Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 595
Contaminant Transport in Non-Uniform Streams and Streambeds
BY
K. JONAS FORSMAN
ACTA UNIVERSITATIS UPSALIENSIS UPPSALA 2000
Dissertation for the Degree of Doctor of Philosophy in Sedimentology presented at Uppsala University in 2000
ABSTRACT
Forsman, K. J., 2000. Contaminant Transport in Non-Uniform Streams and Streambeds. Acta Universitatis Upsaliensis. Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 595. 32 pp. Uppsala. ISBN 91-554-4884-4.
The interplay between hydraulic and chemical processes in streams and adjacent storage zones, e.g. streambed sediments, is of crucial importance for the along-stream transport of contaminants. This thesis presents a methodology for tracer experiments and the development of mechanistic transport models.
We conducted four field tracer experiments using the reactive tracer chromium, 51Cr(III) and/or the conservative tracers potassium iodide (KI) and tritium (3H2O), along 11 km of the Lanna Stream in Skara County, Sweden, and along a 30 km reach of the Säva Stream in Uppland County, Sweden. The field monitoring included sampling of tracer in the surface water as well as in the streambed sediments. A simultaneous injection of tritium and
chromium facilitated an independent evaluation of the hydraulic transport into and out of the sub-surface storage zones. The difference in transport behaviour between the two tracers, were attributed to chemical reactivity.
In terms of idealised transport models we found that the reactivity of chromium could be characterised by simple chemical concepts. The local equilibrium assumption (LEA), the irreversible kinetics assumption (IKA) and the reversible kinetics assumption (RKA) were applied in the experimental evaluations. An independent evaluation of the streambed transport revealed that the impact of reaction kinetics was substantial. Model calculations and results from a chemical extraction procedure indicated that the chemical reactions affecting the chromium transport were to some extent irreversible.
This thesis presents a number of exact analytical solutions to the governing partial differential equations. The main theoretical contribution is the incorporation of variable coefficients for stream discharge and sediment porosity, which were measured in field.
Key words: Sub-surface transient storage, reactive tracers, analytical solutions.
K. Jonas Forsman, Department of Earth Sciences, Uppsala University, Villavägen 16, SE-752 36 Uppsala, Sweden
© K. Jonas Forsman 2000 ISSN 1104-232X
Acknowledgements
With the completion of this thesis, there are many people to whom I owe much thanks.
I express my deep gratitude to...
...Assoc. Prof. Anders Wörman, who engaged me in his projects and introduced me to the interesting field of tracer experiments and mathematical models. His
enthusiasm for experimental and theoretical development has inspired my work, and his comments contributed to improve this thesis.
...Prof. Lars Håkanson, for providing a scientific atmosphere, where interdisciplinary research and pluralism in scientific methods are encouraged.
...Prof. Klas Cederwall at Hydraulic Engineering, KTH, for collaboration and partial financing of the project.
...Håkan Johansson, for having taken the main responsibility for the practical
experimental planning and the extremely time demanding laboratory procedures.
We had many long days and nights of hard work together. His enthusiasm, positive mood and patience have been never-ending.
...Shulan Xu, my former roommate, for the exchange of ideas about everything on earth, and for valuable discussions on scientific matters.
...Karin Jonsson, for experimental collaboration and the exchange of ideas concerning our research and teaching.
...Sören Karlsson, our genius technician, for excellent collaboration on developing new measurement equipment.
...Assoc. Professors Roland Roberts and Lars-Christer Lundin for friendly support and encouragement during the last year.
...The Swedish Natural Science Research Council, for financial support on experiments and scientific conferences.
...All research students at the Programmes of Sedimentology, Hydrology, Meteorology and Environmental Consequence Analysis. You all contributed to a friendly
atmosphere, and a lot of fun.
Finally, I express my deepest gratitude to my dear family, Cecilia, August and Elsa.
Your support, love and joy have carried me each day. Thank you for your patience.
“The mind is not a vessel to be filled, but a fire to be ignited”
Saying attributed to Plutarch
List of papers included in this thesis
This thesis is based on the following six papers, which are appended at the end of the thesis and referred to in the text by their Roman numerals:
I. Forsman, J., Johansson, H., Ulén, B. and Wörman, A., Evaluation of
phosphorus retention in streams by means of radioactive tracer, Proc. of the International Workshop on Sediments and Phosphorus, 9-12 October 1995, Editors B. Kronvang and L.M. Svendsen. Publ.: National Environmental research Institute, Silkeborg, Denmark, NERI Technical Report No. 178, ISSN 0905- 815X, 1996.
II. Wörman, A., Forsman, J., Johansson, H., Modeling retention of sorbing solutes in streams based on a tracer experiment using 51Cr, Journal of Environmental Engineering, 124(1), 122 - 130, 1998
III. Johansson, H., Jonsson, K., Forsman, K. J. and Wörman, A. Retention of conservative and sorptive solutes in streams - simultaneous tracer experiments.
Accepted: Science of the Total Environment, 2000.
IV. Forsman K. J., "Solute transport in non-uniform streams: Exact solutions accounting for spatially varying parameters", submitted, 2000.
V. Forsman, K. J., Johansson, H., and Jonsson, K., The effects of partly irreversible solute exchange: Comparison between conservative and sorptive transport in streams, submitted, 2000.
VI. Forsman, K. J., Johansson, H. Subsurface storage of conservative and reactive solutes in streambed sediments, submitted, 2000.
The papers are reproduced with kind permissions from Ministry of Environment and Energy, NERI, Denmark, American Society of Civil Engineers and Elsevier.
The contribution to the papers by the author is as follows:
I. Significant part of the experimental work, part of the evaluation and writing.
II. Significant part of the experimental work, evaluation and writing.
III. Part of the experimental work and evaluation.
IV. Original idea, theoretical development and writing.
V. Theoretical evaluation and writing, part of the experimental work.
VI. Theoretical development, theoretical evaluation and writing, significant part of original idea and experimental work.
