SOLUTE TRANSPORT THROUGH FRACTURED ROCKS: THE
INFLUENCE OF GEOLOGICAL HETEROGENEITIES AND
STAGNANT WATER ZONES
Batoul Mahmoudzadeh
Doctoral Thesis in Chemical Engineering
Department of Chemical Engineering and Technology
School of Chemical Science and Engineering
Royal Institute of Technology
Stockholm, Sweden 2016
Solute transport through fractured rocks: the influence of geological heterogeneities and
stagnant water zones
Batoul Mahmoudzadeh
Doctoral Thesis in Chemical Engineering
TRITA-CHE Report 2016:15
ISSN 1654-1081
ISBN 978-91-7595-906-1
KTH School of Chemical Science and Engineering
SE-100 44 Stockholm
SWEDEN
Akademisk avhandling som med tillstånd av Kungl Tekniska Högskolan framlägges till
offentlig granskning för avläggande av teknologie doktorsexamen i kemiteknik tisdagen den
26 april 2016 klockan 10.00 i Sal F3, Lindstedtsvägen 26, Kungliga Tekniska Högskolan,
Stockholm. Avhandlingen försvaras på engelska.
Fakultetsopponent: Dr. David Hodgkinson, Ordförande i Quintessa.
 Batoul Mahmoudzadeh 2016
Tryck: US-AB
ii
To my parents
iii
iv
Abstract
To describe reactive solute transport and retardation through fractured rocks, three models are
developed in the study with different focuses on the physical processes involved and different
simplifications of the basic building block of the heterogeneous rock domain. The first model
evaluates the effects of the heterogeneity of the rock matrix and the stagnant water zones in
part of the fracture plane. It accounts for the fact that solutes carried by the flowing water not
only can diffuse directly from the flow channel into the adjacent rock matrix composed of
different geological layers but can also at first diffuse into the stagnant water zones and then
from there further into the rock matrix adjacent to it. The second and the third models are
dedicated to different simplifications of the flow channels. Both account for radioactive decay
chain, but consider either a rectangular channel with linear matrix diffusion or a cylindrical
channel with radial matrix diffusion. Not only an arbitrary-length decay chain, but also as many
rock matrix layers with different geological properties as observed in the field experiments can
be handled.
The analytical solutions thus obtained from these three models for the Laplace-transformed
concentration in the flow channel can all be conveniently transformed back to the time domain
by use of e.g. de Hoog algorithm. This allows one to readily include the results into a fracture
network model or a channel network model to predict nuclide transport through flow channels
in heterogeneous fractured media consisting of an arbitrary number of rock units with piecewise
constant properties. More importantly, the relative impacts and contributions of different
processes in retarding solute transport in fractured rocks can easily be evaluated by simulating
several cases of varying complexity. The results indicate that, in addition to the intact wall rock
adjacent to the flow channel, the stagnant water zone and the rock matrix adjacent to it may
also lead to a considerable increase in retardation of solute in cases with narrow channels.
Moreover, it is elaborated that, in addition to the chain decay, it is also necessary to account for
the rock matrix comprising two or more geological layers in safety and performance assessment
of the deep repositories for spent nuclear fuel. The comparison of the simulation results with
respect to either rectangular or cylindrical channels further suggest that the impact of channel
geometry on breakthrough curves increases markedly as the transport distance along the flow
channel and away into the rock matrix increase. The effect of geometry is more pronounced for
transport of a decay chain when the rock matrix consists of a porous altered layer.
Additionally, a model is developed to study the evolution of fracture aperture in crystalline
rocks mediated by pressure dissolution and precipitation. It accounts for not only advective
flow that can carry in or away dissolved minerals but also the fact that dissolved minerals in
the fracture plane, in both the flow channel and the stagnant water zone, can diffuse into the
adjacent porous rocks. The analytical solution obtained in the Laplace space is then used to
evaluate the evolution of the fracture aperture under combined influence of stress and flow, in
a pseudo-steady-state procedure. The simulation results give insights into the most important
processes and mechanisms that dominate the fracture closure or opening under different
circumstances. It is found that the times involved for any changes in fracture aperture are very
much larger than the times needed for concentrations of dissolved minerals to reach steady state
in the rock matrix, the stagnant water zone and the flow channel. Moreover, it is shown that
diffusion into the rock matrix, which acts as a strong sink or source for dissolved minerals,
clearly dominates the rate of concentration change and consequently the rate of evolution of
the fracture.
v
Sammanfattning
Tre modeller har utvecklats för att beskriva hur radionuklider och andra lösta ämnen
transporteras och fördröjas under inverkan av olika fysikaliska processer när de förs av
sipprande vatten i kanaler i sprickor i berg. De tre modellerna lägger tyngden på olika processer
och utgör byggstenar i mer omfattande modeller där kanalerna sammanlänkas till komplicerade
nätverk i heterogent berg. Den första modellen behandlar hur heterogeniteter i bergmatrisen
och närvaron av stagnant vatten i sprickan vid sidan av vatten i den strömmande kanalen
inverkar på transporten. Närvaron av stagnanta vattenområden kan påverka transporten genom
att ämnena kan diffundera in i det stillastående vattnet och därigenom fördröjas. Dessutom kan
ämnena diffundera vidare in i den porösa bergmatrisen inte bara från det strömmande vatten
utan även från det stillastående och fördröjas ytterligare. Den andra och tredje modellen
behandlar förenklade specialfall av kanalgeometrin, den ena med rektangulärt den andra med
cirkulärt tvärsnitt. I båda fallen behandlas radionuklider som har olika egenskaper och hela
sönderfallskedjor. Modellerna kan hantera kanaler med många lager av sekundära mineral som
kan ha bildats på sprickvägen i såväl kanalen som i den stagnanta området.
De analytiska lösningarna till modellerna i har erhållits med hjälp av Laplacetransformer och
återtransformeras numeriskt till tidsrymden med hjälp av de Hoog–algoritmen. Detta gör det
möjligt att enkelt bygga upp omfattande nätverk av kanaler med olika egenskaper i komplexa
tredimensionella kanalnätverk för att simulera hur kedjor av radionuklider transporteras med
vatten i komplicerade heterogena bergvolymer. Varje kanal eller del av kanal ges konstanta
egenskaper men kan förfinas godtyckligt i kortare kanaldelar om detta skulle krävas för att
beskriva kanaler med varierande egenskaper längs kanalen. Genom att analytiska lösningar till
modellerna tagits fram kan man enkelt se sambandet mellan olika processer och se hur
variationer i parametrar kan påverka transporten. Ett antal simulerade fall tyder på att upptag
och sorption i den porösa bergmatrisen, både från kanaler med strömmande vatten och från det
stagnanta vatten har mycket stor inverkan på fördröjning av radionukliderna. Inverkan av
stagnanta vattnet är särskilt stor då bredden på kanalen med det strömmande vattnet är liten.
Simuleringsresultaten tyder även på att egenskaperna hos och tjocklekarna av de sekundära
mineralen kan ha stor inverkan på fördröjningen av viktiga radionuklider. Jämförelser mellan
cirkulära och rektangulära kanaler visar att den första geometrin visar större fördröjning än den
andra ju längre transportvägen och därmed inträngningen i bergmatrisen blir vid i övrigt lika
förhållanden. Närvaron av en mer porös zon på sprickväggen har olika inverkan på
nuklidkedjors fördröjning i kanaler med olika form på strömningstvärsnittet.
En ytterligare studie presenters där upplösning och återutfällning av de mineral som bygger
upp bergmatrisen kan påverka sprickvidden och hur detta påverkas av de bergspänningar som
sprickorna utsätts för. Detta är av intresse för de långa tider, hundratusentals år, som måste
beaktas vid säkerhetsanalys av ett slutförvar för utbränt kärnbränsle. Samma transportprocesser
och mekanismer som i de tre tidigare modellerna beaktas och samma matematiska teknik
används för att lösa ekvationerna. Simuleringarna visar att det tar mycket längre tider än de
nyss nämnda för att dessa processer nämnvärt skall påverka sprickaperturen och därmed
nuklidtransporten.
vi
Acknowledgements
This work has been carried out at the Division of Chemical Engineering, Department of
Chemical Engineering and Technology, of the Royal Institute of Technology, KTH, in
Stockholm, Sweden. I wish to sincerely thank my colleagues and the staff for all their help and
support during these years. In particular, I would like to express my deepest appreciation to the
supervisor of my research project, Associate Professor Longcheng Liu, for sharing with me his
great knowledge and scientific experience, and for his continuous commitment, support,
guidance and encouragement not only in research work but also in many aspects of my life
from the first day I came to Stockholm. Thank you very much for all your efforts to make this
step a reality.
Special thanks to my supervisor, Professor Ivars Neretnieks, for so many interesting and
enlightening discussions throughout the research work. Your knowledge and encouragement
always gave me driving force not only for the research but also for the life. I highly appreciate
the good times and the interesting discussions in the social events.
My sincere gratitude also goes to Professor Luis Moreno, for his much valuable advices during
these years, scientifically and non-scientifically. I appreciate very much the time you have spent
for questions, discussions and solutions to the problems.
The Swedish Nuclear Fuel and Waste Management Company (SKB) is gratefully
acknowledged for the financial support to this project. It has been very interesting and
stimulating to participate in this project, and for that I would like to thank especially Björn
Gylling.
Thanks to all the staff and colleagues at the divisions of Chemical Engineering and Transport
Phenomena. Many thanks to Associate Professor Joaquin Martinez and Associate Professor
Kerstin Forsberg for all their help and consideration. Big thanks to Ann Ekqvist, Helene Hedin,
Vera Jovanovic for always being friendly to give kind assistance in all administrative matters.
My last, but not least, words belong to my dearest ones, my wonderful parents, for their endless
love, support and encouragement throughout my life. I would like to express my deepest
gratitude for all your understanding and patience. I love you and dedicate this thesis to you.
vii
List of papers included in the thesis
I. Solute transport in fractured rocks with stagnant water zone and rock matrix composed
of different geological layers–Model development and simulations
Batoul Mahmoudzadeh, Longcheng Liu, Luis Moreno and Ivars Neretnieks
Water Resources Research, 49: 1709–1727 (2013).
Contribution: Main author, formulating the model and finding the analytical solutions, making
the simulations and discussing the results.
II. Solute transport in a single fracture involving an arbitrary length decay chain with
rock matrix comprising different geological layers
Batoul Mahmoudzadeh, Longcheng Liu, Luis Moreno and Ivars Neretnieks
Journal of Contaminant Hydrology, 164: 59–71(2014).
Contribution: Main author, formulating the model and finding the analytical solutions, making
the simulations and discussing the results.
III. Solute transport through fractured rock: radial diffusion into the rock matrix with
several geological layers for an arbitrary length decay chain
Batoul Mahmoudzadeh, Longcheng Liu, Luis Moreno and Ivars Neretnieks
Journal of Hydrology, 536: 133–146 (2016).
Contribution: Main author, formulating the model and finding the analytical solutions, making
the simulations and discussing the results.
IV. Evolution of fracture aperture mediated by dissolution in a coupled flow channel–
rock matrix–stagnant zone system
Batoul Mahmoudzadeh, Longcheng Liu, Luis Moreno and Ivars Neretnieks
Submitted for publication (2015).
Contribution: Main author, formulating the model and finding the analytical solutions, making
the simulations and discussing the results.
These papers are appended at the end of the thesis.
Paper I: © 2013 American Geophysical Union.
Paper II: © 2014 Elsevier B.V.
Paper III: © 2016 Elsevier B.V.
viii
Table of contents
Abstract ...................................................................................................................................... v
Sammanfattning ........................................................................................................................ vi
Acknowledgements ................................................................................................................... vii
List of papers included in the thesis ........................................................................................ viii
1
2
3
4
5
Introduction....................................................................................................................... 1
Transport of a stable nuclide ........................................................................................... 7
2.1 Mathematical model................................................................................................. 8
2.2 Solution in the Laplace domain ............................................................................. 13
2.3 Simulations and discussion .................................................................................... 15
Transport of nuclides in an arbitrary length decay chain .......................................... 18
3.1 Mathematical model............................................................................................... 19
3.2 Solution in the Laplace domain ............................................................................. 21
3.3 Simulations and discussion .................................................................................... 23
Evolution of Fracture Aperture..................................................................................... 28
4.1 Mathematical model............................................................................................... 30
4.2 Fracture closure rate ............................................................................................... 34
4.3 Simulations and discussion .................................................................................... 35
Concluding Remarks ...................................................................................................... 39
Notations ................................................................................................................................. 42
References ............................................................................................................................... 47
Appended papers
ix
As far as the laws of mathematics refer to reality, they are not certain,
and as far as they are certain, they do not refer to reality.
Albert Einstein
x
1
INTRODUCTION
In many countries, high-level radioactive wastes will be deposited in deep geological
repositories in crystalline rocks (Li and Chiou, 1993; Neretnieks, 1993; Gylling et al, 1998).
Since the canisters encapsulating the waste may eventually leak due to defect, corrosion or any
other reasons, it is vital to develop models that may be used to efficiently describe the process
of reactive transport of radionuclides through fractured rocks. The aim is to aid in safety and
performance assessment of repositories for spent nuclear fuel and other radioactive wastes.
Consequently, many models have been developed over recent years that simplify the fracturematrix system to different extents but all account for the most important mechanisms governing
nuclide transport in fractured rocks. The differences between these modeling approaches have
been discussed by Selroos et al. (2002). It was emphasized, in particular, that the discrete
fracture network model, DFN, is based essentially on the assumption that some fractures are
fully open over their entire size, and that they form a conductive network by intersecting each
other (Hartley and Roberts, 2012; Dershowitz et al., 1999). However, the premise that rock
fractures are either fully open or closed is not supported by field observations. Instead, these
observations suggested that fracture surfaces are uneven, and groundwater flows mainly
through open parts of fractures whereas there is a negligible flow through the rock matrix
(Birgersson et al., 1992, 1993; Abelin et al., 1991, 1994; Tsang and Neretnieks, 1998; Stober
and Bucher, 2006). The narrow conductive parts will form “flow channels” (Tsang and
Neretnieks, 1998) and subsequently a network for water flow and solute transport from the
repository to the biosphere (Neretnieks, 1993). Based on this concept, the channel network
model, CNM, has been developed and used for solute transport simulations in fractured rocks
(Gylling, 1997; Gylling et al., 1999; Moreno and Neretnieks, 1993a; Moreno et al., 1997). It
accounted for the fact that the low permeability porous rock matrix plays a dominant role on
retarding solute transport. The solutes, which are carried by the flowing water in the channel,
can diffuse in and out of the immobile groundwater in the rock matrix, through “matrix
diffusion” process that makes a much larger water volume accessible for the nuclides to reside
in (Neretnieks, 1980; Sudicky and Frind, 1984). As a result, sorbing nuclides would find more
mineral surfaces in the matrix to be attached and many nuclides could be retarded sufficiently
to decay to insignificant concentrations.
Both DFN and CNM models and many other models (Poteri and Laitinen, 1999; Hartley, 1998;
Dershowitz et al., 1998) did not, however, consider that in heterogeneous fractured media
where the fluid velocity field varies, there are some regions in the fracture plane, in which
1
advective transport is essentially negligible. The formation of these low velocity regions may
be because the fracture has a small aperture there or because it contains more particle fragments
forming a porous medium with low permeability. We call such regions with much less flow,
“stagnant water zones”. Wherever present, the solutes carried by the flowing water in the
channel may first diffuse into the stagnant water zones in the fracture plane and then from there
diffuse into the porous rock matrix. For this reason, Neretnieks (2006) addressed the influence
of the stagnant water zone on solute transport in fractured rocks. It was found that diffusion
into the stagnant water zone could significantly retard solute transport, especially when the
zone is wide. This gives the solute access to a larger surface over which diffusion into the rock
matrix takes place. As a result, considerable amounts of nuclides may be retained for a long
time in the rock matrix adjacent to the stagnant water zone, instead of directly diffusing into
the rock matrix adjacent to the flow channel. The diffusion into the stagnant water zones in the
fracture plane can be accounted for in the CNM, but not in the DFN models because of its main
assumption of fully open or closed fractures.
Furthermore, in most of the models for flow and solute transport through fractured rocks, it is
assumed that the rock matrix is a homogenous medium with either infinite or finite thickness
(Neretnieks, 1980; Moreno and Neretnieks, 1993b; Moreno et al., 1996; Barten, 1996; Grisak
and Pickens, 1980; Tang et al., 1981; Sudicky and Frind, 1982; Davis and Johnston, 1984; van
Genuchten, 1985; Zimmerman and Bodvarsson, 1995; Sharifi Haddad et al., 2013). However,
the structure of the rock matrix, in reality, is more complex as the rock may be altered near the
fracture (Löfgren and Sidborn, 2010b; Cvetkovic, 2010; Piqué et al., 2010). As a result, it is
anticipated that the altered parts of the rock may strongly affect the transport of solutes in
fractured media because they have significantly different retardation capacities than the intact
wall rock (Selnert et al., 2009).
The geological materials in the altered zone could, as shown in Figure 1, be made up of fault
gouge, fracture coating, fault breccia, mylonite, cataclasite, and altered rock etc. (SKB, 2003).
The altered rock appears mostly to be the result of brittle reactivation of old ductile/semi-ductile
deformation zones, while the fault breccia and fault gouge originate mainly from the
movements along fault planes and therefore are distributed in variable amounts and proportions
over the fracture and structure planes (Löfgren and Sidborn, 2010a; Löfgren and Sidborn,
2010b). In spite of this knowledge, however, it is difficult to identify all above-mentioned
matrix layers with distinctively different geological properties in field observations. For this
reason, it is common to simplify the structure of the altered zone somehow and consider it as a
mixture of some layers. In line with this approach, Tsang and Doughty (2003) conceptualized
the rock matrix as an assemblage of the intact wall rock and several layers containing the altered
material with an enhanced porosity, in order to account for their differences in geological
properties. Moreno and Crawford (2009) also considered the rock matrix as being composed
of several geological layers in modeling solute transport in fractured rocks. They found that the
layers close to the fracture surface might be important in determining the behavior of tracer
retardation during site characterization. The more deeply located layers in the rock matrix may,
however, have a large impact on the nuclide transport under the conditions prevailing at the
timescales of the performance assessment.
2
Figure 1. A sketch of a typical conductive structure of different rock layers, based on conceptual model taken by
SKB (2002a).
With this understanding, the work presented in the thesis attempts to get an insight into the
synchronous and the overall effects of both the layered rock matrix and the stagnant water zone
in retarding solute transport. The constructed model combines, in essence, the two models
developed by Moreno and Crawford (2009) and by Neretnieks (2006), respectively, based on
an idealization of the basic building block of a heterogeneous rock domain. It accounts for not
only the stagnant water zone with a finite width lying in the fracture plane but also the rock
matrix, as schematically shown in Figure 2, composed of both the intact wall rock with a finite
thickness and the altered zone that may be made up of altered rock, cataclasite, fault gouge and
fracture coating etc. all at the same time. In addition, the model considers that the rock matrices
adjacent to both the flow channel and the stagnant water zone may have different structures and
geological properties. The effects of physical and chemical heterogeneities, as studied by Deng
et al. (2010), can thus be taken into account by considering the fractured rocks as a series of
blocks with piece-wise constant geological properties.
3
Fracture coating I
Flowpath
Fracture coating II
Fault gauge II
Fault gauge I
Cataclasite II
Cataclasite I
Altered rock II
Altered rock I
Intact wall rock II
Intact wall rock I
Figure 2. Schematic conceptualization of two layered rock volumes.
In addition, it is noticed that the diffusion in the rock matrix is assumed, in most models, to be
one dimensional and orthogonal to the fracture surface where the fracture is conceptualized as
a rectangular channel. This is true when a wide channel is involved in the system. However,
since the flow channels in fractures are most possibly narrow, the matrix diffusion should be
more realistically modeled as radial diffusion from a cylindrical channel, especially when the
diffusion penetration depth grows larger than the width of the channel. Moreover, it should be
stressed that, although fully cylindrical channels are seldom found in nature, there are some
cases where cylinder-like channels may be more adequately used for modeling fluid flow and
solute transport. For instance, circular layered rock formations might be built up during the
completion (rock cracking, mud contamination etc.) of a horizontal well used to inject the
waste. Such cylinder-like channels can also be found at fracture intersections in fractured rocks.
Around these channels it may be expected that over time, chemical changes will lead to
development of circular matrix layers when the diffusion/reaction penetration depth becomes
larger than the channels width.
As a consequence, some theoretical and experimental studies have been performed to describe
the radial diffusion of solutes into a soil or rock matrix from a cylindrical macropore or fracture
for different boundary conditions (van Genuchten et al., 1984; Rasmuson and Neretnieks, 1986;
Carrera et al., 1998; Rahman et al., 2004; Cihan and Tyner, 2011; Chopra et al., 2015). The
radial diffusion has been shown to be more effective than linear diffusion in retarding solute
transport through a channel with the same flow-wetted surface (Rasmuson and Neretnieks,
1986). It has also been shown that radial diffusion gives access to a larger rock matrix volume
than linear diffusion and thus causes larger retardation (Neretnieks, 2015).
The developed model is then extended to study reactive transport through either a rectangular
or a cylindrical channel with the rock matrix composed of several geological layers with
different properties. The aim is to explore the relative impact and contribution of the altered
rock matrix and the decay chain on transport of nuclides through a rectangular/cylindrical
channel with linear/radial matrix diffusion. For simplicity, however, the model ignored the
presence of the stagnant water zones in the fracture plane at this stage. Nevertheless, the
analytical solutions, obtained in the Laplace domain, are applicable to the cases involving an
arbitrary-length decay chain, where the rock matrix consists of any number of different
geological layers. These analytical solutions can be used for prediction of nuclide transport
through complex networks of channels, and serve as verification tools for the numerical codes.
With the use of the developed models, the simulation results obtained from the rectangular
4
channel with linear matrix diffusion are also compared with those obtained from the cylindrical
channel with radial matrix diffusion.
As mentioned above, the flow and transport in a fractured rock mass was conceptualized in the
CNM to take place in a three-dimensional network of channels; the solute from one channel
enters one or more channels at an intersection. This channel network model obviates the need
to account explicitly for “hydrodynamic dispersion” as the conventional advection-dispersion
model does. Instead, as discussed by Neretnieks (1983), hydrodynamic dispersion may be more
adequately considered to be the result of different residence times in different pathways. This
eliminates the problem of the inconsistency between the use of the advection-dispersion
equation and the results obtained from field experiments; the advection-dispersion equation
requires a constant value of the “dispersion coefficient” whereas the observed “dispersion
coefficient” was shown to increase with distance in field experiments. In addition, it is also
found in numerous simulations with the channel network model that the large scale dispersion
in a network is dominated by velocity dispersion and matrix diffusion; and that conventional
“hydrodynamic dispersion” in individual channels has a small impact (Moreno and Neretnieks,
1991, 1993b; Gylling, 1997). For this reason and also because hydrodynamic dispersion in
channels in fractures is not well understood (Molz, 2015; Tsang et al., 2015), this mechanism
was not considered in the models developed for solute transport through individual channels in
the study.
Pressure dissolution
Free-face dissolution/Precipitation
Flow
Figure 3. Schematic view of main processes studied in this paper in control of fracture aperture evolution (left);
top view of fracture plane, showing preferential flow path and stagnant water zones wherein fine-grained
crystals are located (right).
In the models, we commonly assumed that the fracture is made of two parallel surfaces,
meaning that the fracture aperture is constant all the time. However, rock fractures may close
or open due to mechanical and chemical effects, which may considerably influence water flow
and solute transport in fractured rocks (Jing et al., 2013; Zhao et al., 2014). In particular,
dissolution and precipitation of minerals on fracture walls may either seal or widen a fracture.
Evolution of the fracture aperture over time in stressed natural fractures has been
experimentally observed (Polak et al., 2003; Beeler and Hickman, 2004, Liu et al., 2006;
Yasuhara et al., 2006a), and the main causes are still under discussion. Evolution of fracture
permeability has also been studied theoretically and modeled by accounting for dissolution and
5
precipitation (Yasuhara et al., 2003; Revil, 1999, 2001; Revil et al., 2006; Raj, 1982; Yasuhara
and Elsworth, 2006; Elsworth and Yasuhara, 2006, 2010; Yasuhara et al., 2006b; Neretnieks,
2014). The involved mechanisms in modeling are basically that the mineral tends to dissolve
where the higher stress is applied and will precipitate where the stress is lower. As a result, one
may expect that such mechanisms would also contribute to the evolution of natural fractures
where the two surfaces of a fracture are often in contact with each other in some asperities and
form gaps elsewhere, as shown schematically in Figure 3.
At contact points, the local stress is higher than the average stress on the whole fracture since
stress is concentrated on a small part of the entire fracture surface. At points where the local
stress is much larger than the asperity strength, mechanical crushing may occur. This will
produce small mineral fragments within the fracture. When fractured rocks contain water, the
minerals dissolve if the water is under-saturated. At contacting asperities with high
concentrated stress the solubility of minerals is higher than at fracture voids where only the
water pressure acts. The stressed minerals then tend to dissolve there and precipitate where the
stress is lower. This “pressure dissolution” process at fracture contacts may lead to closure of
the fracture as the stressed crystals shrink. In contrast, at fracture voids where the stress is lower
than contact asperities the process of “free-face dissolution” may occur and result in local
fracture widening if the water is under-saturated (Liu et al., 2006). Recent developments and
investigations have shown that a relatively small change of fracture aperture would
significantly influence solute transport in a rock fracture by changing the water flow rate and
the diffusional exchange between the flow channel and the porous rock matrix (Koyama et al.,
2008; Zhao et al., 2011). Therefore, it is worth to investigate how a fracture aperture would
change under the combined effect of stress and flow. The aim is, by developing a mechanistic
model that accounts for the contributions of both the matrix diffusion and the stagnant water
zones, to understand and highlight the most important conditions that govern the evolution of
fracture aperture in fractured rocks in the time scales relevant to the performance assessment
of deep geological repositories for spent nuclear fuel.
The rest of the thesis is organized as follows. In Chapters 2, the system for transport of a stable
nuclide is described conceptually and formulated mathematically in the time domain. The
analytical solutions in the Laplace domain are then derived for the concentration of the stable
nuclide at the outlet of a flow channel. Simulations are subsequently performed to investigate
the relative importance and contributions of the rock matrix layers and the stagnant water zone
in retarding solute transport in fractured rocks. Chapter 3 follows the same structure as Chapter
2, but for transport of nuclides in an arbitrary length decay chain; the simulation results obtained
from a rectangular channel with linear matrix diffusion are then compared with those obtained
from a cylindrical channel with radial matrix diffusion. In Chapter 4, a model is presented and
simulations are carried out to describe the evolution of fracture aperture under the combined
effect of stress and flow. The concluding remarks are given in Chapter 5.
6
2
TRANSPORT OF A STABLE NUCLIDE
In this chapter we consider the transport of a stable nuclide in fractured rocks. The system under
study is schematically shown in Figure 4, and has been well discussed in Paper I. It is an
idealization of the basic building block of a network of intersecting fractures with distributions
of lengths, orientations and positions (Moreno and Neretnieks, 1993b; Gylling et al., 1999). As
discussed previously, the stagnant water zone may exist in the cases involving narrow channels
in the fracture plane. To investigate the effect of the stagnant water zone on solute transport in
fractured rocks, where the rock matrix is composed of several geological layers with different
properties, a reasonable simplifying model is developed.
2Wf
2Ws
2bf
zs
z
δfi
δsi
z=0
zs = 0
x
u
2bs
y
y=0
x=0
Figure 4. Flow in a channel in a fracture from which solute diffuses into rock matrix as well as into stagnant
water in the fracture plane and then further into the rock matrix.
Figure 4 illustrates that the flow is assumed in the model to take place between two smooth
parallel surfaces separated by a distance 2bf, forming a rectangular channel of width 2Wf. The
water velocity, u, is assumed to be uniform across the flow channel. Likewise, the stagnant
water zone next to the flow channel is conceptualized as a rectangular cuboid with the same
length as the flow channel but different aperture and width; they are 2bs and 2Ws, respectively.
The rock matrix adjacent to the flow channel is assumed to consists of nf layers with different
thicknesses of 𝛿𝑓𝑖 , i = 1, 2, …, nf; the layers close to the fracture surface with i < nf constitute
7
the altered zone, while the last one represents the intact wall rock. Similarly, the rock matrix
adjacent to the stagnant water zone is considered to make up of ns layers with different
thicknesses of 𝛿𝑠𝑖 , i = 1, 2, …, ns; the layers close to the stagnant water zone with i < ns also
constitute the altered zone, while the last one stands for the intact wall rock.
When applied for practical use, however, we may simply assume that the rock matrices adjacent
to both the flow channel and the stagnant water zone have identical structures and geological
properties. In addition, the geological materials in the altered zone such as breccia, mylonite
and cataclasite etc. may also be lumped into only one layer to share the same properties. Despite
these simplifications, it should be possible to choose the flow parameters and the geological
parameters such that pessimistic estimates can be obtained for the sake of safety and
performance assessment of a deep repository for spent nuclear fuel.
Based on the idealized geometry of the system, as shown in Figure 4, the model accounts for
advection through the flow channel and diffusion of solute from the flow channel directly into
the adjacent rock matrix. The solute may also at first diffuse into the stagnant water zone and
then from there into the rock matrix adjacent to it. Although in Figure 4 diffusion in both the
stagnant water zone and the rock matrix is shown only in the left and up directions, the model
also considers diffusion in opposite directions. Moreover, linear equilibrium sorption onto both
the fracture surface and the rock matrix has been included. The advection within the rock matrix
is, nevertheless, ignored since the permeability of the geological layers of the rock matrix is
generally low. As a consequence, solute transport in the rock matrix results solely from
molecular diffusion in the model. Furthermore, a complete mixing of the solute across the flow
channel has been postulated (Buckley and Loyalka, 1993) in the model, so that it is unnecessary
to take the transverse dispersion in the flow channel into account. It should be noted that, when
necessary, this phenomenon could be readily incorporated into the model but it was shown to
have a negligible effect on solute retardation compared to the other phenomena considered.
Likewise, the longitudinal dispersion is also neglected as it is generally dominated in fractured
media by the differences in the residence times of solutes transported along different flow paths
and hence it does not need to be treated on the level of individual flow channels (Gylling, 1997).
Nevertheless, the general procedure for combining longitudinal dispersion with any retention
model could be followed (Painter et al., 2008).
2.1
Mathematical model
By using the coupled 1-D approach (Hodgkinson and Maul, 1988), the system under
consideration can be identified as one made up of four subsystems; the flow channel, the
stagnant water zone, the rock matrix adjacent to the flow channel, and the rock matrix adjacent
to the stagnant water zone. As illustrated in Figure 5, these subsystems can be characterized by
nine parameter groups that describe solute transport properties within each subsystem and/or
through the interface between the subsystems. The physical meanings of these characteristic
parameters are given in Table 1, and a complete list of symbols, subscripts, and superscripts
used in the mathematical model is given at the end of the thesis.
The 1-D approach essentially models a 3-D system as an assemblage of 1-D subsystems, in
which solute transport is assumed to take place only in one direction. The exchange of solute
between the subsystems is, then, described as a sink or source located at the interface that
8
couples the 1-D transport equations. Since the mathematical model has been detailed in Paper
I, only the governing equations are presented here for ease of reference.
Altered zone
Intact wall rock
i
 Ds
MPG is
i
 Df
MPG if
Fs
Ff
s
N
Stagnant water
f
Flow channel
Figure 5. Schematic side view of subsystems with their characteristic parameters.
2.1.1 Solute transport in the flow channel
Based on the above considerations, the equation of continuity describing the transport of a nonradioactive solute in the flow channel can be written as,
Rf
C f
C f
b D C s
 u
 s s
t
x b f W f y

