ(s|τ )i (19) i.e. the ensemble average of the conditional PDF (equation (16)). For a distribution of known, deterministic transport times (modelled, measured) in the form of a CDF, equation (18) simplifies to

h(t) =

N

X

i=1

Z

∞ 0

γ(t|τ )f (τ |¯ τ )dτ ∆P ( ¯ τ

i

) (20)

The summation of equation (20) is then performed in the

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10−2 10−1 100 101 102 10−4

10−3 10−2 10−1 100 101

ζ = 0.3

10−2 10−1 100 101 102

10−4 10−3 10−2 10−1 100 101

ζ = 0.75

10−2 10−1 100 101 102

10−4 10−3 10−2 10−1 100 101

ζ = 1

10−2 10−1 100 101 102

10−4 10−3 10−2 10−1 100 101

ζ = 1.5 α = 0.05

α = 0.3 α = 0.5 α = 0.7 α = 0.95

Figure 3: Different TOSS distributions for a single trajectory with

¯

τ = 1. Thick lines show the influence of α and ζ only without mass transfer processes. α = 0.5 corresponds to the classical ADE equation solution. The thin lines are with mass transfer, A = 0.1 and T

0

= 1

Laplace domain h(s) = ˆ X

i

f [s (1 + ˆ ˆ g) ; ¯ τ

j

] ∆P ( ¯ τ

i

) (21)

The choice of both the transport time distribution of the variations, f and the memory function, g and their implica- tions for tracer discharge are two of the important questions of this study. f can be represented by an arbitrary proba- bility distribution, describing the nature of transport along the individual trajectory. Here we use the tempered one- sided stable distribution (TOSS) as described by Cvetkovic &

Haggerty (2002); Cvetkovic (2011). It has also recently been shown (Cvetkovic, 2011) that TOSS is capable of representing the majority of commonly used distributions in hydrological transport (and many in between), making it very well suitable for descriptions of macro-dispersion processes of as well Fick- ian and non-Fickan transport. In the Laplace domain TOSS is defined as

f ˆ

i

(s) = exp [c

i

a

αii

− (a

i

+ s)

αi

] (22) with

¯

τ = cαa

α−1

, ζ ≡ σ

τ

¯ τ =

r 1 − α α

1

ca

α

(23)

where ¯ τ is the mean and ζ is the coefficient of variation of τ ,

and 0 < α < 1. Through the variation of the parameters α, a

(29)

and c, equation (23) can represent a wide variety of commonly used descriptions of transport distributions, from anomalous Levy distributions to plug flow, where α = 0.5 represents the ADE solution and normal Fickian transport. (Cvetkovic, 2011, table 1)

ˆ

g(s) in equation (21), the ”memory function”, represents kinetics, i.e. microscopic dispersion and retention processes.

Several memory functions have been defined in the literature (e.g. Cvetkovic, 2011, Table 2). In this study, effects are ex- emplified by the limited matrix diffusion in slabs (Goltz &

Roberts, 1987), which is defined in the LT domain as

ˆ

g(s) = A q

3

2

sT

0

tanh

r 3 2 sT

0

!

(24)

A is a retention capacity (dimensionless), in this particular study A = 0.1 and T

0

[T ] is a retention time such that 1/T

0

characterises the exchange kinetics. T

0

is related to physical parameters as T

0

= ∆

2

/D

a

; ∆ [L] is a characteristic thick- ness and D

a

[L

2

/T ] is the apparent diffusion coefficient. T

0

is calculated as

T

0

= ∆

2

θ

m−1

D

w

(25)

where m = 1.56 is the Archie’s law cementation exponent and D

w

being the diffusion coefficient for a particular species in water.

α and ζ controls the distribution shape and character from anomalous transport to almost plug flow (Figure 3).

The results from the deterministic model is included in the formulation both in the scaling, ∆P ( ¯ τ

i

), where p = p(¯ τ ; A, a), i.e. the PDF of mean travel times is dependant of the source size (A) and position (a) and in the calculation of each tra- jectory’s pdf f (τ |¯ τ ).