CONTENTS
Abstract
Acknowledgements
List of papers included in this thesis
1. INTRODUCTION...7
1.1 GENERAL...7
1.2 SHORT HISTORICAL BACKGROUND OF STREAM TRANSPORT THEORY...7
1.3 OBJECTIVES...8
2. SCIENTIFIC METHODOLOGY...10
2.1 TRACER EXPERIMENTS...10
Tracers...10
Experimental methodology...10
2.2 HYDRAULIC TRANSPORT MECHANISMS...11
2.3 CHEMICAL AND BIOLOGICAL PROCESSES...11
2.4 STORAGE MODEL CONCEPTS...12
Transient storage...12
Fickian storage...12
2.5 ANALYTICAL SOLUTIONS...13
Transient storage with variable discharge...13
Fickian storage with variable discharge and porosity...15
Streambed model with vertical porosity gradient...17
3. COMPARISON BETWEEN DISPERSION AND TS...18
4. EFFECTS OF SPATIALLY VARIABLE PARAMETERS...20
5. ON THE REVERSIBILITY OF SOLUTE EXCHANGE...23
6. DISCUSSION AND CONCLUSIONS...29
7. REFERENCES...30
Appended papers
1. INTRODUCTION 1.1 General
Various human activities have resulted in the contamination of surface waters and sediments through history, particularly since the industrial revolution. Diffuse (non- point) sources of contaminants have entered the watersheds as a result of e.g. nutrient leakage from agriculture, atmospheric emissions from the traffic and from air polluting industries. One example of a historically abundant Swedish point source is the paper mill effluents, contaminated primarily by mercury and PCB. The main source of mercury was phenyl mercuric acetate that was used as a slimicide. The practice was forbidden in Sweden in 1968. Between 1941 and 1968 about 150 tonnes of phenyl mercuric acetate was released into rivers, lakes or directly into the Baltic Sea. There are 269 identified Swedish lakes, rivers or sea bays containing fibre sediments contaminated by hazardous chemicals (von Post, 1986).
As long as the concentration of the contaminants is higher in the recipient water than in the pore water of the sediments, the net mass transport is directed into the sediments. Hence, the sediments act as a sink of contaminants. After the cease of contaminant release into the recipient, the concentration gradient over the interface between the sediment and the overlying water is reversed. This causes a net transport of contaminants from the sediments into the water, that is, the sediment acts as a source of the contaminating substance.
The background above illustrates the need for an understanding of the processes causing the contaminants to accumulate in sediments (and other storage zones) and subsequently to some extent enter the surface water again. The interplay between the storage zone transport and the surface water transport is important because it governs a large portion of the total transport of contaminants from the continent to the ocean. In the combined tracer experiment and modelling studies reported in this thesis, we analyse the large-scale stream transport and pay particular attention to the exchange between main stream and subsurface storage zones.
1.2 Short historical background of stream transport theory
The modern theory of solute transport in conduits and channels began in 1953, when Sir Geoffrey Taylor theoretically found that the combined effect of molecular lateral diffusion and variation of velocity over the cross section of a circular pipe could be described by the concept of dispersion (Taylor, 1953). He suggested that the concept could be useful for physiologists, who wished to know how a soluble salt is dispersed in blood streams.
However, Taylor's results have since then been extended to a variety of
environmental transport problems. The frequently cited book "Mixing in inland and coastal waters" (Fischer et. al., 1979) essentially deals with problems that can be analysed with the advection-dispersion equation. With the start of the early 70s, environmental engineers and geochemists report on shortcomings of the dispersion
containing circulating or stagnant water, or porous aquifers, see fig. 1. These zones hold onto a portion of the solute as the main bulk passes by in the main stream. There is then a slow, continual release of solute into the main stream until the storage zones are completely clear of solute.
In the transient storage concept, the mass transfer into and out of a storage zone was postulated to depend on the difference in concentration between the stream and the storage zone, and an exchange rate coefficient. The same type of first order mass transfer can be found in the theory of heat conduction in heat exchangers, as well as in the theory for chemical reactions. In the present context, one of the most well known studies is that of Bencala and Walters (1983). They conducted a field tracer
experiment in a highly irregular mountain stream, and showed that the transient storage equations could be used to characterise the solute transport. Furthermore, they found that the effects of the transient storage in some cases were dominating in
comparison with the effects of longitudinal dispersion.
There are recent papers dealing with uncertainties of the most widespread mixing models, e.g. concerning the interface between longitudinal dispersion- and transient storage modelling. Wagner and Harvey (1997) investigated the sensitivity of the TS parameters and found that there were cases where the parameters became “essentially nonidentifiable”, particularly for rapid exchange with surface storage zones. Harvey et al. (1996) discussed the problem of different time-scales of exchange with surface- and subsurface storage zones, respectively. In their simulations of sub-surface exchange they made the common assumption (Fischer et al., 1979) that the main stream/surface water exchange was accounted for by the longitudinal dispersion concept. Hunt (1999) goes even further and suggests that the increased tail concentrations may be caused by a dispersion coefficient that increases linearly with distance downstream.
A continuing evaluation of the reliability/sensitivity of mixing concepts used in the present context seem to be warranted, both by empirical and theoretical methods.
1.3 Objectives
An overall objective with the present study was to increase the understanding of the contaminant transport in natural streams, in terms of basic hydraulic and chemical mechanisms/processes. To achieve this we developed idealised theoretical models as well as a methodology for stream tracer experiments. The success of the combined theoretical and experimental approach relied on an interplay between the two, because experimental planning is theory dependent and observations lead to improvements of the theories.
The main ideas behind the four experiments presented in this thesis were focussed on the impact of the stream/streambed solute exchange on the along-stream transport.
To facilitate a mechanistic understanding of the exchange we decided to conduct direct measurements in the main stream water as well as in the sediments.
The theoretical framework used in the present study was a combination of the transient storage concept of Hays (1966), later applied by e.g. Thackston and Schnelle (1970) and Bencala and Walters (1983), and the concept of pressure-induced advection of Thibodeaux and Boyle (1987). Particularly the simplifying assumption of Elliot (1990), that the pressure-induced advection could be represented by a simple diffusion
process, has been utilised in the present work as well as in Wörman (1998) and
Wörman (2000). The analysis of Elliot (1990) was restricted to friction material, which makes a direct application of his theory on fine-grained alluvial sediment somewhat uncertain. However, the diffusion concept has found widespread use in the modelling of early diagenesis in lake/sea sediments (Berner, 1980 and Boudreau, 1996) as well as in stream transport studies of e.g. Jackman et al. (1984) and Richardson and Parr
(1988).