y 0
Def1 C 1pf
bf
z
(1)
z 0
where y and z are the coordinates directed into the stagnant water zone and into the rock matrix
adjacent to the flow channel, respectively, and both are perpendicular to the channel. The
subscripts f and s have been used to indicate the flow channel and the stagnant water zone,
respectively.
The first term at the right-hand side of Equation (1) describes the advection, while the last two
terms describe the mass transfer at the interfaces between the flow channel and the stagnant
water zone and between the flow channel and the first layer of the rock matrix adjacent to it,
respectively.
The initial condition is,
C f ( x,0)  0
(2)
while the boundary condition for a time dependent concentration at the inlet is given by,
C f (0, t )  Cin (t )
(3)
9
Table 1. Characteristic parameters.
Notation
(dimension)
Definition
Physical significance
MPGif
MPGif  Defi Ripf  ipf
The material property group, for the rock matrix layers
adjacent to the flow channel, measures the ease with which
solute diffuses into the pore water, but mostly the capacity
to retain solute, within the matrix layer. A high value of it
means that the matrix layer can hold a large amount of a
given solute in both the dissolved and the adsorbed states.
MPGis  Desi Rips ips
The material property group for the rock matrix layers
adjacent to the stagnant water zone has exactly the same
meaning as the material property group for the rock matrix
adjacent to the flow channel.
Ff
(TL-1)
Ff   f RSV
The Ff-ratio is the ratio of the flow-wetted surface of the
flow channel to the advection conductance (volumetric
water flow rate). A high value of the Ff–ratio means that a
large amount of solute carried by the flowing water can
quickly be transported into the rock matrix.
Fs
(TL-1)
W
Fs  s
Ds bs
(LT-1/2)
MPG is
(LT-1/2)
2
τf
(T)
f 
τs
(T)
 Dfi
(T)
The characteristic time for advection in the flow channel
which is actually the water residence time but can also be
regarded as the solute travel time along the flow channel
without accounting for any interaction with the
surroundings.
x
u
Ws
Ds
The characteristic time for solute diffusion in the stagnant
water zone which characterizes solute travel time in the
direction perpendicular to the flow channel.
( if ) 2
The characteristic time for solute diffusion in the ith layer
of the rock matrix adjacent to the flow channel.
2
s 
i
 Df

