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Analysis of colloid and tracer breakthrough curves
JOURNAL OF
ELSEVIER
Contaminant Hydrology Journal of Contaminant
Hydrology 21 (1996) 243-253
Analysis of colloid and tracer breakthrough curves Peter Grindrod a,*, Mark S. Edwards a, Jenny J.W. Higgo b, Geoffrey M. Williams b a Intera Information Technologies Ltd., Henley-on-Thames, RG9 IAT, UK b British Geological Survey, Keyworth, NG12 SGG, UK
Received 17 December 1993; accepted 15 December 1994 after revision
Abstract We consider the dispersion and elution of colloids and dissolved nonsorbing tracers within saturated heterogeneous porous media. Since flow path geometry in natural systems is often ill-characterized macroscopic (mean) flow rates and dispersion tensors are utilized in order to account for the sub-model scale microscopic fluctuations in media structure (and the consequent hydrodynamic profile). Even for tracer migration and dispersal this issue is far from settled. Here we consider how colloid and tracer migration phenomena can be treated consistently. Theoretical calculations for model flow geometries yield two quantitative predictions for the transport of free (not yet captured) colloids with reference to a non-sorbing dissolved tracer within the same medium: the average migration velocity of the free colloids is higher than that of the tracer; and that the ratio of the equivalent hydrodynamic dispersion rates of colloids and tracer is dependent only upon properties of the colloids and the porous medium, it is independent of pathlengths and fluid flux, once length scales are large enough. The first of these is well known, since even in simple flow paths free colloids must stay more centre stream. The second, if validated suggests how solute and colloid dispersion may be dealt with consistently in macroscopic migration models. This is crucial since dispersion is usually ill-characterized and unaddressed by the experimental literature. In this paper we present evidence based upon an existing Drigg field injection test for the validity of these predictions. We show that starting from experimental data the fitted dispersion rates of both colloids and non-sorbing tracers increase with the measured elution rates (obeying slightly different rules for tracers and colloids); and that the ratio of colloid and nonsorbing tracer elution rates, and the ratio of colloid and nonsorbing tracer dispersion rates may be dependent upon properties of the colloids and the medium (not the flow regime).
* Corresponding
author.
0169-7722/96/$15.00 0 1996 Elsevier Science B.V. All rights reserved SSDI 0169-7722(95)00051-S
244
P. Grindrod et al. /Journal
of Contaminant Hydrology 21 (1996) 243-253
It is important to realize that even for unretarded species, an earlier peak in the breakthrough curve does not necessarily correspond to a faster mean elution rate, or vice versa. But rather that a colloid may elute faster but disperse less than an equivalent tracer. Hence its peak may be retarded compared to that of the tracer, even assuming no retardation. Hence one must consider a combination of mean elution rate and mean dispersion rate, and not rely on “peak times” to corroborate chromatographic effects. The importance of this lies in the fact that these processes are not independent and yet upscale differently. Thus realistic estimates of effective colloid dispersion rates should be upscaled in a way consistent with that adopted for tracers within the same system.
1. Introduction In previous work (Grindrod, 1993a) a theoretical approach to the dispersion and elution of colloids and tracers within saturated heterogeneous porous and fractured media has been developed. In this paper we shall consider the results of the first investigation in the light of data from field injection experiments (Higgo et al., 1993). Typically flow path geometry is ill-characterized in models of natural systems and as a consequence macroscopic (mean) flow rates and dispersion tensors are utilized in order to account for the of impact microscopic fluctuations in media structure (and the consequent hydrodynamic profile), which are unaccounted for within the transport model considered. Even for tracer migration and dispersion this issue is far from settled. There is field evidence of scale-dependent dispersive effects (Neuman, 1990), and the existence of representative volumes over which processes could be averaged is questionable, and should be validated on a site-specific basis. Given such uncertainty it has been suggested (Grindrod, 1993a) that within any particular flow system (actual or theoretical), colloids and tracer migration phenomena should be treated consistently. As a result of this approach calculations for model flow geometries have yielded two quantitative predictions for the transport of free (not yet captured) colloids with reference to a nonsorbing tracer within the same medium. Briefly they are: (1) the instantaneous average migration velocity of the free colloids is higher than that of the tracer; (2) the ratio of the equivalent hydrodynamic dispersion rates of free colloids and tracer is dependent only upon colloid properties: size (exclusion from a portion of the available pore space), the existence of surface forces between the colloids and the medium surfaces (resulting in deterministic motion reflective of the fluid flow), and so on. The first of these is well known, since even in simple flow paths free colloids must stay more centre stream. Thus mean elution rates for colloids are between 1 and 1.3 times those for the tracer (Grindrod, 1993a). Colloids are either physically excluded from low-permeability zones or are entrapped there (physically or chemically). Hence early colloid breakthrough will be dominated by those colloids which have remained uncaptured during their migration, and have thus been advected along the faster, more permeable, flow paths,