The assumption of a diffusive exchange as a representation of the hydraulic mechanism was tested by independent evaluations of the sub-surface transport, by comparing the results of a simultaneous experiment with a conservative and a reactive tracer, and by interpretation of the main stream breakthrough curves.
One specific objective of the present work was to incorporate the effects of longitudinal discharge variations in the transport models. The discharge increased up to almost a factor of 5 along the experimental reach, which warranted an investigation of the impact of dilution on the solute exchange. Analytical solutions of the transport equations were developed for this purpose in IV.
Another specific objective was to develop a FS model that included a porosity that varied with depth. The purpose was twofold. First, the laboratory analysis of the sediment samples included the determination of vertical porosity gradients, which simply warranted the utilizing of the data in the interpretational models. Second, the dilution study in IV showed that the FS model with an infinitely large storage zone failed to predict the transport of the conservative tracer lithium conducted by Bencala et al. (1990), in that the calculated peak concentrations were significantly lower than the data, in contrast to the results of the TS model.
2. SCIENTIFIC METHODOLOGY 2.1 Tracer experiments
Due to the extreme complexity of most natural streams, tracer experiments constitute a powerful means to study the transport of solutes. The measured concentration
distributions are the result ofthe sum total of all processes affecting the transport of the injected tracers. A well-designed experiment can give valuable information about hydraulic, chemical as well as biological processes. The present studies are, however, focused on basic hydraulic and chemical processes.
Tracers
When designing a tracer experiment, the choice of tracers is of crucial importance.
Factors to take into consideration are health aspects, accuracy of detection, chemical reactivity, biochemical and radioactive decay, possibility of on-site-detection, etc.
In the Säva Stream Tracer Experiment 1997, we used the conservative tracer potassium iodide, KI. The positive characteristics of KI are that it is relatively harmless to the environment and the concentration detection can be done directly in the field with an ion selective electrode, which facilitates a good timing of e.g. the sediment sampling. The main reasons for using the radioactive tracers 3H2O and 51Cr in the other experiments was the good accuracy of the obtained concentration data. To meet the requirements concerning health aspects and environmental impact, the tracer injections were carried out during several hours to ensure that the peak concentrations were sufficiently low. The major problem with these tracers was that the results of the measurements could not be seen directly in the field, which made the timing of the sampling schedule difficult.
Experimental methodology
At the upstream end of the stream-reaches that were studied, we injected one or two tracers at a constant rate for 4-6 hours. Several monitoring stations were distributed along the stream, with the uppermost station located 70-100 m downstream of the injection site. Water samples were collected manually with bottles placed 10-20 cm below the water surface. At each sediment-sampling occasion, three sediment cores were collected along the stream transect with a circular core sampler. Each core was sectioned with depth to give a vertical distribution of concentration, porosity and organic content. The stream discharge was obtained by measuring the velocity in several points in each cross section, multiplying with the corresponding cross sectional
“sub-area ”, and by calculating the mean (according to the USGS stream gauging procedure, e.g. Chow 1959). In one of the experiments we could utilize the
concentration data for the conservative tracer tritium to calculate the discharge at each station according to the dilution gauging technique. This methodology is based on dilution calculations and requires that all the exchanged solute returns to the stream during the sampling period. For details about the laboratory procedures, see papers I- III.
2.2 Hydraulic transport mechanisms
The main hydraulic mechanisms causing the transport and spreading of a substance in streaming surface waters is advection, dispersion, and subsurface storage in porous zones adjacent to the main stream. Advection is the transport with the mean water flow and longitudinal shear dispersion is the spreading caused by the combined effect of differential advection and lateral turbulent diffusion. Shear dispersion is a concept (parameterisation) that is used to describe a three-dimensional transport phenomenon in one dimension. Thus, it should not be considered as a true mechanism. Turbulent dispersion is the spreading caused by random movements of a solute in a turbulent velocity field, and it thus exists even in a uniform flow field. Dilution of a propagating solute cloud is caused by inflow of ground- or surface water (less contaminated than that of the stream), which is mixed with the main stream water. Transient storage is a concept that is used to describe the mass exchange between the main stream and e.g.
adjacent side pockets containing stagnant or slowly circulating water. In papers I and II, the transient storage concept is used to describe the slow exchange with wetlands and ground water aquifers. The concept has also been used to describe the early stage of mixing, that is, when the dispersion theory is not valid because advection and lateral diffusion are not in balance (Fischer et. al., 1979).
In papers I-VI a Fickian, diffusive storage is included, which is caused by exchange of solutes between main stream and adjacent porous zones, e.g. the streambed.
2.3 Chemical and biological processes
A tracer that is introduced to a natural stream will experience different levels of mobility. The mobility may vary in time and space. In addition to the retention effect caused by the hydrodynamic solute exchange with storage zones, there are chemical and biological processes that are of importance. Sorption processes cause a reactive solute to adsorb onto suspended particles in the main stream. The adsorbed solute may thus turn immobile with respect to transport into the porous storage zones. Further, the sorption of solutes onto particulate matter in the storage zones increases the storage capacity and may have a strong retarding effect on the large-scale transport. Chemical and biological degradation causes a loss of solute. Chemical reactions causing
precipitation and co-precipitation in the sediments also appear as a solute loss as viewed from the main stream.
Sorption onto particulate matter is an important process to take into account in the analysis of the reactive tracer 51Cr(III) used in three of the experiments. The sorption kinetics of Cr(III), that is, the effect of time dependent sorption was investigated in a laboratory experiment (II). The results showed that the sorption equilibrium is
obtained after a short time, compared to the time scale of the longitudinal transport.
Because of this, the in-stream sorption processes are considered to be instantaneous (Jonsson and Wörman, 2000).
2.4 Storage model concepts Transient storage
The transient storage has been accounted for in combination with descriptions of advection, dispersion and decay by several authors, as mentioned in the introduction.
The concentration is assumed to be uniformly distributed in the main stream (c) as well as in the storage zone (f), see fig. 3. Modelling of the solute exchange is thus done by assuming an instantaneous mixing in a spatially non-dimensional storage zone, and by utilising a first order transfer function. This description (box model) gives an equation system that is suitable to solve numerically in one dimension (Runkel and Chapra, 1993) or by semi-analytical methods (Schmid, 1995).