i
 Ds
(T)

N
(-)
N
The Fs-ratio is the ratio of the wetted surface of the rock
matrix adjacent to the stagnant water zone (the stagnantwater-wetted surface) to the diffusion conductance of the
stagnant water zone. It describes the competition between
two important processes governing the rate of solute
transport; diffusion into the stagnant water zone and
diffusion from the water into the adjacent rock matrix. A
large value of the Fs–ratio indicates that a large amount of
solute diffusing from the flow channel into the stagnant
water zone can quickly be sucked by the adjacent rock
matrix.
Dafi
( si ) 2
 i
Das
The characteristic time for solute diffusion in the ith layer
of the rock matrix adjacent to the stagnant water zone.
Ds xbs / Ws
ub f W f
The N-ratio is the ratio between the characteristic rate of
diffusion into the stagnant water zone and the characteristic
rate of advection through the flow channel. It quantifies the
fraction of solute that can depart from the flow channel into
the stagnant water zone. A high value of the N–ratio
indicates that the stagnant water zone has a large capability
to capture the solute passing by and then deliver it from
there into the adjacent rock matrix.
i
Ds
10
2.1.2 Solute transport in the stagnant water zone
Assuming that the diffusion in the direction parallel to the flow is negligible, the equation of
continuity describing the transport of solute in the stagnant water zone, Cs, is given by,
Rs
1 C 1
C s
 2 C s Des
ps
 Ds

2
t
b

z
y
s
s
(4)
z s 0
where zs is the coordinate directed into the rock matrix adjacent to the stagnant water zone and
it is perpendicular to the interface between the stagnant water zone and the adjacent rock matrix.
The last term at the right-hand side of Equation (4) describes the mass transfer across the
interface between the stagnant water zone and the rock matrix adjacent to it.
The initial condition is,
C s ( x , y ,0 )  0
(5)
and the boundary conditions are given by,
Cs ( x,0, t )  C f ( x, t )
(6)
and
C s
y
0
(7)
y  2Ws
The processes of solute transport in the flow channel and in the stagnant water zone are thus
coupled through Equation (6), which describes the continuity of the aqueous concentration of
the solute.
2.1.3 Solute transport in the rock matrix adjacent to the flow channel
Accounting for molecular diffusion and linear equilibrium adsorption, the equation of
continuity describing the transport of a non-radioactive solute in the pore water of the ith layer
of the rock matrix adjacent to the flow channel, C ipf , is, i = 1,2,…,nf,
C ipf
t
 Dafi
 2 C ipf
for
z 2
where we have defined z if ,in 
z if ,in  z  z if ,out ,
i
i
k 1
k 1
(8)
  kf 1 , z if ,out    kf and  0f  0 .
The subscripts “in” and “out” indicates the beginning and the end of the matrix layer. The
apparent diffusivity Dafi is defined as, i = 1,2,…,nf,
11
Dafi

Defi
(9)
R ipf  ipf
The initial conditions are, i = 1,2,…,nf,
C ipf ( x, z,0)  0
z if ,in  z  z if ,out ,
for
(10)
while the boundary conditions are given, respectively, by,
C ipf ( x, z, t )  C ipf1 ( x, z, t )
at
z z if ,in ,
(11)
at
z z if ,out ,
(12)
and
Defi
C ipf
z
 Defi 1
C ipf1
z
n 1
where we have defined C 0pf  C f ( x, t ) and Deff
0.
When i = 1, Equation (11) describes the continuity of the aqueous concentration of the solute
at the interface between the flow channel and the first layer of the rock matrix adjacent to it.
This couples the behavior of solute transport in the flow channel and in the adjacent rock matrix.
2.1.4 Solute transport in the rock matrix adjacent to the stagnant water zone
Similar to Equation (8), the equation of continuity describing the transport of a non-radioactive
solute in the pore water of the ith layer of the rock matrix adjacent to the stagnant water zone,
C ips , can be written as, i = 1,2,…,ns,
C ips
t
D
i
as
 2C ips
z s
for
2
where we have defined z si ,in 
z si ,in  z s  z si ,out ,
i
i
k 1
k 1
(13)
  sk 1 , z si ,out    sk and  s0  0 .
The apparent diffusivity Dasi has the same definition as Dafi in Equation (9), except for
changing the subscripts from f to s.
The initial conditions are, i = 1,2,…,ns,
C ips ( x, z s ,0)  0
for
z si ,in  z s  z si ,out ,
(14)
while the boundary conditions are given, respectively, by,
C ips ( x, z s , t )  C ips1 ( x, z s , t )
at
z s  z si ,in ,
and
12
(15)
Desi
C ips
z s
 Desi 1
C ips1
z s  z si ,out ,
at
z s
(16)
where we have defined C 0ps  Cs ( x, y, t ) and Desn 1  0 .
s
When i = 1, Equation (15) describes the continuity of the aqueous concentration of the solute
at the interface between the stagnant water zone and the first layer of the adjacent rock matrix.
It couples, therefore, the behavior of solute transport in the two neighboring subsystems.
It should be noticed that the mass transfer across the interface between the rock matrix adjacent
to the flow channel and the rock matrix adjacent to the stagnant water zone is not accounted for
in the model. The reason for this is that, the diffusion in the stagnant water zone is fast compared
to that in the low porosity rock matrix and therefore the latter can be neglected. In other words,
diffusion in the x and y directions within the rock matrix is too small compared to diffusion in
the z direction.
2.2
Solution in the Laplace domain
The governing partial differential equations for solute transport given above can be solved by
applying the Laplace transformation approach (Watson, 1981). It removes the time variable
leaving a system of ordinary differential equations, the solutions of which yield the transform
of the aqueous concentrations as a function of space variables.
Since the Laplace transform technique allows one to get the aqueous concentration in the rock
matrix in the Laplace domain without obtaining the concentration profiles within the flow
channel and the stagnant water zone, it is preferable to begin with the Laplace-transformed
equations for solute transport in the rock matrices and then continue with the equations in the
stagnant water zone and the flow channel. The detailed mathematical procedure can be found
in Paper I, and the results is given in what follows.
In the case involving only a non-radioactive solute, the solution for the Laplace-transformed
concentration in the flow channel is found to be,