Numerical 3D model

The groundwater code used in this study, Darcy Tools (DT)

(Svensson et al., 2010), has been developed for the Swedish

Nuclear Fuel and Waste Company (SKB) and has been tested

and verified for several applications (Svensson, 2010). DT

uses a continuum description and solves the groundwater flow

equations (10), using finite volume methods. The unsaturated

zone is modelled, using an iterative method where the ground-

water level is located and horizontal conductivities above that

level are reduced. This is efficient in the way that the ground-

water level is allowed to vary, without having to solve the

non-linear Richards equations. DT further includes several

important features for flow in fractured aquifers, making it

useful for modelling the kinds of hydrological systems as in

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Figure 4: Forsmark area and model domain.

Colours show topography in meter above sea level.

Scatter points indicate locations of measurement sta- tions. SFM (groundwater wells) in black and PFM (discharge stations) in blue.

central Scandinavia. Fractures are assumed to follow a power- law distribution; it is assumed that the number of fractures per unit volume, n, in the length interval dl is

n = I a

 l + dl l

ref



a

 l l

ref



a



(26) where I is the intensity, l

ref

is a reference length and a = −2.6 the power law exponent. The discrete features of the fracture network are then parametrised into the continuum model of DT. (Svensson et al., 2010) Particle transport can be modelled by several different methods in DT; in this study we have used the commonly used method, letting particles simply follow velocity flow vectors.

Forsmark catchment

The Forsmark catchment (Figure 4) is located in northern

Uppland, approximately 120 km north of Stockholm (N 60

23’, E 18

12’). As being the candidate area for the final

repository of spent nuclear waste in Sweden it has been stud-

ied thoroughly and long. The catchment has an area of ap-

proximately 35 km

2

and contains several small streams, bogs

and lakes. The overburden consists mainly of till soils with

some clay and peat layers. In a shallow ground water system,

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Figure 5: The main fracture zones of Forsmark and their transmissivity values.

mainly consisting of the overlying quaternary deposits, the groundwater surface is highly correlated to the topography.

Another system is found in the underlying bedrock, where the groundwater pressure generally coincides with a tectonic lens. (Johansson, 2008; Lindborg, 2008)

Data has been extensively collected, through numerous long term surveying. For this study, a detailed digital eleva- tion model with a resolution of 20 meters, a fracture zone net- work of the large-scale hydrogeological features in the bedrock (Figure 5) and data from flow measurement stations in streams (named PFM) and groundwater level wells (named SFM) have been used for model set-up and calibration.

The numerical model built up for DT consists of several hydrological units. The overburden is conceptualized as a layer of uniform depth with an exponentially decreasing hy- draulic conductivity in the vertical direction, to a minimum value, K

soil

= 10

−6

m s

−1

.

K

soil

(d) = K

top

10

−d/3

; if K

soil

≥ 10

−6

, (27) where d is the depth from the soil surface and K

top

is set to 5 · 10

−3

m s

−1

. In practice this means that the effective soil layer in the model is approximately 8 meters thick.

The bedrock consists of the known, deterministic fracture zone domain (Figure 5), one stochastic domain of smaller frac- tures and the matrix. (Follin, 2008)

The numerical model consists of a domain with the outer

limits as defined in figure 4 and vertically constrained by the

soil surface and 100 meters depth. Streams and lakes are rep-

resented as highly conductive volumes in the domain. The

domain is discretised in an unstructured Cartesian grid with

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Figure 6: A detail of the unstructured grid, showing the finer details around streams, lakes and the coast line.

Figure 7: A 3D illustration of the computational

grid and permeability. The finer resolution in the

top layer is clearly visible as well as the fracture

network in the bedrock regions.

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Table 1: Grid discretisation scheme. All numbers are in meters. There is also a smoothing effect in the grid generation, producing a continuous transition between grid sizes. See figure 6.

∆x ∆y ∆z

Bedrock 128 128 8

Soil overburden 32 32 1

Coast 16 16 1

Stream 2 2 1

Lake 2 2 1

Table 2: The different source areas. Position of the lower left corner (LLC) of the square, size of the source and number of particles (nbp) injected.

Case LLC (X;Y) area nbp

A N/A N/A 5169

B (9000;5000) 1 km

2

160

C (2000;6000) 1 km

2

125

D (7000;4000) 4 km

2

623

E (4000;6000) 4 km

2

561

smaller cell sizes around details, such as streams, lakes and the coastline. In total the computational grid consists of 951 887 cells. (Table 1 and Figure 6) The stochastic fracture net- work is generated, using l = 128 m, dl = 372 m, I = 0.2, l

ref

= 1 m. These values generated a total number of 236 942 fractures. A 3D illustration of the grid and the conductive domain shows that the fractures produce conductive zones in the bedrock (Figure 7). The forcing used to solve the station- ary groundwater level and flow field is a constant infiltration rate of P − E = 150 mm yr

−1

.