P. Grindrod et al. /Journal
of Contaminant Hydrology 21 (1996) 243-253
245
The later breakthrough behaviour may show effects of physical filtration (clogging) and release, as well as electrostatic or chemical capture and release. Hence in considering the dispersal of free colloids one must focus on the rising limb of the breakthrough curves, avoiding the behaviour in the tails of the curves. Similarly, one may heuristically argue that the dispersion of such free colloids must be less than that of the tracer owing to the same processes: the transfer of tracer into dead-end pores or regions of relatively low permeabilities providing a large dispersive effect. The relative dispersion rate of free colloids to that of tracer is thus governed by those colloid properties which control exclusion or capture. This idea is again borne out by the mathematical computation for simple geometries (Grindrod, 1993a1, where predictions were made for advection and dispersion rates solely based on theoretical considerations. In addition, dispersion rates were shown to be flux dependent. Thus for both colloids and tracer alike it was anticipated that the elution rates and dispersion rates would be inter-related: dispersion rates increasing with elution rates. Specifically, for identical flow systems, it was shown in Grindrod (1993a) that the dispersion rates should increase with the measured elution rates (obeying slightly different rules for tracers and colloids); and that the ratio of colloid and nonsorbing tracer elution rates, and the ratio of colloid and nonsorbing tracer dispersion rates are dependent solely on properties of the colloids and the medium (not the flow regime). We shall examine some of these predictions in the light of breakthrough data given in Higgo et al. (19931, for a number of migration tests carried out within a shallow confined sand aquifer at Drigg, Cumbria, U.K. For four distinct types of colloid, injections were made at a single borehole, and breakthrough data were subsequently collected at other boreholes (Fig. 1). At each collecting borehole there are a number of ports, providing a vertical stratification of the tracer and colloid migration within the aquifer. The analysis below uses data from two downstream boreholes, and considers two types of colloids, fulvic acid and an artificial polymer, sodium polystyrene sulphonate. Thus for each injection experiment we are able to monitor the breakthrough of both colloids and a nonsorbing tracer at 11 specific locations. Examples of breakthrough curves are given in Fig. 2a and b. Considering curves such as Fig. 2a and b it is important to realize that even for an unretarded species, where the breakthrough curve is of the form of Eq. 3 below, the time to peak breakthrough is dependent upon both the mean velocity (or mean elution rate), U say, and the effective dispersivity, D say. In this case it is given explicitly by:
Tpeak
22’3( -2D2 =
3u
-
21/3w1/3
+ 7DLU - 2L2U2) 3u2w’13
+
where W= (-4D3
+ 21D2LU+
+33/2U&
6DL2U2 + 4L3U3
9D4L2 + 46D3L3U - 16D2L4U2 + 8DL5U3 >
and L is the breakthrough
distance.
3u2
246
P. Grindrod et al. /Journal
of Contaminant Hydrology 21 (1996) 243-253
Water recirculated between D 221 and D 13.00 D221
1
D225
D224
D227
D???
0226
D219
D220
- Ground level 11.00
10.50 i
_Silty
Clay
_ Fine to medium stratified sands Laminated silts ant /very fine sands
dilty
5.50
Clay
5.00
HORIZONTAL
4.50
SCALE
SAME
AS
D220
PI;
@
Jiew
Fig. 1. Details of the Drigg borehole array.
This is a decreasing function of D: (plot it!). Clearly overall earlier elution ahead of the peak does not necessarily imply a higher mean velocity (mean elution rate), and vice versa. All too often the debate about colloid mobility has focussed upon chromatographic effects, and has ignored the subtleties of
P. Grindrod et al. /Journal
247
of Contaminant Hydrology 21 (1996) 243-253
I l.OOE-04 l.OOE-05~
- : 4.00
16.00
31.90
46.06
61.00
76.00
96.00
b
Time (lx)
Polystyrene sulphonate
l.OOE-01 -. l.OOE-02 . .
i.OOE-05 4 3
9
19
29
41
61
110
146
k
Time (hr) Fig. 2. a. Tritium and fulvic breakthrough at D226 port 9.
acid breakthrough
at D226 port 9. b. Tritium
and polystyrene
sulphonate
the impact of dispersion. Hence it does not follow from Fig. 2a, that the tritium has a higher mean elution rate than the fulvic, but rather that the earlier breakthrough and earlier peak may be due to a larger effective dispersion rate. Although the results presented here are preliminary, we feel it is important to address both aspects, since the upscaling of dispersion is problematic - even for tracers. Since colloids may become physically filtered by, or chemically bound to, the medium, it is important to separate the hydrodynamic dispersion of free colloids from dispersion due to capture and release - which are controlled by separate terms in a macroscopic model. For this reason our analysis of breakthrough pulses concentrates on early arrival times, where advection and hydrodynamic dispersion processes dominate. Typically, the initial breakthrough of colloids was faster than that of the tracer, but tails were much longer owing to colloid sorption, capture and release. Consequently, the breakthrough curves observed were not always complete. For this reason the analysis presented below should be considered preliminary, and is concentrated only on the relatively early behaviour, dominated by a proportion of the colloids which passed freely
248
P. Grindrod et al. /Journal
of Contaminant Hydrology 21 (1996) 243-253
through the most permeable pathways, so as to avoid a discussion possible effects of colloid retardation, filtration and re-release.