(
cd fd)
Flux = α −
z f
concentration
S tream
S torage zon e c
Fig. 3. Illustration of the transient storage concept.
Fickian storage
A number of recent investigations have contributed to a mechanistic understanding of the processes involved in the sub-surface solute exchange, particularly in coarse streambeds consisting of friction material. Irregularities along the streambed (Elliot, 1990) or changes of the cross sectional dimensions of the stream (Elliot and Brooks, 1997) cause steady pressure variations along the bed surface, which induces a seepage flow into and out of the streambed. The laboratory observations of Elliot (1990) as well as Elliot and Brooks (1997) revealed that the mass flow into the porous bed, caused by the potential flow, was approximately proportional to the square root of time for a constant boundary condition. This is a typical feature of a diffusive process, for the given experimental boundary- and initial conditions. Further, field experiment observations in I and II indicate that the exchange of a reactive tracer (51Cr) with the alluvial sediment was diffusive, partucularly during the uptake phase. The one- dimensional storage model is based on Fick's law, which is illustrated in fig. 4. Thus, the mass flux depends on the local concentration gradient and an apparent diffusion coefficient D.
Wörman (1998) presented an exact solution for advective stream transport and transversal Fickian storage in an infinitely large storage zone. An alternative way of describing essentially the same processes was done by Gupta and Cvetkovic (2000),
by means of a Lagrangian (“observer following the motion”) concept. The models presented in the present work are all Eulerian (“observer at fixed location”).
0 z d
z D c Flux
∂ =
= ∂
g(z) z
concentration
M a in stre a m
S to ra g e zo n e
cFig. 4. Illustration of the diffusive storage concept.
2.5 Analytical solutions
An analytical solution of a differential equation (or system of equations) is an algebraic expression describing the dependent variable as an explicit function of the independent variables. In this thesis the dependent variable is a concentration, e.g.
c(x,z,t), and the independent variables describe the in-stream water flow, geometry, sediment porosity etc. If a closed Laplace-transformed solution (e.g. for c(x,z)) can be found it may be transformed back to the time-domain by solving a contour integral in the complex space. Analytical Laplace inversions can also be found in tables, e.g., Erdélyi, 1954. The analytical solutions presented in this thesis are exact and time- domain solutions.
To facilitate the derivation of an analytical solution one often has to simplify the governing equations. The models in IV and V were derived with the aim to analyse the impact of in-stream parameter variations and subsurface storage on the in-stream transport. To facilitate this the longitudinal dispersion was neglected, which
mathematically implies that the order of the in-stream equation is reduced from second to first order. This is not to say, however, that it often makes physical sense to neglect dispersion, but it may in some special cases (IV).
The main mathematical development in IV is the incorporation of a discharge that may increase downstream in a linear, as well as in a non-linear manner. In paper VI, the governing equation is of second order, which allows a less flexible concept with respect to the variable porosity.
Transient storage with variable discharge
In paper IV we solved the governing equations in dimensionless form to facilitate a comparison between different model concepts in terms of characteristic dimensionless
Main stream equation:
0 c Da ) f c ( A Da A x Q A c x c A Q t c
d dm d
d em s d
d
d + − + =
∂ + ∂
∂ + ∂
∂
∂ (1)
Storage zone equation:
0 f Da ) f c ( dt Da
df
d ds d
d
d − es − + = (2)
in which t=(t’U)/L is time, x=x’/L is longitudinal co-ordinate, cd=cd’/C0 is the main stream concentration, fd=fd’/C0 is the storage zone concentration, A=A’/A0 is the cross sectional area of the stream and Q=Q’/Q0 is the main stream discharge. L is the
longitudinal length of the stream, index 0 denotes location at the upper boundary, subscript d denotes dissolved mass phase, superscript ‘ denotes dimensional quantity (see IV) As=volume of storage zone per unit length of main stream, α is a transfer rate coefficient (s-1) and λ = rate coefficient of first order decay (s-1). The dimensionless numbers that appear in equations (1) and (2) are Daem=αL/U(1+Km) and Dadm=λmL/U, the Exchange Damköhler number and Decay Damköhler number, respectively, in the main stream. Daes=αL/U(1+Ks) and Dads=λsL/U are similar dimensionless numbers referring to the storage zone. Subscripts m and s refer to main stream and storage zone, respectively. These dimensionless numbers are utilised to conduct a general
comparison between the two concepts presented herein. The initial conditions for the main stream and storage zone are
cd(x,t=0) = 0 fd(x,t=0) = 0 (3)
and the boundary condition
cd(x=0,t > 0) = 1 (4)
which implies that the boundary concentration is kept constant for times greater than zero. Utilizing the Laplace transform facilitates the solution of (1) to (4). A major difficulty with the solution is to find spatially varying coefficients that are possible to transform from the Laplace-space and give a correct physical meaning to the time- domain solution. Paper IV shows that it is possible to find an exact solution for a discharge variation expressed by
) ax 1 ( Q
Q= 0 + b (5)
The solution reads
( )
[ ]
( ) [ ( ) ]
Da Da t x/UQ~ 2 A I Da Da exp Da
Da
t Da Q~ Da
2 A I t Da Da ) exp
ax 1 (
Q~ ) Da A Da
exp ) t , x ( c
t
0
es s em
0 ds es ds
es
es s em
0 ds b es
em s dm
d
>
ïï þ ïïý ü τ ú∂ û ê ù
ë
é τ
τ +
− +
+
ïï î ïïí ì
ú+ û ê ù
ë + é
+ −
úû ê ù
ë
é− −
=
ò
∗
∗
∗
(6)
and cd(x,t)=0 for t<x/U, because the main stream equation is hyperbolic (storage zone equation is parabolic). I0 is the modified Bessel function of zero order and t*=t–x/U.