Deterministic transport times are calculated by using par-

ticle tracking through a stationary flow solution. Five dif-

ferent cases (A-E) are studied, where source location and size

were varied from the whole surface of the domain to quadratic

sources of 1 km

2

size. (Table 2) For all cases the sea has been

functioning as a control plane (CP), ensuring that mass break-

through is registered for diffuse loading along the whole shore

line. Two special cases are then studied more thoroughly; A

and B, the introduced particles in the whole domain and the

small source close to the coast. All particles are introduced

a certain distance from details, such as stream, lakes and the

coastline. This means that the modelled travel times are all

conditional on the injection in the soil domain, which gives a

bias towards longer transport times on one hand but repre-

sents the nature of diffuse sources as transport of e.g nutrients

on the other hand.

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Figure 8: Station names, locations, measurement periods and de- vice names of the SMHI wave measurement stations in the Baltic Proper basin. The figure also shows the extent of hypoxic zones in 2010 (modified from SMHI).

Wave power and floating breakwaters

Wave data from four available wave measurement stations of the Swedish Meteorological and Hydrological Institute (SMHI) was analysed to quantify the wave fields and wave power po- tential in the Baltic Proper basin (Figure 8). The data in- cludes one-hour time interval series of significant wave height (H

s

) and mean wave period (T). The functionality of the buoy- driven upwelling device will be assessed through calculation of the linear wave power from the data, whereas the overtopping flow will be calculated through relations derived by Frigaard et al. (2004b).

Calculation of linear wave power for buoy clusters

To estimate the number of devices needed it is assumed that buoys are able to capture most of the available linear wave power (in W m

−1

) if deployed in circular clusters as in the analysis of Bernhoff et al. (2006). One single cluster contains 389 individual devices and has a diameter of 600 m. This type of set-up is assumed to be able to capture most of the energy in incoming wave fronts. We calculate the linear wave power from the available wave data, seeking the total number of devices required to produce the needed 60 MW from the available waves. The power per meter wave crest, the linear wave power, P [W m

−1

] is calculated as

P = c

g

E (28)

where c

g

is the wave group velocity in m s

−1

, E is the en-

ergy density in J m

−2

. The group velocity, c

g

, is calculated

assuming linear wave theory and deep water wave conditions

(35)

as c

g

= gT

4π (29)

where T is the wave period and g is the gravitational acceler- ation. E is related to the significant wave height, H

s

, through the surface displacement, η [m]

E = ρ

w

ghη

2

i (30)

where

H

s

= 4 hη

2

i 

1/2

(31) This results in an expression of the linear wave power

P = c

g

E = g

2

ρ

w

T H

s2

64π (32)

where ρ

w

is the water density (set to 1007 kg m

−3

, representing typical Baltic Proper conditions).

Calculation of overtopping flow rates

Overtopping flow capacities of waves on offshore structures is not a well-documented field. For this study, we have used a suggested expression from the above-mentioned Wave Dragon project for a linear breakwater with 45 degrees slope (Frigaard et al., 2004b):

q = 0.017c

d

exp −48 R

c

H

c

r S

op

!

· pgH

s3

q

Sop

L (33)

where q is the overtopping flow rate in m

3

s

−1

, c

d

= 0.9 is a reduction coefficient for spreading effects, L is the length of the breakwater ramp (L = 1 m to calculate specific discharge), S

op

= H

s

/L

op

where is a steepness factor, R

c

is the crest freeboard height, and T

p

is the peak wave period calculated as 1.2T . The overtopping discharge results are sensitive to the chosen freeboard height. For the current study, a freeboard height of R

c

= 0.4 m was chosen as an example.

Results

Modelling of fluxes and transport times Calibration

The groundwater model is calibrated by ensuring that the

groundwater level corresponds to the large-scale topographic

features as indicated by litterature (Johansson (as indicated

(36)

Table 3: Calibration results for flow. The interval of measured values indicate the different means for four different time series. (Johansson, 2008)

Location X Y Simulated [l s

−1

] Measured [l s

−1

]

PFM005764 5659 6747 33 25-33

PFM002667 5594 6264 13 12-17

PFM002668 6066 5476 7 9-12

PFM002669 3379 7044 17 12-16

Figure 9: Modelled groundwater levels in Forsmark.