 
C f ( x, s)  Cin exp   f exp  N s tanh(2 s )

(17)
where and henceforth the over-bar is used to denote the Laplace-transformed quantities, and s
stands for the Laplace transform variable.
In Equation (17),  f is defined as,
 f  R f  f s  Ff P f s
(18)
where we have introduced three characteristic parameters τf, Ff and N. For ease of reference,
these parameters together with other characteristic parameters are presented in Table 1 with
their definitions and physical significance. Likewise,  s is defined for the stagnant water zone
in the same way as Equation (18) except for changing the subscripts from f to s.
The Pf-parameter in Equation (18) is given by,
13
Pf 
k
| M kf 1 |
MPG kf (1) k tanh( kf )  if cosh( if )
f |
k 1
i 1
nf
 |M

(19)
where Mf is a matrix that characterizes solute transport properties within each layer of the rock
matrix and through the interface between adjacent layers, and it is defined as,
i 1
 i
i
j


sinh(

)
sinh(

)
cosh( lf ) j  i
 f
f
f
l  j 1

  if cosh( if )
j i

j
 i MPG f
j  i 1
 f MPG i
f

0
otherwise

 
M f  m if, j
m if, j
with
n f n f
(20)
In Equation (19), Mfk+1 stands for the sub-matrix of Mf whose first to the kth rows and also
columns have been deleted and |Mf| and |Mfk+1| are the determinant of Mf and Mfk+1,
respectively.
The first exponential function in Equation (17) describes, then, the contribution of the
subsystem composed of the flow channel and the adjacent rock matrix in retarding solute
transport and retention, as  f given in Equation (18) is determined solely by the parameters
Rf, τf, Ff and Pf which are all related to the flow channel and the adjacent rock matrix. By the
similar token, the second exponential function describes the contribution of the subsystem
composed of the stagnant water zone and the adjacent rock matrix in retarding solute transport
and retention with the N-ratio accounting for the fraction of solute that can depart from the flow
channel into the stagnant water zone.
Since Equation (17) expresses the ratio of the outlet and the inlet concentrations (in the Laplace
domain) as a product of two exponential functions, it can be easily extended to the case where
the transport path consists of a series of piece-wise constant channels, along each of which the
surrounding media have constant geological properties. The result is, for L successive channels,

C f  Cin exp 



 if  exp 
i 1


L

 N
L
i 1
i


 is tanh(2  is ) 

(21)
It signifies that, when applied for practical use such as in predicting the behavior of nuclide
transport in fractured media from a repository to the biosphere, it is unnecessary to calculate
solute concentration at every outlet of the channels by performing the inverse Laplace
transform of Equation (17). Rather, one only needs to follow the flow channels along which
solute travels and document the involved characteristic parameters. The information would then
be sufficient to obtain solute concentration, i.e. the breakthrough curve, once at the final end of
the transport path, by use of De Hoog algorithm (De Hoog et al., 1982) to numerically transform
Equation (21) back to the time domain. This would not only save computation time dramatically
but also give accurate predictions, provided that the assumptions underlying Equation (21) are
justified to be fair for the purpose of safety assessment of a deep geological repository for spent
nuclear fuel.
14
2.3
Simulations and discussion
In this section, several cases of varying complexity are simulated to investigate the relative
importance and contributions of different processes and mechanisms on solute transport and
retardation in fractured rocks. In the simulations, we define C f (0, t )  c0 (t ) , for a point-source
boundary, where δ is the Dirac delta function, and c0 is not a concentration but the ratio between
the instantaneous mass injected and the volumetric flow rate through the channel. This gives
C f (0, s )  c0 , and as a result the breakthrough curve (the response function) C f c0 to the pulse
injection, can readily be obtained by use of the inverse Laplace transform (Watson, 1981) of
Equation (17).
The data used in the base case are presented in Tables 2, 3 and 4 for the flow channel, the
stagnant water zone and the adjacent rock matrix composed of five layers, respectively. The
solute-dependent properties in MPG such as the diffusion coefficient and the distribution
coefficient are those for cesium, while the geometry of the system and the geological properties
of the rock matrix are representative of those from field observations (SKB, 2006). The
database available for transport and sorption properties of the rock matrix, such as thickness
and porosity, can also be found in SKB report (2003). To simplify the simulation without losing
generality, however, we assume that the rock matrices adjacent to both the flow channel and
the stagnant water zone have identical structures and properties.
Table 2. Flow channel properties.
Symbol (unit)
bf (m)
Rf (–)
u (m/yr)
Wf (m)
x (m)
Value
1.0e-04
1
5
0.1
50
Definition
Rectangular channel half-aperture
Retardation coefficient
Groundwater velocity
Rectangular channel half-width
Channel length
Table 3. Stagnant water zone properties.
Symbol (unit)
bs (m)
Ds (m2/yr)
Rs (–)
Ws (m)
Value
1.0e-04
0.0315
1
0.5
Definition
half-aperture
Diffusion coefficient
Retardation coefficient
half-width
Table 4. Rock matrix properties for cesium.
Rock zones
Intact wall rock
Altered rock
Cataclasite
Fracture coating
Fault gouge
MPG (m2/yr)0.5
0.0057
0.0138
0.0179
0.1200
0.6399
15
τD (yr)
1.43e06
3.81e04
128
0.02
0.7
A step-by-step procedure is, then, adopted to carry out sensitivity analysis, by changing one of
the characteristic parameters at a time. As detailed in Paper I, it starts from a simple case where
the rock matrix does not consist of any layers of altered zone with the aim of exploring the
effect of the intact wall rock. The rock matrix is thereupon extended to include one or four more
layers in order to facilitate understanding of the effect of the altered zone. Following this, the
stagnant water zone is taken into account but assuming that the rock matrix consists only of the
intact wall rock. Thereafter, the results with respect to the effects of several matrix layers and
also stagnant water zone are shown.
In most cases, solute transport in fractured rocks would take place in such a system that consists
of not only the flow channel, the stagnant water zone, the intact wall rock, but also the altered
zone which may be composed of altered rock, cataclasite, fault gouge and fracture coating etc.
all at the same time. This, or even more complex, system would certainly make it hard to clarify
the effects of different subsystems in retarding solute transport. However, based on sensitivity
analysis made in Paper I, it is anticipated that each of the subsystems would play similar roles
in determining the fate of solute transport even when everything is put together into the system.
To illustrate this, we compare in Figure 6 the breakthrough curves obtained for different cases
where the rock matrix consists of either none, one or four layers of altered zone, in addition to
the intact wall rock, with and without accounting for the existence of the stagnant water zone.
It is assumed that the altered zone consists solely of the altered rock in the case where only one
matrix layer of altered zone is present; otherwise it is composed of altered rock, cataclasite,
fault gouge and fracture coating.
Figure 6. Comparisons of breakthrough curves when there is/there is not stagnant water zone and rock matrix is
composed of two/five layers.
It is seen in Figure 6 that when the stagnant water zone is not accounted for, the altered rock
itself not only weakens the peak value of Cf (Cf,peak) but also delays the time at which the
response function reaches the peak (tpeak). When other layers of altered zone are also included,
however, the breakthrough curve only slightly shifts to the right, resulting in a minor decrease
in Cf,peak. This indicates that the layers other than the altered rock have a moderate effect in
16
retarding solute transport, although they have large material property groups, they are too thin
to have a significant influence on solute retardation.
More importantly, it is noted that the effects of the stagnant water zone in determining the
behavior of the breakthrough curves are always similar irrespective of how the rock matrix is
structured. It always results in a considerable decrease in Cf,peak and a significant increase in
tpeak, even if the altered zone consists solely of cataclasite, fault gouge or fracture coating
instead of the altered rock. The reason for this is that the structure of the rock matrix only
influences the equivalent material property group to some extent, if we take it as a whole as an
equivalent intact wall rock, but not the N–ratio, the Fs–ratio and the diffusion time  s . It is,
however, the latter characteristic parameters that play the key roles in determining the effects
of the rock matrix adjacent to the stagnant water zone in retarding solute transport under the
condition that the intact wall rock is thick enough to act as a deep trap for solute. Only in the
cases with a wide channel and a narrow stagnant water zone so that F𝑠 MPG1𝑠 is far smaller than
F𝑓 MPG𝑓1, the influence of the stagnant water zone in retarding solute transport might become
marginal. The smaller the F𝑠 MPG𝑠𝑚 , the less the rock matrix adjacent to the stagnant water zone
would contribute to the retardation of solute transport.
Therefore, it can be concluded that the key process in retarding the solute transport is nothing
but the matrix diffusion. The presence of the stagnant water zone allows the solutes to access a
larger fracture surface from which diffuse into the rock matrix continues. An additional space
becomes, then, available for solute transport in the rock compared to the simple system without
stagnant water zone. Although the results are shown and discussed only for a stable species in
this chapter, the same effects are expected to hold also for the transport of radionuclides in
fractured rocks.
17
3
TRANSPORT OF NUCLIDES IN AN
ARBITRARY LENGTH DECAY CHAIN
In this chapter, we consider migration of nuclides in an arbitrary length decay chain for two
cases of channel geometry: rectangular flow channel with linear matrix diffusion (LMD) and
cylindrical flow channel with radial matrix diffusion (RMD). The rock matrix is assumed in
both cases to be composed of nf geological layers with different thicknesses of 𝛿𝑓𝑖 , i = 1, 2, …,
nf, as schematically shown in Figures 7 and 8, respectively. To simplify the problem, somehow,
the stagnant water zone and its adjacent rock matrix are assumed to be absent in the system
studied, in order to facilitate comparison of the modeling results with those from existing
conventional models.
2Wf
2bf
z
δfi
z=0
x
u
x=0
Figure 7. Flow in a flat channel in a fracture from which solute diffuses into the adjacent rock matrix layers.
Noticeably, the channel width has been found to vary typically from a few mm to several tens
of cm (Moreno and Neretnieks, 1991; Abelin et al. 1983, 1985, 1994; Tsang and Neretnieks,
1998). This fact seems to favor the application of the model with RMD for practical use.
However, in field experiments the observation times are usually short, which makes the impact
of radial matrix diffusion on solute retardation to be small or negligible for wide channels.
Therefore, the model with LMD is more applicable in these cases. The strong effect of RMD
will become more pronounced mostly after tens to thousands of years, which is the time scale
of interest of this work for the sake of performance assessment of deep repositories for spent
18
nuclear fuel. In this study, comparisons between the results obtained from the model with LMD
and those from the model with RMD will be made, to explore the relative impact and
contributions of the altered rock matrix and the decay chain on transport of radionuclides in
fractured rocks.
δf3
δf2
δf1
rr0
f
Matrix
Diffusion
x=0
Figure 8. Flow in a cylindrical channel from which solute diffuses into the adjacent rock matrix layers.
3.1
Mathematical model
Again, by using the coupled 1-D approach, as was done by Hodgkinson and Maul (1988) and
Neretnieks (2006), two subsystems can be identified in the systems shown in Figures 7 and 8,
respectively; the flow channel and the rock matrix composed of different geological layers, for
which the equations of continuity describing the transport processes can be formulated
individually. These two subsystems can also be characterized by those parameters given in
Table 1, as detailed in Papers II and III.
Without further discussions, the following sections simply present the systems of governing
equations, which are valid for both models involving either rectangular or cylindrical channels.
3.1.1 Solute transport in the flow channel
Given a radioactive nuclide decay chain of length m, the equation of continuity describing the
transport of the jth nuclide in the flow channel can be expressed as, j = 1,2,…,m,
Rf ,j
C f , j
t
 u
C f , j
x
 RSV De1, j
C 1p, j
r
 R f , j  j C f , j  R f , j 1 j 1C f , j 1
(22)
r rf
where we have defined λ0 = 0, and RSV as the ratio of the flow-wetted surface of the flow channel
to its volume. For the rectangular channel where the flow takes place between two parallel
surfaces separated by a distance 2bf (see Figure 7), the parameter RSV would become 1/bf, while
19
for the flow channel with right cylindrical geometry of radius rf (see Figure 8), the entity RSV
would be equal to 2/rf.
The first term at the right-hand side of Equation (22) describes the advection through the flow
channel, whereas the second term indicates the mass transfer at the interface between the flow
channel and the rock matrix adjacent to it. The third term represents the mass loss of member j
by decay, and the forth term denotes the mass ingrowth of member j by radioactive
disintegration of member j-1. Noticeably, this equation is valid for both cases involving either
rectangular or cylindrical channels. In the former case, the variable r will be changed to z, and
consequently the boundary r = rf will be replaced by z = 0.
Equation (22) is to be solved with the initial condition that, j = 1,2,…,m,
C f , j ( x,0)  0
(23)
and the boundary condition,
C f , j (0, t )  Cin, j (t )
(24)
3.1.2 Solute transport in the rock matrix
Accounting for diffusion and linear equilibrium adsorption, the transport equation describing
the transport of nuclide for the species j in the pore water of the ith layer of the rock matrix,
C ip, j , is given by, j = 1,2,…,m, i = 1,2,…,nf,
R ip , j
C ip , j
t
D ip , j   q C ip , j
r
 q
r
r r 