For t*<0, cd=0, which implies that the shortest possible arrival time for the leading edge of the breakthrough is x/U, that is for pure advection and no retardation caused by transversal exchange. As could be expected, (5) causes a scaling of the entire solution. In other words, if the discharge is increased two times along a reach, the solution will be divided by 1+axb = 2, irrespective of other processes. The dilution affects the exchange only if Q~
is variable, or chosen as a mean value as an approximation. In the present work we use the reach-weighted average
ò
=
L
0
dx ) x ( L Q
Q~ 1 (7)
Fickian storage with variable discharge and porosity
The dimensionless equation system for the longitudinal transport model with a Fickian, diffusive storage (FS) reads
Main stream equation:
0 c z Da
g R Pe 1 x Q A c x c A Q t c
d dm 0
z em d
d d
d + =
∂
− ∂
∂ + ∂
∂ + ∂
∂
∂
=
(8)
Storage zone equation:
0 g z Da
g dz
d Pe 1 t g
t d ds
d es ÷+ =
ø ç ö
è æ
∂ ϕ∂
− ϕ
∂
∂ (9)
in which Peem=LD/[U(R0)2(1+Km)] is the Exchange Peclét number for the main stream
hydraulic radius and Pe= φ0P is the exposed wetted perimeter (m), i.e. the cross
sectional length of the open boundary between the main stream and the storage zone, φ is an exponentially depth varying porosity, φ0 is the surface porosity (at z=0). The Decay Damköhler numbers have the same definitions as in the TS model.
We need a solution to the system (8) and (9) satisfying the initial conditions
cd(x, t=0)=0 gd(x, z, t=0)=0 (10)
and the boundary conditions
cd(x=0, t>0)=1 (11)
gd(x, z=0, t>0) = cd(x, t>0) (12)
and gd(x, z=∞,t>0)=0 (13)
The real-domain solution is obtained as a factor and a convolution integral in IV as
( )
τ τ ∂
π +
= +
ò
∗
τ
÷ø ç ö è
æ +
− τ β + β −
+ +
− t
0
Pe 4
R z Pe~ Pe Da
3 es em
b ) R z Pe~ ( Da
d es
2 em es 2 ds dm em
Pe e 2
R~ z Pe ax
1 ) e t , z , x (
c (14)
In which β=-dφ/(2φdz). R~
should vary in the same way asQ~
for a constant velocity U and a constant wetted perimeter Pe. Equation (14) equals zero for t∗<0, as in (6).
Although this solution is two-dimensional, it is simpler than the one-dimensional TS solution. An even simpler solution can be found by utilizing a boundary condition that varies exponentially with time and a porosity that is constant with depth. A radioactive solute that is introduced at a constant rate at the upper boundary requires the boundary condition
t d(x 0,t 0) e
c = ≥ = −λ (15)
for which the Laplace transform is
λ
= +
= p
) 1 0 x (
cd (16)
The description of the variable boundary is analogous with the description of a first order decay that operates on the entire model domain. Thus, if only radioactive decay is accounted for, Dadm=Dads, that is, the decay is the same in both domains. It is however possible to account for an additive first order loss, whatever the type, in the main stream. If we keep Dadm as rate coefficient for the main stream, and recall that it now denotes the sum of Dads and a possible additional loss, the solution reads
[ ]
úú ú û ù
êê ê ë
é +
+
−
= −
∗
∗
t Pe 2
R~ z Pe ax erfc
1
t Da Da
) exp t , z , x ( c
es em
b ds
d dm (17)
The absence of a convolution integral makes (17) easier to evaluate than (14), particularly for large times. Equation (14) has however to be used whenever Dads is larger than Dadm and/or if the porosity is spatially variable.
Streambed model with vertical porosity gradient
In paper VI we developed three diffusion-reaction/adsorption models with the aim to explicitly study the streambed transport. The main theoretical development consists in the incorporation of a vertical porosity gradient in the second order PDE and deriving analytical solutions for the LEA and IKA as well as a semi-analytical solution for the RKA. The porosity gradient was measured in connection to the experiments and we found that an exponential variation could be useful (fig. 5).
z φ (z) = φ0e-bz
p o rosity
M a in stre a m
S to ra g e zo n e
0 z d
z D c Flux
∂ =
= ∂
Fig. 5. Schematic of porosity model for streambed sediments.
Including a vertically variable porosity (Boudreau, 1996) leads to following one- dimensional model
t 0 g z g D z
1 t
g d a
d z =
∂ +∂
÷ø ç ö
è æ
∂ ϕ∂
∂
∂
−ϕ
∂
∂ (18)
For the case of linear first order irreversible kinetics assumption (IKA) we specify the adsorption rate as
d IKA
a Rg
t
g =
∂
∂ (19)
in which R is a reaction rate (s-1). The linear first order reversible kinetics assumption
÷÷øö ççèæ
ϕ ϕ
− −
∂ =
∂
a d
RKA r RKA
a (1 )g
g K t k
g (20)
in which kr (s-1) is the reaction rate of first order kinetics, and KRKA is the equilibrium partition coefficient for RKA. Analytical solutions are derived in VI for the LEA and IKA cases, with initial- and boundary conditions (10), (12), (13) (but without
dependence on x) and they read
{ } [ ] [ ]
þý úûü êë ù
éΘ +Ψ ΨΘ
î + íì
úûù êëéΘ −Ψ ΨΘ
− β
=
=
t t 2 erfc z z exp t t
2 erfc z z exp
z 2 exp c
) t , z ( g
d d
(21)
in which
z 1 2 1
∂ ϕ
∂
− ϕ
=
β (22)
(
LEA)
2z
LEA 1 K
D β
= +
Ψ (23)
and
( )
21z LEA
LEA D
K 1 ÷÷øö ççèæ +
=
Θ (24)
are coefficients for the LEA case and corresponding coefficients to the IKA case read R
Dz 2
IKA = β +
Ψ (25)
2 1
IKA Dz
1 ÷÷øö ççèæ
=
Θ (26)
The solutions based on (21) are used in a subsequent section to evaluate the reversibility of chromium exchange.
3. COMPARISON BETWEEN DISPERSION AND TS
As mentioned in the introduction, Taylor (1953) introduced the concept of
dispersion as a convenient way of parameterizing the combined effect of differential advection and lateral diffusion in pipe flow. The one-dimensional advection-dispersion equation reads
x 0 K c x U c t c
2
2 =
∂
− ∂
∂ + ∂
∂
∂
in which K is the dispersion coefficient. The Peclét number characterising this
equation is defined as Pe=UL/K. For a constant boundary concentration c(0,t)=C0 and zero initial concentration c(0,x)=0, the solution can be found in Fischer et al. (1979).
In this section the solution of Fischer et al. will be compared with the TS solution of (6).