The groundwater surface follows the large-scale to-

pographic features (Figure 4) The four wells that

the model reproduce less well (SFM0004-06,09 and

0106) are coloured in red with black edges.

(37)

by 2008)). Where possible, calibration is made against data of mean groundwater levels in 39 groundwater wells (Figure 9, Table 4) and flow rates in streams with time series of flow rates (Figure 3). Most of the values are in good agreement, whereas some show deviations (mainly the wells SFM0004, SFM0005, SFM0006, SFM0009 and SFM0106). Given the many uncertainties in these kinds of modelling and the objec- tive of the current study to exemplify how to combine different modelling efforts, and not to show a fully calibrated model, the solution has been considered agreeable.

Basic configuration

The distributions of mean travel times, ¯ τ

i

, from the groundwa- ter model indicate asymmetric forms, reflecting the different flow paths of the different source sizes and locations. (Figure 10, 11 and Table 5). PDFs of the mean travel time distribu- tions are difficult to obtain directly from the model results, due to the discrete nature of the modelled CDFs. Slightly modified distributions that are thought to represent the de- terministic distributions are calculated by summation of equa- tion (20) with α = 0.5 and ζ = 0.3 (Figure 12). These sym- metric and low values of dispersion acts more as a filter on the deterministic PDF, producing smooth, continuous distri- butions, without changing it significantly, except in the most extreme cases. The complex and, from commonly used sta- tistical distributions, different shape of the PDFs for the dif- ferent source cases are quite apparent. The PDFs are highly skewed (Table 5) and in some cases multimodal (Figure 12).

The poor representation of the MRT (¯ τ ) as a general catch- ¯ ment descriptor, is also clearly shown, for example by the comparison of the mean and the median. The last column in table 5 shows that for all five cases the MRT corresponds to the time for 80-85 % breakthrough of mass.

Effects of source scale and position

Generally, the source scale (area) influences the shape of the distribution (Figure 12). Although not unequivocal, increase in source size seems to increase the irregularity of the differ- ent modelled distributions. Multimodality is also present and important in e.g. case A and E. This result can be explained by the fact that larger sources captures more of the hetero- geneity of the catchment, both in terms of pathway length and structural differences.

The source distance to the coast or streams is reflected in

the mean travel time (¯ τ ) and the effect of proximity to streams ¯

in the early arrivals (¯ τ

1

). (Table 5, Figure 11 and 12) For ex-

ample, case E (purple colour), which is both close to the coast

and most particles are released relatively close to a stream or

lake, shows both a shift towards shorter mean travel times and

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Table 4: Calibration results for groundwater levels for 39 groundwater wells (Figure 9). The four wells that the model reproduce less well (SFM0004-06,09 and 0106) are coloured in red with black edges.