of the nature and
2. Results Taking data from Higgo et al. (1993) we analyzed the early time breakthrough behaviour by considering only the first half of the normalized arrival pulse. Such early breakthrough is assumed to be dominated by colloids which have remained free, unsorbed and unfiltered, whereas later breakthrough behaviour must show some effects of capture and re-release (particularly in the tail). Assuming that advection and hydrodynamic dispersion processes dominate during this phase we fit equivalent average velocities, U, and dispersion rates, D according to the behaviour of solutions for the macroscopic one-dimensional equation: ac
2
_ZDKJE
(1)
at
where C(x, t) denotes the colloid or tracer concentration. Of course for later times the behaviour is more complex and Eq. 1 should be amended to include filtration, sorption and release. We calculated D and U using a quasi-Newton minimization procedure [DavidonFletcher-Powell (DFP) algorithm (see Press et al., 1987)]. This procedure requires a function to be minimized together with its partial derivatives. The function used was:
5 [Co(ti) i=
(2)
-RC,(U,D,ti>]’
1
where C,(t,) denotes the observed concentration at time t, (i = 1, . . . , m, where m is the number of observations considered); R is a scale factor (accounting for the fact that the final arrival mass is not necessarily known); and C,(U,D,t,) denotes the theoretical (normalized) flux across x = 1 (found by the solution of Eq. 1) at time t, given by: [-(IC,(U,D,t,)
Utl)2/4Dt,](l+
= exp /z+
Ut,)
(3)
where 1 is the breakthrough length, given; and D and U are the dispersion and velocity rates, respectively, to be calculated. The problem, therefore is to fit D, U and R so that our function (2) is minimized. With the same notation as is used in Higgo et al. (1993) (experiments, boreholes, ports) we derived early time D- and U-values for both tracers and colloids. The results are shown in Figs. 3-8. Figs. 3 and 4, 6 and 7 depict fitted dispersion rates plotted against fitted advective velocities for both colloids and tracers at both boreholes. For each borehole the separate points correspond to breakthrough curves normalized at particular ports. The lines indicate the general trends of the monotonic dependence of D upon U: their slopes may be interpreted as dispersion lengths for tracer and colloid species. Figs. 3 and 4
P. Grindrod et al. /Journal
of Contaminant Hydrology 21 (1996) 243-253
249
Irn
0.007
0.002
0.006
I
:
Velocity Fig. 3. Fitted effective
dispersion
(m* h- ‘) vs. fitted mean velocity
(m h-l),
experiment
correspond to experiment 1 - fulvic acid colloid and tritium tracer, correspond to experiment 4 sodium polystyrene p-sulphonate tracer (experiments 2 and 3 did not yield enough data). As anticipated, we see (from Figs. 3 and 4, 6 and 7) that in both sodium polystyrene p-sulphonate experiments, early colloids: (1) migrate faster than the nonsorbing tracer; (2) disperse less than the nonsorbing tracer. Also we see for both the colloids and tracers, that the dispersion observed velocities, according to different observed relationships squares “line up” differently).
Fig. 4. Fitted effective
dispersion
(m2 h _ ‘) vs. fitted mean velocity
1, borehole
D224.
whilst Figs. 6 and 7 colloid and tritium the fulvic acid and
rates increase with (the triangles and
(m h-’ ), experiment
1, borehole
D226.
‘pzza
aloya~oq ‘p luaqtadxa
‘( 1
q “) .hrqa~ umm pall!3 ‘s* (, q zm) uoyads!p
agaa33a
pall!d ‘9 %d
Al!DoleA LOO
90’0
so.0
VO‘O
CO’0
20’0
10’0
-UK)‘0
-IO’0
-800’0
-SW0 .
‘(azwzlsrp q%o_y~yaalq
aDuaq) aIoyaloq
sr jey$ ‘yled 30 ssalpIe8al)
qzwa .I03 UWsu03 &E3 s! 3e1’fl/T10Jfl (2) fhoayl ayl dq palzpald se ‘(a[oqaloq
add1 pro1103 q%a 103 welsuo:, Aw3 s! “““a/“““~ (I) :s~o1103 $1 8 pue s ‘s%!~dayj uIor$ ‘p luauryadxa 103 ydel8 awes aql sP!dap
8
%d ‘palw!pur se aIoyaloq e U~I!M vod e w! sJatatue.wd pan!3 aql30 sow ayl 01 spuods -a1103 lu!od y~es ‘1 luaurnadxa 103 3e’Lfl/1105fl‘sh 3e”~/1103g MOqS afi g ‘%d uI
9aa
V
tua m
ESZ-EPZ (9661) 12 ~“PP~H
l”~“!‘W“~3~~
P‘l”of-/
OS2
‘10 la pO.+‘Uf13 ‘d
P. Grindrod et al. /Journal
251
of Contaminant Hydrology 21 (1996) 243-253
: ii
k a
:
m
c:2
s
o.ooo, 0.01
-
,
0.02
.