  R ip , j  j C ip , j  R ip , j 1 j 1C ip , j 1


for
i
rini  r  rout
(25)
i 1
i
k 0
k 0
i
   k and 𝛿 0 = 𝑟𝑓 .
where we have defined rini    k , rout
In Equation (25) the exponent “q” is equal to 0 in the case of rectangular channels and 1 in the
𝑖
𝑖
case of the cylindrical channels. Correspondingly, the boundaries 𝑟𝑖𝑛
and 𝑟𝑜𝑢𝑡
for the case of
𝑖
𝑖
cylindrical channels will be replaced by the boundaries 𝑧𝑓,𝑖𝑛 and 𝑧𝑓,𝑜𝑢𝑡 , respectively, in the case
of rectangular channels.
Equation (25) is to be solved with the initial condition that, j = 1,2,…,m, i = 1,2,…,nf,
C ip, j ( x, r,0)  0
for
i
,
rini  r  rout
(26)
at
r  rini
(27)
and the boundary conditions,
C ip, j ( x, r, t )  C ip,1j ( x, r, t )
and
20
Dei , j
C ip , j
r
 Dei , j1
C ip,1j
r
at
i
r rout
(28)
n 1
where we have defined C 0p, j  C f , j ( x, t ) and De,fj  0 .
Comparison of Equations (22) to (24) with Equations (1) to (3), and Equations (25) to (28) with
Equations (13) to (16), shows that, when radioactive decay is accounted for, the only
modification is to add one decay term and one ingrowth term into the governing equations. The
initial and the boundary conditions keep invariant for the subsystems involved. This principle
applies also to the case when the stagnant water zone and its adjacent rock matrix are taken into
account. As mentioned before, for the moment, we only consider a simplified system that
consists of a flow channel and an adjacent rock matrix. The aim is to explore the effect of
different geological layers on the transport of radioactive nuclides, and to compare the
difference between the radial matrix diffusion from a cylindrical flow channel and the linear
matrix diffusion from a rectangular flow channel.
3.2
Solution in the Laplace domain
The governing differential equations for solute transport given in the mathematical model can
be solved by applying the Laplace transformation approach (Watson, 1981), as was done also
for a stable nuclide in the previous chapter. The detailed mathematical procedure can be found
in Papers II and III. The final solution is presented in the following.
In the case involving a radioactive decay chain, the Laplace-transformed concentration vector
at the outlet of the flow channel where the effects of the stagnant water zone and its adjacent
rock matrix have been neglected can be written as,

~
Cf  Θ exp T Θ 1 Cin
(29)
which can be transformed back to the time domain numerically by use of e.g. the De Hoog
algorithm (De Hoog et al., 1982).
The solution given in Equation (29) is valid for both rectangular and cylindrical channels. In
~
~
this equation, the matrices Τ , Θ 1 and Θ are obtained from similarity transformations. Τ is a
diagonal matrix, Θ is a lower triangular matrix, and Θ 1 , which is the inverse of Θ matrix, is
also a triangular matrix. The reader is referred to Papers II and III for the definition of these
matrices and the general procedure to get the Laplace-transformed concentration in the rock
matrix and through the flow channel. The explicit analytical expressions for these matrices are,
in general, difficult to obtain. Nevertheless, they are available in some simplified cases. For
instance, in the case involving only one geological layer of the rock matrix and one single
~
nuclide, it can be shown that Θ  1 , Θ 1  1 , and the matrix Τ simplifies to,
~
T  R f,1τ f s  λ1   Ff Z11MPG11 s  1
(30)
The parameter Z11 in Equation (30) is defined, for the rectangular channel, as,
21
Z11  tanh(  1D,1 ( s  1 ) )
(31)
1
The two characteristic parameters 𝜏𝐷,1
and MPG11 follow the general definitions given in Table
𝑖
1. The characteristic diffusion time, 𝜏𝐷,𝑗
, and the material property group, MPG𝑗𝑖 , for the jth
nuclide through the ith layer of the rock matrix are then defined, respectively, as,

i
D, j
 

i 2
f
Dai , j
(32)
and
MPG ij  Dei , j R ip , j  ip , j
(33)
For the cylindrical channel, however, the parameter Z11 in Equation (30) becomes,
Z11
~
~ 1
~
~ 1
K1 (h11,1rin1 ) I1 (h11,1rout
)  I1 (h11,1rin1 ) K1 (h11,1rout
)

~1 1
~1 1
~1 1
~1 1
K 0 (h1,1rin ) I1 (h1,1rout )  I 0 (h1,1rin ) K1 (h1,1rout )
with
~
h11,1 
R1p ,1 s  1 
D1p ,1
(34)
where Iα and Kα are the modified Bessel functions of the first and the second kind of order α,
respectively.
It follows from Equations (29) and (30) that,

C f  Cin exp  R f ,1 f s  1   Ff Z11MPG11 s  1

(35)
This equation describes the Laplace-transformed concentration of a single nuclide along a
rectangular/cylindrical flow channel with the rock matrix composed of only one geological
layer. The first term inside the exponential function in Equation (35) describes the contribution
of advection in the flow channel to solute transport, as  f is the water residence time through
the flow channel. The second term describes the contribution of diffusion and adsorption in the
rock matrix to solute transport and retardation, since the entity Z11 , as given in Equations (31)
and (34) for the rectangular and cylindrical channels, respectively, is related only to the rock
matrix properties.
It can, thus, be justified that, by performing the inverse Laplace transform of Equation (29), the
outlet concentration in general cases involving not only decay chain but also different
geological layers for the rock matrix would be of the following form,
C f  f ( R f  f , Ff , MPG ij , Di , j ,  j , t )
(36)
22
This suggests that the characteristic parameters, as defined in Table 1, and the decay constants
would be sufficient to describe the fate of nuclide transport through a discrete fracture in
crystalline rocks.
On the other hand, it is noted that Equation (29) can easily be extended to a more complex case,
where the solutes transport through a series of channels with the properties of adjacent rock
matrices varying from channel to channel. The result for the outlet concentration from L
successive channels would, then, be given by,
Cf 
~
 Θ expTΘ 
L
1
(37)
L i 1 Cin
i 1
This indicates that, in assessing nuclide transport from a repository to the biosphere, the
characteristic parameters recorded for each of the channel-matrix blocks would suffice to get
the solute concentration at the final end of the transport path.
3.3
Simulations and discussion
To account for the effect of radioactive decay, we have simplified the system conceptualized
in Figure 4 to the one shown in Figure 7, where the stagnant water zone and its adjacent rock
matrix have been ignored. As detailed on Papers II and III, the difference between the effect of
the matrix diffusion of solutes from a rectangular flow channel and from a cylindrical flow
channel was particularly investigated. Similarly for a stable nuclide in the previous chapter, in
the simulations in this section a pulse-source boundary is considered for all nuclides at the
channel inlet. Analysis of the response function, defined as C f , j c0, j , at the channel outlet
would, then, give an insight into different mechanisms in retarding nuclide transport in
fractured rocks.
Table 5. Rock matrix properties.
Symbol (unit)
Dp (m2/s)
De (m2/s)
Rp (–)
δ (m)
εp (%)
Altered rock
9.28e-12
5.57e-14
6.66e03
0.2
0.6
Intact wall rock
6.13e-12
1.84e-14
1.33e04
1.0
0.3
Definition
Pore-water diffusivity
Effective diffusivity
Retardation factor
Thickness of layer
Porosity
Table 6. Radionuclides properties.
Symbol (unit)
Kd (m3/kg)
Rf (–)
t1/2 (yr)
Am243
1.48e-02
1.0
7.37e03
Pu239
1.48e-02
1.0
2.41e04
23
Definition
Volumetric sorption coefficient
Surface retardation factor in fracture
Half-life
To be specific, we consider a case where a two-member decay chain, Am243→Pu239, migrates
through a rectangular/cylindrical flow channel with simultaneous linear/radial diffusion into
the rock matrix comprising two geological layers. The geometrical data and physical properties
of the flow channel and the rock matrix layers are listed in Tables 2 and 5, respectively (SKB,
2006; SKB, 1999). The half-width of the rectangular channel is Wf = 0.05 m and the radius of
the cylindrical channel is rf = 2Wf/π. The nuclide properties for the two-member decay chain
are given in Table 6 (SKB, 1999; SKB, 2010). The other data have been chosen to resemble
the conditions in water-saturated granite rocks hosting a nuclear waste repository at about 500
m depth underground. The bulk density of the rock matrix is assumed to be 2700 kg/m3.
A stepwise procedure is also used to carry out sensitivity analysis, by changing or introducing
a parameter value at a time. As detailed in Papers II and III, we start from a simple case, where
the rock matrix consists of only one layer (intact wall rock), and decay is not considered for the
only nuclide involved, i.e. plutonium. The rock matrix is thereafter extended to have one more
layer (the altered rock), to explore the effect of the geological layer on radial matrix diffusion
(RMD) and also on linear matrix diffusion (LMD). Following this, the decay chain,
Am243→Pu239, is taken into account to explore in detail the difference between the effect of
radial and linear matrix diffusions on nuclide chain migration.
For the purpose of comparison, we kept the water velocity in both channels for radial and linear
matrix diffusions identical (Neretnieks, 2015) so as to have the same water residence time
 f  x u in channels. In general, however, the effect of  f on retarding solute transport is
negligible when the effect of matrix diffusion is large.
Matrix Diffusion
Matrix Diffusion
bf
rf
2bf
x=0
2Wf
x=0
Figure 9. Rectangular channel with linear matrix diffusion, LMD (left) and cylindrical channel with radial
matrix diffusion, RMD (right); r f  2W f  .
To keep the same flow-wetted surface and water velocity, the cylindrical channel can be
conceptualized as concentric cylinders, as shown in Figure 9, where the inner shell (dashed
line) is impermeable and the outer shell is of radius rf = 2Wf/π with Wf being the width of the
corresponding rectangular channel. The entity RSV, the ratio of flow-wetted surface of the
channel to its volume, in Equation (22) would then become 1 b f for both rectangular and
cylindrical flow channels.
24
As discussed above, in a simple case where only one layer involved in the homogeneous rock
matrix and only one single nuclide is considered, the outlet concentration is given in Equation
(35), where the only difference between rectangular and cylindrical channels is in the parameter
Z11 . In the case of linear matrix diffusion, it is simply a hyperbolic tangent term given in
equation (31). As a result, Z11 will approach unity in the limiting case as  1D ,1   for an infinite
extent of the intact wall rock. Subsequently, for a point-source boundary, the breakthrough
curve (the response function), C f c0 , can readily be obtained by applying the inverse Laplace
transform (Watson, 1981) on Equations (35) and (31), for t  R f ,1 f , as,
Cf
 (Ff MPG 11 ) 2

c0 
exp 
 1t 
 4(t  R f ,1 f )

2  (t  R f ,1 f ) 3


F f MPG 11
(38)
In the case of radial diffusion, however, the entity Z11 given in equation (34) will approach
~
~
K1 (h11,1rin1 ) K 0 (h11,1rin1 ) for infinite thickness of the intact wall rock. This term can be
approximated as, by using series expansions of the modified Bessel functions (Abramowitz and
Stegun, 1968) for arguments greater than 0.2,
Z11
~
K1 (h11,1rin1 )
1

~1 1  1  ~1 1
K 0 (h1,1rin )
2h1,1rin
(39)
Therefore, for short time periods, i.e. when s goes to infinity, the approximate analytical
solution for a cylindrical channel, can be obtained as, for t  R f ,1 f ,
C f c0 
 (Ff MPG 11 ) 2 De1,1 (Ff ) 2

exp 

 1t 
 4(t  R f ,1 f )

4 f
2  (t  R f ,1 f ) 3


F f MPG 11
(40)
Comparison of Equations (38) and (40) shows that the difference between linear and radial
matrix diffusion for a single nuclide makes itself felt only when exp  De1,1 (Ff ) 2 4 f starts to
deviate significantly from 1. This means that, when a rock matrix with large MPG is concerned,
the radial matrix diffusion will retard solute transport more efficiently than the linear matrix
diffusion.