The result of a tentative comparison between the effect of dispersion and transient storage is shown in fig. 6. The comparison was made by generating breakthrough curves for both models and by fitting the TS solution to that of the dispersion model.
The dimensionless numbers so obtained was plotted in fig. 6. Paper IV quantifies the error of the curve fitting in terms of the relative error (%) in the second temporal moment (Fischer et al. 1979). All effects other than advection, dispersion and zero dimensional exchange are neglected.
0 500 1000 1500 2000 2500 3000
0 2 4 6 8 10 12 14 16 18 20 22
Peclet number [−]
Exchange Damköhler number [−]
Fig. 6. Power law relation between the effects of non-dimensional exchange and that of dispersion.
The dead zone ratio As/A equaled unity, and the boundary concentration was kept at C0 for 1 h (which was achieved by means of the superposition principle (e.g. Fischer et al. 1979).
The calibration underlying the result in fig. 6 was to some extent subjective. If the calibration procedure had been made by consequently minimizing the relative error of the second moment, the "fit by eye" would have been poor in some cases. Almost perfect visual fit between the two models is obtained for K=10 with a relative error of
The main implication of this result is that the effect of transient storage is almost identical to that of longitudinal dispersion within the range of 1≤Daem≤21, which is a direct result of an exchange that has reached a state of equilibrium. The result agrees fairly well with the findings of Wagner and Harvey (1997), and may be useful to consider in the design of experiments.
The theoretical result is not surprising if one recalls that an important condition for the validity of the dispersion concept is a well-mixed cross section, that is, the
transversal mixing must be at equilibrium. It is evident that this condition also can be satisfied if one considers a rapid exchange with a storage zone that is instantaneously well mixed.
4. EFFECTS OF SPATIALLY VARIABLE PARAMETERS
Hart et al. (1999) conducted conservative tracer experiments in a small woodland stream for different discharges. They found that there was a marked increase in
exchange rates for increasing discharge. This indicates that dilution in the main stream is important for the exchange rates, and dilution is thus an important factor to
incorporate in the exchange description.
Both the TS and the FS models contain an exchange description that is dependent on the main stream concentration. For logical reasons the exchange description should account for concentration changes caused by dilution in highly non-uniform streams.
Mathematically this implies that all coefficients describing variations of discharge and cross sectional dimensions should be spatially variable, which is difficult to achieve analytically. To the author’s knowledge this has not been done previously in terms of exact solutions. The work of Bencala et al. (1990), however, included the effects of dilution in their numerical model, which also incorporated inflow of contaminated water. Analytical models often need to be simpler for mathematical reasons.
Therefore, this thesis suggests a simplified methodology to deal with dilution dependent exchange.
The flow in a stream may vary linearly with distance, or nonlinearly in various ways, depending on the hydrological characteristics of the watershed. The variation of stream discharge is expressed by (5). For moderate non-uniformity, the coefficients of
Q~
and R~
may be chosen as constant for simplicity. One should, however, note that this may give rise to a physical contradiction (e.g. if Q varies, and A and P are constant) and that the exchange terms become independent of the dilution according to (6), (14) and (17). This section investigates the relevance of the variation of Q~ and R~ for three specific cases, by inserting the reach-weighted mean values as
ò
= L
0
dx ) x ( L Q
Q~ 1 and R~ = L1
ò
L0R(x)dxThe discharge increases two times along a 5000m long reach according to three imaginary cases. In the first case, [1], the inflow from the watershed to the stream is constant with x, that is, the discharge in the stream increases linearly with x, which is
shown in fig. 7. In the case [2], the water flow from the watershed increases with x, which implies that the stream discharge increases nonlinearly with x. The case [3]
implies that the inflow from the watershed increases along approximately the first 1000 m, and then decreases with x along 4000 m. The velocity is assumed to be constant U=0.1 m/s, which means that the cross sectional area A(x) must increase at the same rate as Q(x). Further, the solute exchange with the streambed of a uniform channel is considered, that is, the wetted perimeter of the storage zone is constant (impermeable channel walls). The apparent diffusivity Ds=2*10-6 m2/s, the TS exchange rate α=5*10-4 and the capacity ratio As/A=1.
0 1000 2000 3000 4000 5000
1 1.2 1.4 1.6 1.8 2
Longitudinal distance [m]
Discharge Q [−]
[1]
[2]
[3]
Figure 7. Hypothetical discharge variation of three different watersheds.
Four cases are illustrated in each of figures 8 and 9 in terms of concentration distributions at x=5000 m. If both Q and A increase two times along L (according to [1], [2] or [3]), and Q~
and R~equal the values at x=0, we get the solutions represented by the solid lines. If, instead, the reach-weighted average values are used according to
[ ]1 3
Q~
− in fig. 7, the main stream concentration increases as a result of the decreased dilution-dependent exchange.
[ ] 1.78 Q~
3 =
[ ] 1.15 Q~
2 =
[ ] 1.50 Q~
1 =
10 20 30 40 50 60 0
0.1 0.2 0.3 0.4 0.5
Time [h]
Concentration [−]
Fig. 8. Breakthroughs for different discharge variations according to the FS model.
Figures 8 and 9 reveal that the effective velocity is sensitive to the storage parameters in the TS model, but not in the FS model (for these specific cases). The retardation in the FS model seems to be significant only for large storage zones in combination with very high diffusion rates. In the TS model the retardation seems to be small for cases when As/A is small. Further, for cases when the retardation is small in the TS model, the sensitivity to changes in Q~
is similar to changes of a variableR~ in the FS model. It is, however, uncertain to draw general conclusions concerning the general relationship between effective velocity and exchange parameters, based on tentative simulations. The main difficulty with the tentative approach is to deal with the high degree of freedom that characterizes the model concepts. Tentative
simulations do, however, give the analyst important insight on the “real-domain”
behaviour of the models.
A possible way of deriving general relationships could be by means of the
parameter optimisation method of Wörman (2000). This is, however, outside the scope of the present work.
FS simulations
Solid line: Q~ =1
Dash-dotted line: Q~=Q~[ ]2 Dash-dotted line: Q~[ ]1
Q~
= Dotted line: Q~[ ]3
Q~
= φ(z)=exp[-z]
10 20 30 40 50 60 0
0.1 0.2 0.3 0.4 0.5
Time [h]
Concentration [−]
Figure 9. Breakthroughs for different discharge variations according to the TS model.