,
,
0.03
0.04
_
,
_
0.05
0.0I
Velocity Fig. 7. Fitted effective dispersion (m2 hK‘) vs. fitted mean velocity (m h-l),
experiment
4, borehole
D226.
-_
iA A
1.1s 1.1-
A ?? 1.4.
+
I'
12. m 1.0 .I
A
1 1
dcollldtrac Fig. 8. Colloid velocity/tracer
velocity vs. colloid dispersion/tracer
The graphs do not give an exact constant both experiments the ratios are clustered.
3. Validation and implications
dispersion,
experiment
for either Dco,,/Dtrac or U,ol,/~rac
4
but in
for colloid transport parameters
If we were required to describe the migration of, say, the colloids in experiment 4, as well as tracer or solute species within a similar aquifer over a larger distance, then on the basis of these results an elution rate of UCO,,= 3/2U,,,, and a dispersion rate of D con = 4/5%, would be supportable. Of course, the colloid submodel still requires
252
P. Grindrod et al. /Journal
of Contaminant Hydrology 21 (1996) 243-253
capture (or filtration) and, possibly, re-release terms to be calibrated. This issue has been addressed in Grindrod (1994). The results given here should be particularly important where parameters are to be inferred (upscaled) to consider transport over large distances (e.g., performance assessment scales). Scale dependence is a well-known problem affecting solute dispersion, and a number of studies have indicated channelling models or channel network models (Grindrod and Impey, 1993) should be considered for fractured rock. Even for porous media, the scale of resolution of the model geometry assumed imposes the need for scale-dependent dispersivities to be set for solutes: there is a lot of evidence for this (Neuman, 1990). Hence we must be consistent: considering upscaling issues, colloid transport parameters may best be described relative to those for solutes. In general when modelling key migration processes and their mathematical representation we consider alternative theoretical or conceptual models, and then try to make predictions for experimental and field behaviour which can be tested directly. This should include input to the experiment design phase so that the data collected are relevant to testing modelling hypotheses. In considering colloid retardation mechanisms for example, we might consider a number of processes: dynamic chemical retention and re-release; migration into stagnant water-filled pore space; instantaneous adsorption; nonlinear effects such as partial monolayer limited adsorption; etc. (Grindrod, 1993b). Theoretically, each process has a certain signature, and could be discriminated in well-designed experiments. The results of such model validation studies must feed back to the models by: (1) supporting or denying existing approaches, on the basis of their ability to predict or represent observed behaviour; (2) refining model descriptions. In summary, we have proposed and utilized a method by which free colloid migration and dispersion rates can be calculated from early experimental breakthrough curves. We have obtained results which support the modelling predictions made previously on the basis of theoretically upscaling microscopic conditions (Grindrod, 1993a). This is a preliminary attempt to test such model terms and predictions, and is a key step in our ongoing model ualidation programme. Acknowledgements Part of this work (P.G. and M.S.E.) was carried out within the project “The role of colloids in the transport of radionuclides in geological media”, jointly funded by HMIP UK and the CEC Radioactive Waste Programme. The field experiments at Drigg (Higgo et al., 1993) were funded by HMIP UK and the authors are grateful to BNFL for use of their Drigg site. The paper is published by permission of the Director of BGS (Natural Environmental Research Council). References Grindrod, P., 1993~ The impact of colloids on the migration and dispersal of radionuclides within fractured rock. In: J.I. Kim and G. de Marsily (Editors), Chemistry and Migration of Actinides and Fission Products. J. Contam. Hydrol., 13: 167-182 (special issue).
P. Grindrod et al. /Journal
of Contaminant Hydrology 21 (1996) 243-253
253
Grindrod, P., 1993b. Identification and validation of model processes controlling colloid migration. Proc. MIRAGE Proj., 3rd Phase Progr. Meet., Brussels, Oct. 1993. Grindrod, P., 1994. How should the migration and retardation of colloids be characterized? Proc. 5th Ann”. Conf. on High Level Radioactive Waste Management, Las Vegas, NV, 1994. Proc. Am. Sot. Civ. Eng. Grindrod P. and Impey, M.D., 1993. Channelling and Fickian dispersion in fractal simulated porous media. Water Resour. Res., 29(12): 4077-4089. Higgo, J.J.W., Williams, G.M., Harrison, I., Warwick, P., Gardiner, M. and Longworth, G., 1993. Colloid transport in a glacial sand aquifer: Laboratory and field studies. Colloids Surfaces A: Physiochem. Eng. Aspects, 73(1/3): 179-200. Neuman, S.P., 1990. Universal scaling of hydraulic conductivities and dispersivities in geological media. Water Resour. Res., 26(g): 1749-1758. Press, W.H., Flannery, B.P., Teukolsky, S.A. and Vetterling, W.T., 1987. Numerical Recipes - The Art of Scientific Computing. Cambridge University Press, Cambridge, 307 pp.