On the other hand, in the limiting case as  1D,1  0 or equivalently as the penetration depth
 1D,1 (s  1 ) . As a result,
the cylindrical and rectangular channels would make no difference in retarding nuclide
transport and consequently we can write, by applying the inverse Laplace transform (Watson,
1981) on Equation (35),
1
 1f  0 , the entity Z1 in both Equations (31) and (34) will approach
C f c0  exp( 1t ) (t  R f ,1 f  Ff MPG 11  1D,1 )
(41)
This is the response function to the pulse injection for both rectangular and cylindrical channels
when the characteristic diffusion time into the rock matrix is very small, i.e.,  1D,1  0 .
This analysis indicates that the difference between the two models comes mainly from the role
of matrix diffusion in retarding solute transport. With small penetration depth, the role of rock
25
matrix on drawing the solutes off from the flow channel is trivial compared to the role of
flowing water on carrying the solutes along the channel. As a consequence, the two models
would give identical results. To better understand this, one may also conclude from Figure 9
that, when only one matrix layer (intact wall rock) is involved, the matrix volume in the linear
diffusion case is 4𝑊𝑓 𝛿 1 ∆𝑥 while it becomes 4𝑊𝑓 𝛿 1 ∆𝑥 + 𝜋(𝛿 1 )2 ∆𝑥 in the radial diffusion case.
Hence, when 𝑊𝑓 ≫ 𝛿 1 , or equivalently 𝜏𝐷 ⁄𝑊𝑓 → 0, the two cases would give essentially the
same matrix volume accessible for nuclide transport, leading to nearly identical outlet
concentrations.
With large penetration depth (compared to the width Wf or rf of the channel), however, the
radial matrix diffusion from a cylindrical flow channel would give solutes a much larger water
volume to transport and reside than the linear matrix diffusion from a rectangular flow channel.
As a result, it is not surprising to see in the simulations that much more solutes would be
retained in the rock matrix when radial diffusion takes place. This is especially true when the
diffusion penetration depth has become comparable to or exceed the width of the channel such
that the term De1,1 (Ff ) 2 4 f in Equation (40) begins to influence the result.
Figure 10. Comparison of the response function, for Pu239, between linear matrix diffusion (LMD) and radial
matrix diffusion (RMD) where the rock matrix is composed of one/two geological layers and decay chain is
not/is accounted for.
To further illustrate what happens in more general cases, we shown in Figure 10 the response
function for Pu23 as either a stable species or a nuclide in a first-order radioactive decay chain
of Am243→Pu23. A pulse-source boundary is considered for both nuclides at the inlet. The outlet
concentration of plutonium when the rock matrix is composed of either one (intact wall rock)
or two layers (an altered rock zone and the intact wall rock) is shown for the both cases of LMD
and RMD. It is seen, from the left figure, that the 0.2 m layer of altered rock plays an important
role in both rectangular and cylindrical models not only in decreasing the outlet concentration
but also in delaying the arrival time of nuclide at the channel outlet. The right figure illustrates
that accounting for two matrix layers and when decay chain is considered, the outlet
concentration of plutonium obtained from the cylindrical model is much lower than that
obtained from the rectangular model.
The difference between the output concentrations from the two cases will increase strongly by
either increasing the travel distance or decreasing the water velocity in the flow channel. As
26
discussed above, increasing the travel distance or decreasing the flowing water velocity will
make much more volume of the rock matrix accessible for nuclides to diffuse into and to retard
there in both the dissolved and adsorbed states for a longer time. This again indicates the
important role of matrix diffusion, which has a larger impact in radial diffusion since the
cylindrical channel, compared to the rectangular channel, will allow a larger volume of the rock
matrix to become available for retarding nuclide transport.
27
4
EVOLUTION OF FRACTURE APERTURE
In this chapter, we wish to explore how the aperture of a channel could evolve over time and
along the channel by dissolution/precipitation of the minerals of the rock, and to explore how
far along a channel changes in solute concentration in infiltrating water can propagate over
time. For this purpose, we develop a reasonably simple model in order to understand and
highlight some important conditions and phenomena governing the evolution of fracture
aperture. The system under study is schematically shown in Figure 11, and has been discussed
in more detail in Paper IV. A real flow path is obviously much more complex. The fracture
aperture varies not only along it but also across it. Therefore, complex flow paths rather than
straight flow channels are expected to form in fractures with variable apertures (Liu and
Neretnieks, 2005).
As a first attempt, however, we assume that the flow initially takes place between two parallel
surfaces separated by 2bf, forming a rectangular channel of width 2Wf, with a constant mean
velocity of u. As shown in Figure 11, the stagnant water zone next to the flow channel is
similarly conceptualized as one rectangular channel of 2bs aperture and 2Ws width. Moreover,
it is assumed that the channels in the fracture are held open by contact of the fracture asperities
on the fracture surfaces. The minerals at the asperities are subject to rock stresses, which are
considered to be present only in the stagnant water zone adjacent to the flow channel.
The rock matrix is porous and hence the mineral grains there can also dissolve or act as
precipitation surfaces depending on local conditions of over- or under-saturation. . In practice,
it has been shown that the rock matrix is actually a strong source/sink for dissolved minerals
such as quartz in the absence of flow, when the matrix porosity is larger than a few tenths of
percent (Neretnieks, 2014). In addition, it was found that when the stressed crystals are small,
less than a few mm, the diffusion in the thin water film between the crystal and the wall is fast,
leading the concentration to be essentially equal to that surrounding the crystal.
On the other hand, the volume of water film between the dissolving stressed crystal and the
fracture surface is so thin that it can be neglected compared to the crystal size (Revil, 2001).
The study of Neretnieks (2014) also suggested that there is no need to consider the details of
the transport of dissolved minerals in this thin film because the concentration in the thin film
between the crystal and the fracture plane is essentially the same everywhere and the same as
the concentration in the water surrounding the crystal when there is a strong sink/source by
matrix diffusion. Under these conditions, the dissolution rate of stressed faces of the crystal
depends practically only on the stress and the concentration in the water just outside the thin
28
film. In the present model the stressed crystals make up a stressed surface a fraction, α. Should
the crystals or the contact surface be very large, it may be necessary to account for the radial
concentration gradient in the thin water film.
Cpf (t,x,-,z)
2bs
Matrix
diffusion
z=0
z1 = 0
Diffusion
2bf
Lz
Cf (t,x,-,-)
Ly
z
Lx
x
y
x=0
2Wf
2Ws
y=0
Dissolving
free faces
Dissolving
free faces
Matrix
diffusion
Cps(t,x,y,z1)
Dissolving
free faces
2bs
Dissolving
stressed faces
Cs(t,x,y,-)
Cfilm(t,x,y,-)
σP
side view
Figure 11. Flow in a channel in a fracture where crystals are stressed in the stagnant water zone in the fracture
plane. The mass dissolved by pressure dissolution, can precipitate on the unstressed surfaces and also diffuse
into the adjacent porous rock matrix. Diffusion in the rock matrix and the stagnant water zone is shown only in
the upward and right direction but the model considers also diffusion in opposite directions.
With these understandings, the model accounts for free-face dissolution/precipitation of
minerals that make up the surfaces of the flow channel. In the stagnant water zone, pressure
dissolution occurs on the stressed surfaces and free-face dissolution/precipitation on the
unstressed surfaces. Dissolved minerals in flowing water and in stagnant zone can also diffuse
into and out of the adjacent rock matrix. Additionally, there is mineral exchange between the
flowing water and the stagnant water zone by diffusion. The effect of any hydrodynamic
dispersion is, however, neglected at this stage.
29
4.1
Mathematical model
In order to devise a model that can be solved analytically, a starting point is to consider a case
where the stressed area and free-face area are constant in time, or at least the rate is so slow
that pseudo- steady-state conditions can be used as an approximation. This simple model will
allow us to gain some insights into which processes and mechanisms will have the larger impact
on evolution of fracture aperture under different circumstances. The strength of the analytical
approach is that the rates of different competing processes can be summarized in a limited
number of parameter groups. The analytical approach can only handle linear processes and
nonlinear processes have to be handled in a different way. The analytical solution for
concentration will be used in a pseudo-steady-state approach, PSSA, to study the rate of fracture
closure/opening due to stress and the conditions that may lead to the growing of the flow
channel aperture.
We start from a case with constant aperture in the flow channel and the stagnant water zone
over time. According to the coupled 1-D approach, the system shown in Figure 11 can be
identified as one composed of four subsystems; the flow channel, the stagnant water zone, the
rock matrix adjacent to the flow channel, and the rock matrix adjacent to the stagnant water
zone. The equations of continuity for each of subsystems can be formulated individually, and
be coupled at boundaries between the subsystems. The mathematical model has been detailed
in Paper IV. The main governing equations are, however, presented in the following sections.
The equations describing the transport of the dissolved mineral in different subsystems are
similar to those describing the transport of solutes presented in Chapter 2. They differ only in
additional terms indicating dissolution processes in different subsystems.
4.1.1 Transport in the flow channel
The equation describing the transport of dissolved minerals in the flow channel, C f , can be
written as,
C f
t
 u
C f
x

 1   pf
 bk C
f
eq , F

Cf  
bs Dws C s
b f W f y
  pf
y 0
D pf C pf
bf
z
(42)
z 0
with
   1 
C eq , F  C eq ,1 expVm F

RT 

(43)
Where Ceq,1 and Ceq,F are the equilibrium concentrations of the mineral at the reference stress
 1 and at the stress  F , respectively.
Equation (43) shows the influence of stress on solubility (Stumm and Morgan, 1996). The stress
enhancement factor is the exponential term in which Vm is the molar volume of the mineral, R
and T are the gas constant and temperature, respectively.  1 is the reference stress of 1 bar (0.1
MPa) and  F is the stress applied on the free faces of the crystals which is practically the pore
water pressure at the prevailing depth, i.e.,
30
 F  Pp
(44)
In equation (42), x, y and z are coordinates along the flow channel, into the stagnant water zone
and into the rock matrix adjacent to the flow channel, respectively.  is the porosity of the
stagnant water zone. k is the mass transfer coefficient, which is the same in all four subsystems
(see Appendix A in Paper IV).
The term on the left side of equation (42) accounts for mineral accumulation in the water in the
flow channel. On the right side, the first term accounts for transport of dissolved mineral by
advection, the second term shows free-face dissolution of the fracture wall into the flow
channel, the third term represents the diffusion of dissolved mineral into the stagnant water
zone in the y direction and the last term describes exchange of dissolved mineral with adjacent
rock matrix by diffusion in the z direction.
Equation (42) is to be solved with the initial condition that
C f (t  0, x)  Ceq, F
(45)
and the boundary condition that
C f (t , x  0)  C f ,in
(46)
4.1.2 Transport in the stagnant water zone
Assuming that crystals in the fracture plane are rectangular cubes of size Lz and Ly = Lx and
have two stressed sides parallel to the fracture surfaces, and that diffusion in the direction
parallel to the flow is negligible, the equation describing the transport of dissolved minerals in
the stagnant water zone, C s , is given by,
C s
 2 C s 1  4
1   
 k
 Dws

 1   ps
k C eq , F  C s 
C eq , P  C s
2


t

L
b

b
y
y
s
s


1   D ps C ps
  ps
 bs z1 z 0






(47)
s
with
   1 
Ceq, P  Ceq,1 expVm P

RT 

(48)
Where Ceq,P is the equilibrium concentration of the mineral at the stress applied on the stressed
faces of the crystal,  P , given by
P 
e
 PP

(49)
with the effective stress  e defined as,
31
 e   T  PP
(50)
which is basically the difference between the overburden total vertical stress  T from the
weight of the rock and the pore water pressure PP (Terzaghi, 1925). In equation (49), φ is the
ratio of the contact area between stressed crystals and fracture surface to the total crosssectional area normal to the applied total stress, and it varies from 0 to 1.
In equation (47), the term on the left side accounts for mineral accumulation in the stagnant
water zone. On the right side, the first term accounts for transport of dissolved mineral by
diffusion, the second term represents the contribution of free-face dissolution/precipitation of
fracture faces as well as crystal free faces into the stagnant water zone, the third term defines
the contribution of pressure dissolution of stressed faces of crystals into the stagnant water zone
assuming that the fracture wall is inert where the crystal is in contact, and the last term describes
exchange of dissolved mineral with adjacent rock matrix by diffusion. The ratio  is considered
as a fraction of fracture wetted surface in the stagnant water zone representing all lumped
stressed surfaces of crystals and it is related to  by    1  W f Ws  . In the case where there
is no flow channel and only stagnant water zone is present in the fracture plane, the two ratios
φ and α would be identical.
The initial condition is
C s (t  0, y)  Ceq, F
(51)
and the boundary conditions are
C s (t , y  0)  C f
(52)
and
C s
y
0
(53)
y  2WS
The transport processes in the flow channel and in the stagnant water zone are, thus, coupled
through equation (52) describing the continuity of the aqueous concentration of dissolved
minerals.
4.1.3 Transport in the rock matrix adjacent to the flow channel
The equation describing the transport of dissolved minerals in the pore water in the rock matrix
adjacent to the flow channel is
C pf
t
 D pf
 2 C pf
z 2