Well X Y measured simulated ∆

SFM0001 5335 7713 0.5 0.9 0.4

SFM0002 5378 7586 1.2 1.2 -0.1

SFM0003 5487 7615 1.2 1.1 -0.1

SFM0004 7441 6866 2.9 0.7 -2.2

SFM0005 7252 6648 4.9 1.7 -3.2

SFM0006 8502 5747 4.6 1.8 -2.7

SFM0008 8623 5931 0.5 1.3 0.8

SFM0009 7224 6578 4.0 1.6 -2.4

SFM0010 4735 5314 12.5 10.8 -1.8

SFM0011 4711 7117 1.9 2.1 0.2

SFM0013 5123 6699 1.2 2.0 0.8

SFM0014 5716 5027 5.3 5.4 0.1

SFM0016 6174 4976 5.2 5.2 0.0

SFM0017 6138 4505 5.5 5.5 0.0

SFM0018 5950 4558 5.3 5.3 0.1

SFM0019 6118 5701 3.1 2.5 -0.6

SFM0020 6994 6127 1.4 1.4 0.0

SFM0021 6493 7706 1.0 0.7 -0.3

SFM0026 8152 4703 0.7 0.6 -0.1

SFM0028 7589 6508 0.2 0.1 -0.1

SFM0030 5663 6678 1.1 1.1 0.0

SFM0033 5728 6839 0.5 0.7 0.2

SFM0034 5859 7757 0.5 0.6 0.1

SFM0036 5746 7992 0.3 0.6 0.2

SFM0049 4533 8028 2.3 1.6 -0.7

SFM0057 4949 6980 3.4 2.3 -1.1

SFM0058 5740 7349 1.5 0.9 -0.6

SFM0059 9777 6464 0.1 0.2 0.1

SFM0061 9924 6377 0.0 0.4 0.4

SFM0077 4389 7921 2.7 2.7 0.0

SFM0078 4765 7704 3.5 2.1 -1.4

SFM0079 4568 7691 2.7 2.8 0.0

SFM0084 6406 7868 0.5 0.6 0.0

SFM0091 5491 7746 0.4 0.7 0.3

SFM0095 4438 6018 10.5 10.3 -0.2

SFM0104 5275 7592 1.0 1.2 0.3

SFM0105 6465 7710 1.6 0.7 -0.9

SFM0106 8043 6321 3.0 0.6 -2.4

SFM0107 4769 8187 0.9 0.6 -0.2

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Figure 10: Trajectories for the different source cases A to E (sub-

figures a-e) and 3D structure of pathways (f ). There is a maximum

of 500, randomly chosen, particles shown for each case.

(40)

10−1 100 101 102 103 10−4

10−3 10−2 10−1 100

τ [years]

cdf/ccdf

A B C D E

Figure 11: CDFs and CCDFs for the deterministic, numerically modelled transport times. The power law dependency in the tails of all cases is strong, whereas the early travel times are quite different be- tween the cases.

Table 5: Statistics of the CDF of mean travel times (¯ τ ) of the dif- ferent source cases. The different nature of the source sizes and positions are quite visible from the higher order moments (vari- ance (s

2

), skewness (γ

1

) and kurtosis (γ

2

)) and the percentiles ¯ τ

x

. Interestingly, the huge difference between the mean and median is present in all distributions. The mean corresponds to the time for 80-86 % of tracer breakthrough (last column, CDF(¯ τ )). ¯

¯ ¯

τ s

2

s γ

1

γ

2

CV τ ¯

1

τ ¯

10

τ ¯

50

τ ¯

90

τ ¯

99

CDF(¯ τ ) ¯ A 16.3 7058 84.0 22.4 672 5.16 0.27 1.87 4.56 25.0 194 86%

B 12.1 525 22.9 4.5 28 1.90 1.46 2.54 4.85 25.0 124 80%

C 14.9 808 28.4 4.5 25 1.91 0.87 3.38 5.71 29.5 178 81%

D 15.2 3642 60.4 15.2 292 3.98 1.04 2.19 4.60 24.2 188 85%

E 9.9 736 27.1 8.0 84 2.74 0.23 1.08 3.63 19.4 135 84%

(41)

10−2 10−1 100 101 102 0

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

α = 0.5, ζ = 0.3

t [years]

ccdf

Figure 12: PDFs for low dispersion cases that rep- resent slightly filtered deterministic PDFs. Note that the shape of the curves seems to deviate more from regular distributions with the source shape and size.

less long-term transport. In case C (red), with a 1 km

2

source, located far from the coast, we can see a shift towards longer travel times. However, the mean is not larger, due to the lack of very long and slow pathways. In the last case, B (dark green), the 1 km

2

source is close to the coast but not within the reach of any stream one can see the effects of proximity to the coast by the slightly shorter ¯ τ and effects of the diffuse loading in the absence of shorter travel times (¯ τ

1

=1.46 years).

Finally, it is worth to mention the full domain source (case A, blue), where effects of both short-time loading and long-term transport are present, producing a CDF of a quite peculiar shape. The continuous shift between long-term transport and short-term transport is markedly pronounced by the deviation from the turquoise and green curve (cases B and D) towards the purple curve (Figure 11, case E). Thus, the full domain encompasses a more complete representation of travel times, as expected, especially at smaller time scales, whereas the 1 km

2

sources has steeper curves at the early stages. However the distributions do not seem to follow any commonly used statistical descriptions.

In the tail (Figure 11, CCDF), a power-law dependency

is present in all distributions. The relationship seems to hold

for several orders of magnitude.

References

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