ELSEVIER
Contaminant Hydrology Journal of Contaminant
Hydrology 21 (1996) 243-253
Analysis of colloid and tracer breakthrough curves Peter Grindrod a,*, Mark S. Edwards a, Jenny J.W. Higgo b, Geoffrey M. Williams b a Intera Information Technologies Ltd., Henley-on-Thames, RG9 IAT, UK b British Geological Survey, Keyworth, NG12 SGG, UK
Received 17 December 1993; accepted 15 December 1994 after revision
Abstract We consider the dispersion and elution of colloids and dissolved nonsorbing tracers within saturated heterogeneous porous media. Since flow path geometry in natural systems is often ill-characterized macroscopic (mean) flow rates and dispersion tensors are utilized in order to account for the sub-model scale microscopic fluctuations in media structure (and the consequent hydrodynamic profile). Even for tracer migration and dispersal this issue is far from settled. Here we consider how colloid and tracer migration phenomena can be treated consistently. Theoretical calculations for model flow geometries yield two quantitative predictions for the transport of free (not yet captured) colloids with reference to a non-sorbing dissolved tracer within the same medium: the average migration velocity of the free colloids is higher than that of the tracer; and that the ratio of the equivalent hydrodynamic dispersion rates of colloids and tracer is dependent only upon properties of the colloids and the porous medium, it is independent of pathlengths and fluid flux, once length scales are large enough. The first of these is well known, since even in simple flow paths free colloids must stay more centre stream. The second, if validated suggests how solute and colloid dispersion may be dealt with consistently in macroscopic migration models. This is crucial since dispersion is usually ill-characterized and unaddressed by the experimental literature. In this paper we present evidence based upon an existing Drigg field injection test for the validity of these predictions. We show that starting from experimental data the fitted dispersion rates of both colloids and non-sorbing tracers increase with the measured elution rates (obeying slightly different rules for tracers and colloids); and that the ratio of colloid and nonsorbing tracer elution rates, and the ratio of colloid and nonsorbing tracer dispersion rates may be dependent upon properties of the colloids and the medium (not the flow regime).
* Corresponding
author.
0169-7722/96/$15.00 0 1996 Elsevier Science B.V. All rights reserved SSDI 0169-7722(95)00051-S
244
P. Grindrod et al. /Journal
of Contaminant Hydrology 21 (1996) 243-253
It is important to realize that even for unretarded species, an earlier peak in the breakthrough curve does not necessarily correspond to a faster mean elution rate, or vice versa. But rather that a colloid may elute faster but disperse less than an equivalent tracer. Hence its peak may be retarded compared to that of the tracer, even assuming no retardation. Hence one must consider a combination of mean elution rate and mean dispersion rate, and not rely on “peak times” to corroborate chromatographic effects. The importance of this lies in the fact that these processes are not independent and yet upscale differently. Thus realistic estimates of effective colloid dispersion rates should be upscaled in a way consistent with that adopted for tracers within the same system.