1   pf
 pf

ka f Ceq, F  C pf

(54)
Where Ceq,F is the equilibrium concentration of the mineral at the stress  F in the rock matrix
and a f is the specific surface area of crystals in the rock matrix adjacent to the flow channel,
i.e., the surface area per unit volume of crystals.
32
In equation (54), the term on the left side accounts for accumulation of dissolved mineral in the
pore water in the rock matrix. On the right side, the first term represents diffusion within the
rock matrix and the second term defines free-face dissolution/precipitation of crystals in the
porous rock matrix adjacent to the flow channel. Pressure dissolution of crystals in the rock
matrix is not included in equation (54). The reason is that the rock matrix has usually such a
low porosity, on the order of a few percent, that makes the contact (stressed) area of crystals to
be large and consequently a small pressure dissolution rate which will have minor effect on the
change of mineral concentration in the rock matrix. Thus, for simplicity we may assume that
the whole surface of crystals in the rock matrix is subject only to free-face dissolution.
The initial condition is
C pf (t  0, z)  Ceq, F
(55)
and the boundary conditions are
C pf (t , z  0)  C f
(56)
and
C pf
z
0
(57)
z  f
4.1.4 Transport in the rock matrix adjacent to the stagnant water zone
Similar to equation (54), the equation describing the transport of dissolved minerals in the pore
water in the rock matrix adjacent to the stagnant water is
C ps
t
 D ps
 2 C ps
z s
2

1   ps
 ps

kas Ceq, F  C ps

(58)
The initial condition is
C ps (t  0, z s )  Ceq,F
(59)
and the boundary conditions are
C ps (t , z s  0)  Cs
(60)
and
C ps
z s
0
(61)
z s  s
33
4.2
Fracture closure rate
The set of equations of continuity along with initial and boundary conditions, as described
above, can be solved by applying the Laplace transformation approach (Watson, 1981). The
detailed mathematical procedure can be found in Paper IV. The analytical solution for the
concentration obtained in the Laplace space is then used to study evolution of the fracture
aperture in a pseudo-steady-state, PSSA, procedure.
The evolution of the fracture aperture is, as discussed in Paper IV, a complicated process
governed by different mechanical deformation and chemical changes in the fracture subjected
to the normal stress and fluid flow. In our conceptual model, we model the crystals to be cubes
with initial size Lx = Ly = Lz and have two stressed sides parallel to the fracture surfaces as
illustrated in Figure 12. The two stressed surfaces have the area 2LxLy and the four unstressed
surfaces have the area 4LyLz. In the lower part of Figure 12, solid lines show the datum line for
aperture of both flow channel and stagnant water zone at initial time t0 and dashed lines indicate
the new size of 2bf and 2bs in different regions after a certain time t1. The closure rate of fracture
in the stagnant water zone will be determined by the size Lz of the shrinking (dissolving) crystal
assuming that part of the fracture wall which is in contact with the crystal is inert, i.e., not
dissolving (this assumption can be made when the crystal is in contact with another mineral
with much lower solubility). However, we may consider a case where the fracture wall consists
of the same mineral and dissolves with the rate enhanced by the stress.
2Wf
2Ws
2Ws
Flow channel
Stagnant water zone
Stagnant water zone
z
Lx = Ly
x
2bf (t0)
2bs (t1)
2bf (t1)
Lz
2bs (t0)
y
Figure 12. Schematic illustration of variable fracture aperture in the flow channel and the stagnant water zone
due to pressure dissolution and free-face dissolution/precipitation. Full lines show the location of the fracture
surface at time t0 and dashed lines at t1.
In the following examples, we assume that part of the fracture wall, which is in contact with
the stressed crystal, is inert. At the same time the free-face dissolution/precipitation in the
fracture voids in the stagnant water zone may open/close the stagnant water zone to some
extent. The fracture closure rate in the stagnant water zone is assumed to be mainly governed
by the rate of the shrinking stressed crystals. Therefore, by neglecting the very thin water film
between the stressed crystals and the fracture wall, and given that PSSA is applicable (see
Appendix B in Paper IV), the fracture closure rate in the stagnant water zone can be
approximated by the rate of change of Lz dissolving from two sides as,
34

dbs 1 dLz

 Vm k Ceq, P  C s
dt
2 dt

(62)
The flow channel aperture, which determines the hydraulic conductivity, will not only
close/open with the change of the aperture in the stagnant water zone at the interface, y = 0, but
also simultaneously change with the free-face dissolution/precipitation of the fracture wall in
the flow channel. Consequently, the aperture evolution in the flow channel can be determined
by,
db f
dt