5. ON THE REVERSIBILITY OF SOLUTE EXCHANGE
This chapter deals with the reversibility of the solute exchange between main stream and storage zones. Reversibility in the current context means that all solute that enters the storage zone is eventually returned to the main stream. The theoretical models in IV-VI take into account an irreversibility that is caused by chemical reactions leading to the strong binding of Cr(III) to the sediment. (If, however, the chemical conditions are drastically changed e.g. by a lowering of the pH, the strongly bound solutes may dissolve again.)
Fine-grained sediments, such as those of the Lanna- and Säva Stream, contain large surface areas available for sorption processes. Sorption onto these surfaces is often believed to be rapid in comparison to the hydraulic transport in the porewater. This is the main reason to why sorption is commonly treated as an instantaneous equilibrium reaction (Berner, 1980, Malcolm and Kennedy, 1970).
Several investigations in natural waters show that an immobilisation of Cr(III) at oxidising conditions also can be obtained by co-precipitation with Fe oxides and hydroxides (Rai et al., 1987), which generally are abundant in alluvial sediments in coniferous forests. Harvey and Fuller (1998) studied the transport of dissolved manganese in a small stream contaminated by copper mining. They found that
TS simulations Solid: Q~ 1
=
Dash-dotted: Q~[ ]2 Q~
= Dash-dotted: Q~[ ]1
Q~
= Dotted: Q~[ ]3
Q~
=
There are numerous references in the literature to adsorption/desorption hysteresis, or adsorption irreversibility from solution (Adamson 1990). Davis et al. (2000)
injected Cr (VI) into a sand and gravel aquifer. They found that a fraction of the
Cr(VI) was reversibly adsorbed to the gravel where oxic conditions prevailed. Because of a negative gradient of dissolved oxygen along the transport path, Cr(VI) was
reduced to Cr(III) and was subsequently “very strongly sorbed” to the gravel, and the dissolved concentrations became undetectable. Cerling et al. (1990) found that the sorption of 137Cs by streambed sediments was irreversible, that is, no desorption from the sediment could be observed. This operational definition of irreversibility is used in the present thesis.
In paper V we analysed the in-stream transport of tritium and chromium in the Säva Stream. Fig. 10 shows the breakthrough for the conservative tritium at 16 100 m downstream of the tracer injection site. The solid line represents (17) with all reaction parameters equal to zero, that is, for a conservative (and reversible) case.
40 45 50 55 60 65 70
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
Time [h]
3 H 2O concentration [−]
[m]
+16100 [m]
+16100 [m]
Fig. 10. Breakthrough for tritium in the Säva Stream. Solid line:
Equation (17), for Ds=2.2*10-6. Markers: Data.
In fig. 11 a similar exchange model is applied on the results from the conservative KI transport (VI). The reversible Fickian concept represented by (21) fits the spatial concentration distribution in the sediment as good as the temporal distribution in the main stream in fig. 10.
0 1 2 3 4 5 6 7 8 x 10−5 0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Depth [m]
KI concentration [Mol/L]
Fig. 11. Conservative exchange of potassium iodide. Data from the Säva Stream Tracer Experiment 1997, station A (VI).
The model calibration underlying the result in fig. 10 yielded a transversal exchange coefficient of Ds=2.1*10-6 for tritium. As this result was assumed to characterise a pure hydraulic exchange, the calibration for chromium in fig. 12 is based on the same
exchange coefficient and additional parameters for chemical reactivity. The
assumption of two simultaneous reactivites based on LEA and IKA, respectively (V), resulted in the best fit represented by the solid line in fig. 12.
Data: Time: Theory:
x 1.2 h (solid line)
∗ 3.8 h (dashed line)
+ 10.3 h (dash-dotted line)
o 28.6 h (dotted line)
40 45 50 55 60 65 70 0
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
Time [h]
51 Cr(III) concentration [−] +16100 [m]+16100 [m]+16100 [m]
Fig. 12. Data (x) from the Säva Stream, and theoretical (eq. 14) breakthroughs for 51Cr(III). Dotted line: Rs=0, Ks=6. Dash-dotted line: Rs=2.4*10-4, Ks=0.
Best fit: Rs=2.0*10-5, Ks=3.7 (solid line).
The application of LEA and IKA was tested in the independent evaluation of streambed transport in paper VI. We found that both concepts could be used to
describe the concentration profiles during the uptake phase (figures 13 and 14, for 2.5 and 5.5 h), that is, during times of high main stream concentrations. The reaction coefficient Rs was fairly constant, but Ks increased markedly with time. We may thus conclude that the reactivity of chromium during the uptake phase is clearly kinetic and sufficiently rapid to allow a representation by LEA (if one is aware that Ks is far from reflecting true equilibrium).
0 1 2 3 4 5 6 x 104 0
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
Depth [m]
Cr(III) radioactivity [Bq/l]
Fig. 13. Reactive exchange of chromium with IKA. Data from the Lanna Stream Tracer Experiment 1996, station A (II).
0 1 2 3 4 5
x 104 0
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
Depth [m]
Cr(III) radioactivity [Bq/l]
Fig. 14. Reactive exchange of chromium with LEA. Data from the
Data: Time: Theory:
x 2.5 h (solid line)
∗ 5.5 h (dashed line)
+ 29 h (dash-dotted line)
o 50 h (dotted line)
Data: Time: Theory:
x 2.5 h (solid line)
∗ 5.5 h (dashed line)
+ 29 h (dash-dotted line)
o 50 h (dotted line)
However, simulations based on LEA failed to describe the washout process, which is shown by the poor fit for 29 h and 50 h in fig. 14. The washout simulations based on the IKA (fig. 13) and RKA (fig. 15) revealed that the desorption/dissolution processes were very slow. The reaction rate coefficients both for IKA and RKA were fairly constant, which makes it uncertain to draw any excluding conclusions concerning the
reversibility/irreversibility based entirely on the modelling.
0 1 2 3 4 5 6 7
x 104 0
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
Depth [m]
Cr(III) radioactivity [Bq/l]
Fig. 15. Reactive exchange of chromium with RKA. Data from the Lanna Stream Tracer Experiment 1996, station A (II).