1. Introduction In previous work (Grindrod, 1993a) a theoretical approach to the dispersion and elution of colloids and tracers within saturated heterogeneous porous and fractured media has been developed. In this paper we shall consider the results of the first investigation in the light of data from field injection experiments (Higgo et al., 1993). Typically flow path geometry is ill-characterized in models of natural systems and as a consequence macroscopic (mean) flow rates and dispersion tensors are utilized in order to account for the of impact microscopic fluctuations in media structure (and the consequent hydrodynamic profile), which are unaccounted for within the transport model considered. Even for tracer migration and dispersion this issue is far from settled. There is field evidence of scale-dependent dispersive effects (Neuman, 1990), and the existence of representative volumes over which processes could be averaged is questionable, and should be validated on a site-specific basis. Given such uncertainty it has been suggested (Grindrod, 1993a) that within any particular flow system (actual or theoretical), colloids and tracer migration phenomena should be treated consistently. As a result of this approach calculations for model flow geometries have yielded two quantitative predictions for the transport of free (not yet captured) colloids with reference to a nonsorbing tracer within the same medium. Briefly they are: (1) the instantaneous average migration velocity of the free colloids is higher than that of the tracer; (2) the ratio of the equivalent hydrodynamic dispersion rates of free colloids and tracer is dependent only upon colloid properties: size (exclusion from a portion of the available pore space), the existence of surface forces between the colloids and the medium surfaces (resulting in deterministic motion reflective of the fluid flow), and so on. The first of these is well known, since even in simple flow paths free colloids must stay more centre stream. Thus mean elution rates for colloids are between 1 and 1.3 times those for the tracer (Grindrod, 1993a). Colloids are either physically excluded from low-permeability zones or are entrapped there (physically or chemically). Hence early colloid breakthrough will be dominated by those colloids which have remained uncaptured during their migration, and have thus been advected along the faster, more permeable, flow paths,
P. Grindrod et al. /Journal
of Contaminant Hydrology 21 (1996) 243-253
245
The later breakthrough behaviour may show effects of physical filtration (clogging) and release, as well as electrostatic or chemical capture and release. Hence in considering the dispersal of free colloids one must focus on the rising limb of the breakthrough curves, avoiding the behaviour in the tails of the curves. Similarly, one may heuristically argue that the dispersion of such free colloids must be less than that of the tracer owing to the same processes: the transfer of tracer into dead-end pores or regions of relatively low permeabilities providing a large dispersive effect. The relative dispersion rate of free colloids to that of tracer is thus governed by those colloid properties which control exclusion or capture. This idea is again borne out by the mathematical computation for simple geometries (Grindrod, 1993a1, where predictions were made for advection and dispersion rates solely based on theoretical considerations. In addition, dispersion rates were shown to be flux dependent. Thus for both colloids and tracer alike it was anticipated that the elution rates and dispersion rates would be inter-related: dispersion rates increasing with elution rates. Specifically, for identical flow systems, it was shown in Grindrod (1993a) that the dispersion rates should increase with the measured elution rates (obeying slightly different rules for tracers and colloids); and that the ratio of colloid and nonsorbing tracer elution rates, and the ratio of colloid and nonsorbing tracer dispersion rates are dependent solely on properties of the colloids and the medium (not the flow regime). We shall examine some of these predictions in the light of breakthrough data given in Higgo et al. (19931, for a number of migration tests carried out within a shallow confined sand aquifer at Drigg, Cumbria, U.K. For four distinct types of colloid, injections were made at a single borehole, and breakthrough data were subsequently collected at other boreholes (Fig. 1). At each collecting borehole there are a number of ports, providing a vertical stratification of the tracer and colloid migration within the aquifer. The analysis below uses data from two downstream boreholes, and considers two types of colloids, fulvic acid and an artificial polymer, sodium polystyrene sulphonate. Thus for each injection experiment we are able to monitor the breakthrough of both colloids and a nonsorbing tracer at 11 specific locations. Examples of breakthrough curves are given in Fig. 2a and b. Considering curves such as Fig. 2a and b it is important to realize that even for an unretarded species, where the breakthrough curve is of the form of Eq. 3 below, the time to peak breakthrough is dependent upon both the mean velocity (or mean elution rate), U say, and the effective dispersivity, D say. In this case it is given explicitly by:
Tpeak
22’3( -2D2 =
3u
-
21/3w1/3
+ 7DLU - 2L2U2) 3u2w’13
+
where W= (-4D3
+ 21D2LU+
+33/2U&
6DL2U2 + 4L3U3
9D4L2 + 46D3L3U - 16D2L4U2 + 8DL5U3 >
and L is the breakthrough
distance.
3u2
246
P. Grindrod et al. /Journal
of Contaminant Hydrology 21 (1996) 243-253
Water recirculated between D 221 and D 13.00 D221
1
D225
D224
D227
D???
0226
D219
D220
- Ground level 11.00
10.50 i
_Silty
Clay
_ Fine to medium stratified sands Laminated silts ant /very fine sands
dilty
5.50
Clay
5.00
HORIZONTAL
4.50
SCALE
SAME
AS
D220
PI;
@
Jiew
Fig. 1. Details of the Drigg borehole array.
This is a decreasing function of D: (plot it!). Clearly overall earlier elution ahead of the peak does not necessarily imply a higher mean velocity (mean elution rate), and vice versa. All too often the debate about colloid mobility has focussed upon chromatographic effects, and has ignored the subtleties of
P. Grindrod et al. /Journal
247
of Contaminant Hydrology 21 (1996) 243-253
I l.OOE-04 l.OOE-05~
- : 4.00
16.00
31.90
46.06
61.00
76.00
96.00
b
Time (lx)
Polystyrene sulphonate
l.OOE-01 -. l.OOE-02 . .
i.OOE-05 4 3
9
19
29
41
61
110
146
k
Time (hr) Fig. 2. a. Tritium and fulvic breakthrough at D226 port 9.