dbs
dt

 Vm k Ceq, F  C f

(63)
y 0
There may, thus, be free-face dissolution or precipitation depending on the local conditions of
saturation, and the aperture as a whole may close if the stressed crystals in the stagnant water
zone dissolve faster than the free-face of crystals of the fracture wall. As a result, the aperture
changes differently at different locations along the flow path. The rate of change of the aperture
varies with time and can be approximately followed using the PSSA. We will, however, only
consider a series of snapshots in time in this study to follow the aperture evolution.
4.3
Simulations and discussion
In this section, a series of simulations is presented and discussed to explore the effects of
different processes and mechanisms on the rate of change of the fracture aperture and mineral
concentration in the water in different locations. The data used in the examples are shown in
Table 7. The geometrical and physical properties of the flow channel, the stagnant water zone
and the rock matrix are representative of those from field observations (SKB, 2006; SKB,
1999). The pore diffusivity in the stagnant water zone modeled as a 2-D porous medium is
calculated according to Archie’s law, Dws = Dw β0.6. The reacting mineral of the rock is quartz.
The solubility dissolution/precipitation related data are taken from Rimstidt and Barnes (1980),
and the other data have been chosen to resemble the conditions in water-saturated granite rocks,
hosting a nuclear waste repository at about 500 m depth underground. The grains in the rock
matrix are assumed to have the same size as the stressed mineral crystals in the stagnant water
zone. It should be noted that the fracture aperture in both the flow channel (2bf) and the stagnant
zone (2bs) change with time, but in a PSSA we use time intervals which are long enough for
steady state concentration profile to establish. The relative rate of the aperture change is small
enough compared to the rate of concentration change for the PSSA to be used. The series of
simulations will give insights into the processes dominating the stress-mediated
closure/opening of fractures.
The results, by using the data in Table 7 and as detailed in Paper IV, show that the time to reach
steady state concentrations in the rock matrix, the stagnant water zone and the flow channel is
on the order of half day to a few days, and the mineral concentration reaches steady state a few
cm into each of the subsystems. More importantly, it is found that the contributions from freeface dissolution and pressure dissolution to concentration build-up in both the flowing and the
stagnant water zone are nearly negligible compared to the effect of the matrix diffusion, and
therefore matrix diffusion is the dominant process by 99.5% compared to the other
mechanisms.
35
Table 7. Data used in the simulations
Symbol (unit)
af = as (m2/m3)
bf (m)
bs (m)
Ceq,1 (mol/m3)
Ceq,F (mol/m3)
Ceq,P (mol/m3)
Dw (m2/s)
Dpf = Dps (m2/s)
k (m/s)
Value
6.0e04
1.0e-04
0.5e-04
0.56
0.58
1.12
1.0e-09
5.2e-10
8.54e-12
Symbol (unit)
Vm (m3/mol)
Wf (m)
Ws (m)
α (–)
β (–)
δf = δs (m)
εpf = εpf (–)
σ1 (Pa)
σe (Pa)
Value
2.26e-05
0.05
0.5
0.1
0.9
0.03
0.01
0.1e06
8.1e06
Furthermore, simulations made in Paper IV suggest that the concentration of dissolved minerals
in a fracture where both free-face dissolution and pressure dissolution are present will always
be between Ceq,F and Ceq,P. In the absence of pressure dissolution, the concentration will finally
reach Ceq,F while when stressed crystals are dissolving the concentration will go above Ceq,F
but stay below Ceq,P. It might, therefore, be concluded that the free-face dissolution generally
dominates over the pressure dissolution since the available area for free-face dissolution is
much larger than that for pressure dissolution. This is shown schematically in Figure 13 for the
concentration profile of dissolved minerals in the stagnant water zone.
Figure 13. Schematic concentration profile in the stagnant water zone.
By using the PSSA, the system under study is now used to evaluate how the fracture aperture
will change in the stagnant zone and in the flow channel. The simulations account for the fact
that the stress applied on stressed faces changes over time because the stressed area changes by
free-face dissolution/precipitation of crystals on their unstressed surfaces. In a fracture in which
aperture changes, the stress decreases considerably if either the stressed surface grows or more
stressed crystals get involved. It may be noted that in the model studied in this work, all the
stressed crystals in the stagnant water zone have the same size and are stressed in the same way.
The stressed surface area of crystals is initially 10% of the fracture surface area in the stagnant
water zone, but will change with time. The change in the stressed area will consequently
increase or decrease the stress applied on the crystals.
36
The closure/opening rate of fracture can be obtained by equations given in previous sections.
At the inlet of the flow channel with incoming fresh water, the free-face crystals of the fracture
wall would dissolve with a rate of Vm kC f ,in which has the value of 1.12e-16 m/s (3.5e-09 m/yr).
This indicates that the channel aperture would grow by 0.1 mm in about 15,000 yr by free-face
dissolution, unless the fracture in the stagnant water zone closes and brings the flow channel
mechanically down with it. Therefore, the flow channel aperture increases due to free-face
dissolution at least near the fresh water inlet. However, further downstream where the
concentration has reached equilibrium the fracture would close slowly everywhere due to the
dissolution of shrinking stressed crystals in the stagnant water zone. The rate of change of the
fracture aperture is shown in Figure 14. It is seen that the fracture in the stagnant water zone
will completely close in about 15,000 yr, and simultaneously the fracture aperture in the flow
channel will decrease from 0.2 mm to 0.1001 mm. The aperture of the flow channel will close
at the rate that the stressed crystals shrink but will grow at the rate that the free-face dissolution
of crystals of the fracture wall determines. The solubility of crystals in the stressed faces
increases by nearly a factor of two compared to the solubility in the free-face sides as well as
those crystals in the rock matrix and the fracture wall. Therefore, the fracture aperture would
be mainly determined by stress-induced dissolution of crystals in the stagnant water zone. This
indicates that the fracture aperture in the flow channel will essentially decrease by mechanical
effects resulting from pressure dissolution of stressed crystals in the stagnant water zone.
Figure 14. Aperture evolution with Cf,in = 0 at x = 1 m and y = 1m.
The fracture aperture of the flow channel determines the hydraulic conductivity, which affects
the rates of water flow and solute transport. Thus, for safety and performance assessment of
radioactive waste repositories, it can be of importance to account for the change of fracture
aperture, if any, in the solute transport modeling and simulations. In the examples studied, the
0.1 mm crystals in the stagnant water zone that makes up the largest part of the fracture would
have dissolved in less than 20,000 yr and no further closing will take place. In the flow channel,
any change in the inlet concentration would reach steady state after less than a year and after
less than 0.1 m flow distance. This indicates that essentially nothing happens more than a very
short distance into the flow channel in less than 20,000 yr under the conditions considered.
Therefore for the assumed conditions, and for the time scales of interest for the repositories for
spent nuclear fuel, the described processes have a negligible impact on the evolution of fracture
37
aperture. However, there might be conditions under which a noticeable impact on
closing/opening of a fracture over a time scale of interest in crystalline rock can be found. One
example could be the increase in temperature to several hundred degrees and use of much larger
flow velocities such as could be of interest in geothermal heat extraction from hot dry rock.
Under these or other possible conditions, the influences of crystal dissolution, matrix diffusion,
and the stagnant water zone on fracture aperture need to be addressed and studied in the future.
38
5
CONCLUDING REMARKS
In this study, four models are developed. The first three models attempt to describe reactive
solute transport in channels in fractured rocks, while the fourth model is dedicated to study how
fracture aperture change under combined effects of stress and flow in crystalline rocks.
The first model accounts for the fact that groundwater flows only in a small part of a fracture
to form one or more flow channels, while at least part of the fracture is occupied by more or
less stagnant water. As a result, solutes can diffuse directly from the flow channel into the
adjacent heterogeneous porous rock matrix that consists of different geological layers. They
may also at first diffuse into stagnant water zones and then from there into the adjacent rock
matrix layers. It is, therefore, expected that both the rock matrix and the stagnant water zone
will play important roles in determining the fate of solute transport in fractured rocks.
The next two models take the radioactive decay chain into account, in the cases involving either
linear or radial diffusion through the rock matrix composed of different geological layers. The
models allow for not only an arbitrary-length decay chain but also any number and
combinations of the rock matrix layers with different properties. The solutions describing the
solute transport in the flow channel, which are analytical in the Laplace domain, can be
numerically inverted by use of e.g. the de Hoog algorithm (de Hoog et al., 1982) to provide the
solutions in the time domain. The models can, therefore, be readily implemented in both the
discrete fracture network models (Dershowitz et al., 1998) and the channel network models
(Gylling et al., 1999) to describe reactive transport of solutes through channels in
heterogeneous fractured media consisting of an arbitrary number of rock units with piecewise
constant properties.
The three models have been used to simulate a series of problems of increasing complexity to
understand the collaborative effect of different processes on retarding nuclide transport in
fractured rocks. The results show that, if the rock matrix consists of several layers (often more
porous than the intact wall rock) with different properties, an additional space for diffusion and
sorption in series to the intact wall rock will be provided for radionuclides and therefore, in the
case of pulse injections, the outlet concentration would significantly change not only in
decreasing the peak value but also in increasing the arrival time and the peak time. This is
particularly true when the altered zone is present. Furthermore, it is shown that the presence of
the stagnant water zone has a major impact on retarding solute transport. The presence of the
stagnant water zone and the rock matrix adjacent to it constitute an additional space for solute
39
transport in parallel to the flow channel and its adjacent rock matrix. The governing process in
retarding the solutes, when accounting for stagnant water zones, is again matrix diffusion.
The simulation results for the case involving a radioactive decay chain, where the effects of the
stagnant water zone and its adjacent rock matrix have been neglected, are compared for two
cases involving either a rectangular flow channel with LMD, or a cylindrical flow channel with
RMD. The comparison suggests that the outlet concentration from the rectangular channel is
practically the same as that from the cylindrical channel in the cases where the solute
penetration depth is much smaller than the channel width. For narrow channels, the diffusion
into the rock matrix becomes more radial by increasing the penetration depth. This leads to a
larger amount of solutes to be drawn off from the flow channel, and consequently to lower the
concentration at channel outlet. This indicates that RMD is more efficient in retarding solute
transport than LMD when the channel width is very small compared to the solute penetration
depth into the rock matrix. Thus, the two models will have the same performance when the role
of matrix diffusion on retarding solute transport is negligible. The difference between the
results obtained from the cylindrical channel and those from the rectangular channel increases
strongly by increasing the water residence time in the flow channel. Furthermore, the
simulations show that, if a thin porous altered rock layer is considered between the flow channel
and the main intact rock matrix, the outlet concentration would significantly decrease,
especially for the cylindrical channel case. When the rock matrix consists of an additional layer
with larger porosity, an additional water volume is provided for diffusion and sorption of
solutes into the rock matrix, which will in turn decrease the concentration at the outlet of the
flow channel. Certainly, this additional water volume would be larger in the case involving a
cylindrical channel than otherwise. In addition, it is confirmed that ignoring decay chains can
cause a major overestimation of the concentration of the daughter nuclide at the flow channel
outlet. This effect is again more obvious in the case of cylindrical channels, particularly when
accounting for the presence of altered rock layer in the rock matrix.
To summarize, the Laplace-transformed solutions obtained in the first three models require
simple input parameters, and they are very efficient computationally. The results of simulated
examples indicate that the solutions can easily handle the problems of reactive transport of
solutes in multilayered heterogeneous systems, such as transport from a potential repository for
radioactive wastes through fractured rocks. The solutions also provide a simple means for
giving insight into system sensitivities to important parameters, and they are particularly useful
for validating numerical codes describing reactive transport in fractured rocks. More
importantly, the simulation results highlight the fact that it is necessary to account for decay
chain, stagnant water zone and also rock matrix comprising at least two different geological
layers (i.e. the altered rock and the intact wall rock) in safety and performance assessment of
the deep repositories for spent nuclear fuel. In addition, quantitative comparisons of the results
obtained from the cases of cylindrical and rectangular channels suggest that the impact of the
system geometry on breakthrough curves increases noticeably as the transport distance along
the flow channel and away into the rock matrix increases. The effect of the system geometry
would be more pronounced for transport of a decay chain when the rock matrix contains a
porous altered layer.
The fourth model is developed to evaluate the evolution of fracture apertures mediated by
dissolution and precipitation of a rock mineral such as quartz. The aim is to get some basic
insights into how a fracture aperture changes under combined effects of stress and flow when
accounting for matrix diffusion and stagnant water zones. The model includes advective flow
in the fracture that can carry in or away dissolved minerals. The model also accounts for the
fact that dissolved minerals in the fracture plane, in both the flow channel and the stagnant
40
water zone, can diffuse into the adjacent porous rock matrix. The analytical solution obtained
in the Laplace space is then used to study evolution of the fracture aperture under combined
influence of stress and flow, in a pseudo-steady-state procedure. The simulation results give
insights into some important processes and mechanisms that dominate the fracture closure or
opening under different circumstances. It is found that the times involved for any changes in
fracture aperture are very much larger than the times needed for concentrations of dissolved
minerals to reach steady state in the rock matrix, the stagnant water zone and the flow channel.
This suggests that the pseudo-steady-state model can be used to assess the evolution of
concentration of dissolved minerals in the rock fracture. Moreover, it is shown that diffusion
into the rock matrix, which acts as a strong sink or source for dissolved minerals, clearly
dominates the rate of concentration change and consequently the rate of evolution of the
fracture aperture. For the condition considered, it is found that the described processes would
have a significant impact on the evolution of fracture aperture, and therefore they should be
considered in safety and performance assessment of repositories for spent nuclear waste. This
is an important issue not only for radioactive waste repositories, but can also be used in other
areas such as geological storage of CO2 or geothermal heat extraction from hot dry rock.
In short, the thesis provides some insights for understanding the effects of stagnant water zones
and matrix diffusion on solute transport as well as on evolution of fracture aperture in fracture
rocks. However, to simplify the problems, some assumptions were made that may cause some
limitations for the application of the models and open some new possibilities for further
improvements. Obviously the use of the models under different conditions (by means of e.g.
comparisons between the model outputs and the results of field experiments) would provide
valuable information with respect to the important mechanisms and processes that govern
solute transport in fractured rocks. Attempts to improve the models should, in the long run,
continue as more experimental data and field observations become available.
It follows that further investigations are required to get more information regarding e.g. the size
distributions of stagnant water zones in fracture plane in fractured rocks. More field
observations should also be made in order to better recognize the complex structure of all rock
matrix layers with distinctively different geological properties. In addition, small-scale tracer
tests are expected to investigate, in particular, if radial matrix diffusion has a significant impact
on retarding solute transport in fractured rocks in comparison with linear matrix diffusion.
The model developed for evaluation of fracture aperture should also be validated by carrying
out some experiments, and field observations might be needed to give more information on
saturation conditions and crystals distributions in the fractured media. Last, but not least, the
overall effects of stagnant water zones, rock matrix layers and radioactive decay chain in
retarding nuclide transport in the cases of both linear and radial matrix diffusion should be
carefully modeled and compared with field observations and experimental results, if any, in
order to aid properly for the safety and performance assessment of repositories for spent nuclear
fuel.
41
NOTATIONS
Dimensions are given in terms of mass (M), length (L), time (T), moles (N) and dimensionless
(–).
af
Specific surface of dissolving crystals in the rock matrix adjacent to the flow
channel (L-1)
as
Specific surface of dissolving crystals in the rock matrix adjacent to the
stagnant water zone (L-1)
bf
Half aperture of the flow channel (L)
bs
Half aperture of the stagnant water zone (L)
c0
Instantaneous inlet concentration into the flow channel (NL-3)
Ceq,1
Equilibrium solubility of mineral at reference stress 1 bar (NL-3)
Ceq, F
Equilibrium solubility of mineral at stress  F (NL-3)
Ceq, P
Equilibrium solubility of mineral at effective stress  P (NL-3)
Cf
Concentration in the flow channel (NL-3)
Cf,j
The jth nuclide concentration in the flow channel (NL-3)
C in
Concentration at the inlet of the flow channel (NL-3)
Cin, j
The jth nuclide concentration at the inlet of the flow channel (NL-3)
C ipf
Pore-water concentration in the ith layer of the rock matrix adjacent to the
flow channel (NL-3)
42
C ips
Pore-water concentration in the ith layer of the rock matrix adjacent to the
stagnant water zone (NL-3)
C ip, j
The jth nuclide pore water concentration in the ith layer of the rock matrix
(NL-3)
Cs
Concentration in the stagnant water zone (NL-3)
Dafi
Apparent diffusivity in the ith layer of the rock matrix adjacent to the flow
channel (L2T-1)
i
Das
Apparent diffusivity in the ith layer of the rock matrix adjacent to the stagnant
water zone (L2T-1)
Dai , j
Apparent diffusivity of the jth nuclide in the ith layer of the rock matrix (L2T-1)
Defi
Effective diffusivity in the ith layer of the rock matrix adjacent to the flow
channel (L2T-1)
Desi
Effective diffusivity in the ith layer of the rock matrix adjacent to the stagnant
water zone (L2T-1)
Dei , j
Effective diffusivity of the jth nuclide in the ith layer of the rock matrix (L2T-1)
Dip, j
Pore diffusivity of the jth nuclide in the ith layer of the rock matrix (L2T-1)
Ds
Diffusivity in the water in the stagnant water zone (L2T-1)
Dws
Pore water diffusivity in stagnant water zone (L2T-1)
Ff
Ratio of the flow-wetted surface of the flow channel to the volumetric water
flow rate (TL-1)
Fs
Ratio of the stagnant-water-wetted surface to the diffusion conductance of the
stagnant water zone (TL-1)
k
Mass transfer coefficient (LT-1)
L0
Initial size of cubic crystals in both z and y directions (L)
Ly
Size of cubic crystals in y direction in stagnant water zone (L)
Lz
Size of cubic crystals in z direction in stagnant water zone (L)
Mcr
Molar mass of minerals, silica (MN-1)
MPG if
Material property group of the ith layer of the rock matrix adjacent to the flow
channel (LT-1/2)
43
MPG is
Material property group of the ith layer of the rock matrix adjacent to the
stagnant water zone (LT-1/2)
MPGij
Material property group of the of the jth nuclide in the ith layer of the rock
matrix (LT-1/2)
nf
Number of the geological layers of the rock matrix adjacent to the flow channel
(–)
ns
Number of the geological layers of the rock matrix adjacent to the stagnant
water zone (–)
N
Ratio between the diffusion rate into the stagnant water zone and the mass flow
rate through the channel (–)
r
Distance into the rock matrix (L)
rf
Radius of the cylindrical flow channel (L)
Rf
Surface retardation coefficient in the flow channel (–)
Rf,j
Surface retardation coefficient for the jth nuclide in the flow channel (–)
Ripf
Retardation coefficient of the ith layer of the rock matrix adjacent to the flow
channel (–)
Rips
Retardation coefficient of the ith layer of the rock matrix adjacent to the
stagnant water zone (–)
R ip , j
Retardation coefficient for the jth nuclide in the ith layer of the rock matrix (–)
Rs
Surface retardation coefficient in the stagnant water zone (–)
RSV
The ratio of flow-wetted surface of the channel to its volume (L-1)
s
Laplace transform variable (T-1)
t
Time (T)
u
Groundwater velocity (LT-1)
Vm
Molar volume of silica (L3N-1)
Wf
Half-width of the flow channel (L)
Ws
Half-width of the stagnant water zone (L)
x
Distance along the flow direction (L)
y
Distance into the stagnant water zone (L)
z
Distance into the rock matrix adjacent to the flow channel (L)
zs
Distance into the rock matrix adjacent to the stagnant water zone (L)
44

Fraction of wetted surface in stagnant water zone demonstrating all lumped
stressed surfaces (–)

Porosity of the stagnant water zone (–)
 if
Thickness of the ith layer of the rock matrix adjacent to the flow channel (L)
 si
Thickness of the ith layer of the rock matrix adjacent to the stagnant water
zone (L)
 ipf
Porosity of the ith layer of the rock matrix adjacent to the flow channel (–)
 ips
Porosity of the ith layer of the rock matrix adjacent to the stagnant water zone
(–)
 ip, j
Porosity of the ith layer of the rock matrix adjacent to the stagnant water zone
for the jth nuclide (–)
λj
Decay coefficient for the jth nuclide (L)
ρcr
Density of crystal, silica (ML-3)
1
Reference stress (ML-1T-2)
e
Effective stress (ML-1T-2)
F
Stress applied on the crystal free faces (ML-1T-2)
P
Effective stress applied on the crystal stressed faces (ML-1T-2)
τf
Characteristic advection time (T)
 Dfi
Characteristic diffusion time through the ith layer within the rock matrix
adjacent to the flow channel (T)
i
 Ds
i
 D,
j
Characteristic diffusion time through the ith layer within the rock matrix
adjacent to the stagnant water zone (T)
Characteristic diffusion time for the jth nuclide in the ith layer of the rock
matrix (T)
τs
Characteristic diffusion time through the stagnant water zone (T)
φ
Ratio of the contact area between stressed crystals and fracture surface to the
total cross-sectional area normal to the applied total stress (–)
Subscripts
cr
Refers to crystals
45
f
refers to the flow channel or the rock matrix adjacent to it
s
refers to the stagnant water zone or the rock matrix adjacent to it
p
refers to the pore water in the rock matrix
Fdiss
Refers to free-face dissolution
Pdiss
Refers to pressure dissolution1
1
Other notations can be found at the end of Papers I, II, III and IV.
46
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