A chemical extraction procedure in paper VI indicated, however, that Cr(III) was strongly and to a some degree irreversibly bound to the sediment in the Lanna Stream.
Chromium was not dissolved at all by porewater, NH4Ac or 0.1 M HCl. 36% was dissolved by 0.04 M NH2OH-HCl, which indicates that this fraction was chemically bound to compounds containing Fe(III)- and Mn oxides. A fraction of 9% was
dissolved by 0.1 M NaOH, which indicates the existence of chemical bounds to humic substances. The largest fraction, 55%, was found to be chemically bound to mineral constituents and was dissolved by 7M HNO3.
Extraction studies conducted in field for different instants during the experiment would be a potential way to obtain increased information about the kinetics of chemical reactions. Such procedure would, however, be unrealistic from a practical point of view because of the time demanding procedures.
Data: Time: Theory:
x 2.5 h (solid line)
∗ 5.5 h (dashed line)
+ 29 h (dash-dotted line)
o 50 h (dotted line)
6. DISCUSSION AND CONCLUSIONS
A basic understanding of the solute transport in a stream can be achieved by means of tracer experiments, evaluated with theoretical models based on physical and chemical principles. The result of the interpretation of a tracer experiment is to some degree dependent on the choice of theoretical model. Particularly the description of the mass exchange between the main stream and the adjacent storage zones is important in the analysis of many natural streams. The different character of different watersheds may justify differences in the storage description. However, the general idea of the
theoretical development should be to facilitate a quantitative description of different streams in a way that allows an objective comparison. Thus, it is of crucial importance that the theoretical models, although by necessity strongly simplified, are as generally formulated as possible. An important scientific contribution in this context is the independent evaluation of various types of storage zones (e.g. Harvey et al., 1996, Elliot and Brooks, 1997 and Wörman et al., 1998).
A complication of the transport modelling presented in this thesis is that
conceptually different model concepts may sometimes give almost the same result. It was shown in paper IV that the effects of longitudinal dispersion and that of transient storage are almost exactly the same within an interval of the dimensionless numbers characterising the two concepts. Such effects should be taken into account in the experimental design (Wagner and Harvey, 1997), because a lack of model sensitivity with regard to mechanisms being studied may make experimental evaluations difficult.
The incorporation of additional parameters generally makes a model concept more flexible in the curve fitting procedure, but does not necessarily lead to an
increased understanding of the transport processes. On the other hand, if an important parameter is missing in a model, this may lead to a loss of important information to the analyst. The ideal situation is that as many parameters as possible are evaluated
independently, to give meaningful results. In other words, the interpretative models should not be over specified, in terms of mechanisms that are impossible to evaluate independently from other mechanisms (Boudreau, 1996).
The purpose of including new parameters for non-uniform streams and
streambeds was not to increase the flexibility of curve fitting procedures, but to utilise independently evaluated data. The extent, to which the including of independently evaluated data leads to an increased understanding of the transport processes, is
heavily dependent on the absence of model bias. That is, if e.g. the basic assumption of a diffusive exchange is physically correct, then including a variable porosity increases the generality of the theoretical concept.
The main theoretical contribution in this thesis was the derivation of a number of exact analytical solutions that were found to be useful in the interpretation of the experimental results. The discharge variation in the models developed in IV is treated in a more general way than in previous models. Including the effects of dilution in IV made it possible to conclude that dilution may have a marked impact on the outcome of evaluations of experiments conducted in highly non-uniform streams. The exchange coefficients may be clearly underestimated for such cases.
River (Bencala et al. 1990). This result warranted the deriving of an exact solution for FS with a finite storage zone, which was achieved by including an exponentially varying porosity. In spite of the restriction of exponential variation with depth, the porosity model fitted the data from the Lanna Stream sediment well. More important was that the finite storage zone makes the general characteristics of the FS solution much more convenient from a physical as well as from a modelling point of view.
Because of the conceptual uncertainties concerning the Fickian diffusion concept (from a mechanistic point of view), the detailed evaluation of the sub-surface transport of potassium iodide and chromium into and out of the alluvial sediment was
surprisingly successful. The FS concept accounted for the difference in conservative transport and reactive transport in a consistent way, and the measured boundary concentrations could be straightforwardly used for the chromium transport. Applying the LEA clearly indicated that the chemical immobilisation of Cr(III) was rapid and kinetic, and showed that the LEA failed to predict the mobilisation process during the washout phase. Both IKA and RKA were successfully applied to both the uptake and washout phase of chromium. It was difficult to draw any general conclusions on the reversibility of chemical processes entirely based on the modelling based on IKA and RKA, primarily because the time scale of the data was not sufficiently large. A
chemical extraction procedure indicated, however, that chromium was to some extent irreversibly bound to the sediments.
One may conclude that successful future scientific contributions in the current context probably will continue to rely on an interplay between controlled, small-scale laboratory experiments, well designed large-scale field experiments and the
development of mechanistic models.
7. REFERENCES
Adamson, A. W., Physical chemistry of surfaces, John Wiley & Sons Inc., 1990.
Bencala, K. E. and Walters, R. A., Simulation of solute transport in a mountain pool-and-riffle stream: a transient storage model, Water Resources Res., Vol. 19, No. 3, 718-724, 1983.
Bencala, K. E., McKnight, D. M. and Zellweger, G. W.,Characterization of transport in an acidic and metal-rich mountain stream based on a lithium tracer injection and simulations of transient storage, Water Resources Res., Vol. 26(5), 989-1000, 1990.
Berner, R. A., Early Diagenesis, Princeton University press, Princeton, N.J., 1980.
Boudreau, B. P., Diagenetic models and their implementation: Modelling transport and reactions in aquatic sediments, Springer-Verlag, 1996.
Cerling, T. E., Morrison, S. J., Sobocinski, R. W. and Larsen, I. L., Sediment-water interaction in a small stream: Adsorption of 137Cs by bed load sediments, Water resources research, vol. 26, no. 6, 1165-1176, 1990.
Chow, V. T., Open Channel Hydraulics, McGraw-Hill, New York, 1959.
Davis, J. A., Kent, D. B., Coston, J. A., Hess, K. M., Joye J. L., Multispecies reactive tracer test in an aquifer with spatially variable chemical conditions, Water
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