acid breakthrough
at D226 port 9. b. Tritium
and polystyrene
sulphonate
the impact of dispersion. Hence it does not follow from Fig. 2a, that the tritium has a higher mean elution rate than the fulvic, but rather that the earlier breakthrough and earlier peak may be due to a larger effective dispersion rate. Although the results presented here are preliminary, we feel it is important to address both aspects, since the upscaling of dispersion is problematic - even for tracers. Since colloids may become physically filtered by, or chemically bound to, the medium, it is important to separate the hydrodynamic dispersion of free colloids from dispersion due to capture and release - which are controlled by separate terms in a macroscopic model. For this reason our analysis of breakthrough pulses concentrates on early arrival times, where advection and hydrodynamic dispersion processes dominate. Typically, the initial breakthrough of colloids was faster than that of the tracer, but tails were much longer owing to colloid sorption, capture and release. Consequently, the breakthrough curves observed were not always complete. For this reason the analysis presented below should be considered preliminary, and is concentrated only on the relatively early behaviour, dominated by a proportion of the colloids which passed freely
248
P. Grindrod et al. /Journal
of Contaminant Hydrology 21 (1996) 243-253
through the most permeable pathways, so as to avoid a discussion possible effects of colloid retardation, filtration and re-release.
of the nature and
2. Results Taking data from Higgo et al. (1993) we analyzed the early time breakthrough behaviour by considering only the first half of the normalized arrival pulse. Such early breakthrough is assumed to be dominated by colloids which have remained free, unsorbed and unfiltered, whereas later breakthrough behaviour must show some effects of capture and re-release (particularly in the tail). Assuming that advection and hydrodynamic dispersion processes dominate during this phase we fit equivalent average velocities, U, and dispersion rates, D according to the behaviour of solutions for the macroscopic one-dimensional equation: ac
2
_ZDKJE
(1)
at
where C(x, t) denotes the colloid or tracer concentration. Of course for later times the behaviour is more complex and Eq. 1 should be amended to include filtration, sorption and release. We calculated D and U using a quasi-Newton minimization procedure [DavidonFletcher-Powell (DFP) algorithm (see Press et al., 1987)]. This procedure requires a function to be minimized together with its partial derivatives. The function used was:
5 [Co(ti) i=
(2)
-RC,(U,D,ti>]’
1
where C,(t,) denotes the observed concentration at time t, (i = 1, . . . , m, where m is the number of observations considered); R is a scale factor (accounting for the fact that the final arrival mass is not necessarily known); and C,(U,D,t,) denotes the theoretical (normalized) flux across x = 1 (found by the solution of Eq. 1) at time t, given by: [-(IC,(U,D,t,)
Utl)2/4Dt,](l+
= exp /z+
Ut,)
(3)
where 1 is the breakthrough length, given; and D and U are the dispersion and velocity rates, respectively, to be calculated. The problem, therefore is to fit D, U and R so that our function (2) is minimized. With the same notation as is used in Higgo et al. (1993) (experiments, boreholes, ports) we derived early time D- and U-values for both tracers and colloids. The results are shown in Figs. 3-8. Figs. 3 and 4, 6 and 7 depict fitted dispersion rates plotted against fitted advective velocities for both colloids and tracers at both boreholes. For each borehole the separate points correspond to breakthrough curves normalized at particular ports. The lines indicate the general trends of the monotonic dependence of D upon U: their slopes may be interpreted as dispersion lengths for tracer and colloid species. Figs. 3 and 4
P. Grindrod et al. /Journal
of Contaminant Hydrology 21 (1996) 243-253
249
Irn
0.007
0.002
0.006
I
:
Velocity Fig. 3. Fitted effective
dispersion
(m* h- ‘) vs. fitted mean velocity
(m h-l),
experiment
correspond to experiment 1 - fulvic acid colloid and tritium tracer, correspond to experiment 4 sodium polystyrene p-sulphonate tracer (experiments 2 and 3 did not yield enough data). As anticipated, we see (from Figs. 3 and 4, 6 and 7) that in both sodium polystyrene p-sulphonate experiments, early colloids: (1) migrate faster than the nonsorbing tracer; (2) disperse less than the nonsorbing tracer. Also we see for both the colloids and tracers, that the dispersion observed velocities, according to different observed relationships squares “line up” differently).
Fig. 4. Fitted effective
dispersion
(m2 h _ ‘) vs. fitted mean velocity
1, borehole
D224.
whilst Figs. 6 and 7 colloid and tritium the fulvic acid and
rates increase with (the triangles and
(m h-’ ), experiment
1, borehole
D226.
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aloya~oq ‘p luaqtadxa
‘( 1
q “) .hrqa~ umm pall!3 ‘s* (, q zm) uoyads!p
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8
%d ‘palw!pur se aIoyaloq e U~I!M vod e w! sJatatue.wd pan!3 aql30 sow ayl 01 spuods -a1103 lu!od y~es ‘1 luaurnadxa 103 3e’Lfl/1105fl‘sh 3e”~/1103g MOqS afi g ‘%d uI
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P. Grindrod et al. /Journal
251
of Contaminant Hydrology 21 (1996) 243-253
: ii
k a
:
m
c:2
s
o.ooo, 0.01
-
,
0.02
.
,
,
0.03
0.04
_
,
_
0.05
0.0I
Velocity Fig. 7. Fitted effective dispersion (m2 hK‘) vs. fitted mean velocity (m h-l),
experiment
4, borehole
D226.
-_
iA A
1.1s 1.1-
A ?? 1.4.
+
I'
12. m 1.0 .I
A
1 1
dcollldtrac Fig. 8. Colloid velocity/tracer
velocity vs. colloid dispersion/tracer
The graphs do not give an exact constant both experiments the ratios are clustered.
3. Validation and implications
dispersion,
experiment
for either Dco,,/Dtrac or U,ol,/~rac
4
but in
for colloid transport parameters
If we were required to describe the migration of, say, the colloids in experiment 4, as well as tracer or solute species within a similar aquifer over a larger distance, then on the basis of these results an elution rate of UCO,,= 3/2U,,,, and a dispersion rate of D con = 4/5%, would be supportable. Of course, the colloid submodel still requires
252
P. Grindrod et al. /Journal
of Contaminant Hydrology 21 (1996) 243-253
capture (or filtration) and, possibly, re-release terms to be calibrated. This issue has been addressed in Grindrod (1994). The results given here should be particularly important where parameters are to be inferred (upscaled) to consider transport over large distances (e.g., performance assessment scales). Scale dependence is a well-known problem affecting solute dispersion, and a number of studies have indicated channelling models or channel network models (Grindrod and Impey, 1993) should be considered for fractured rock. Even for porous media, the scale of resolution of the model geometry assumed imposes the need for scale-dependent dispersivities to be set for solutes: there is a lot of evidence for this (Neuman, 1990). Hence we must be consistent: considering upscaling issues, colloid transport parameters may best be described relative to those for solutes. In general when modelling key migration processes and their mathematical representation we consider alternative theoretical or conceptual models, and then try to make predictions for experimental and field behaviour which can be tested directly. This should include input to the experiment design phase so that the data collected are relevant to testing modelling hypotheses. In considering colloid retardation mechanisms for example, we might consider a number of processes: dynamic chemical retention and re-release; migration into stagnant water-filled pore space; instantaneous adsorption; nonlinear effects such as partial monolayer limited adsorption; etc. (Grindrod, 1993b). Theoretically, each process has a certain signature, and could be discriminated in well-designed experiments. The results of such model validation studies must feed back to the models by: (1) supporting or denying existing approaches, on the basis of their ability to predict or represent observed behaviour; (2) refining model descriptions. In summary, we have proposed and utilized a method by which free colloid migration and dispersion rates can be calculated from early experimental breakthrough curves. We have obtained results which support the modelling predictions made previously on the basis of theoretically upscaling microscopic conditions (Grindrod, 1993a). This is a preliminary attempt to test such model terms and predictions, and is a key step in our ongoing model ualidation programme. Acknowledgements Part of this work (P.G. and M.S.E.) was carried out within the project “The role of colloids in the transport of radionuclides in geological media”, jointly funded by HMIP UK and the CEC Radioactive Waste Programme. The field experiments at Drigg (Higgo et al., 1993) were funded by HMIP UK and the authors are grateful to BNFL for use of their Drigg site. The paper is published by permission of the Director of BGS (Natural Environmental Research Council). References Grindrod, P., 1993~ The impact of colloids on the migration and dispersal of radionuclides within fractured rock. In: J.I. Kim and G. de Marsily (Editors), Chemistry and Migration of Actinides and Fission Products. J. Contam. Hydrol., 13: 167-182 (special issue).
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of Contaminant Hydrology 21 (1996) 243-253
253
Grindrod, P., 1993b. Identification and validation of model processes controlling colloid migration. Proc. MIRAGE Proj., 3rd Phase Progr. Meet., Brussels, Oct. 1993. Grindrod, P., 1994. How should the migration and retardation of colloids be characterized? Proc. 5th Ann”. Conf. on High Level Radioactive Waste Management, Las Vegas, NV, 1994. Proc. Am. Sot. Civ. Eng. Grindrod P. and Impey, M.D., 1993. Channelling and Fickian dispersion in fractal simulated porous media. Water Resour. Res., 29(12): 4077-4089. Higgo, J.J.W., Williams, G.M., Harrison, I., Warwick, P., Gardiner, M. and Longworth, G., 1993. Colloid transport in a glacial sand aquifer: Laboratory and field studies. Colloids Surfaces A: Physiochem. Eng. Aspects, 73(1/3): 179-200. Neuman, S.P., 1990. Universal scaling of hydraulic conductivities and dispersivities in geological media. Water Resour. Res., 26(g): 1749-1758. Press, W.H., Flannery, B.P., Teukolsky, S.A. and Vetterling, W.T., 1987. Numerical Recipes - The Art of Scientific Computing. Cambridge University Press, Cambridge, 307 pp.