Introduction
It is well known that heterogeneities, including biogenic
pores/channels, desiccation cracks, fissures, fractures, nonuniform particle
size distributions and inter-aggregate pores, are widespread in the
subsurface and lead to a range of preferential flow phenomena (Beven and
Germann, 1982; Cuthbert et al., 2013; Cuthbert and Tindimugaya, 2010; Flury
et al., 1994). The coexistence of a relatively high hydraulic conductivity
(K) domain(s) and an impermeable one, often termed dual porosity, results
in a non-Fickian breakthrough curve. Solute transport in such systems is
often characterised by an early arrival of solutes originating from the more
mobile domain (macropores) and a slow approach to the final concentration
caused by diffusion into the immobile domain (matrix or microporous network).
When fitting breakthrough curves, therefore, it is often difficult to
differentiate between contributions from the micro- and macropore transport
mechanisms. As a consequence, in recent years there has been much research
into the development of effective empirical and modelling techniques to
characterise solute transport processes for dual porosity systems. One method
investigated has been the use of interrupted-flow solute-breakthrough
experiments. Amongst the original work on this topic Murali and Aylmore
(1980) discussed the influence of nonconstant flow on solute transport in
aggregated soil. Brusseau et al. (1989) developed a flow-interruption method
for use in measuring rate-controlled sorption processes in soil systems,
which was subsequently applied by Koch and Fluhler (1993) to investigate
advection and diffusion phenomena occurring for nonreactive solute transport
in aggregated media. The idea proposed was that, by interrupting flow during
nonreactive tracer breakthrough, the degree of nonequilibrium between any
fast- and slow-flow pathways can be determined. Central to this hypothesis is
that the magnitude of the change in nonreactive tracer concentration in
effluent samples taken immediately after a no-flow period is indicative of
such nonequilibrium. Subsequent work within this field has included
determination of physical (e.g. diffusive mass transfer between advective and
nonadvective water) and chemical (e.g. nonlinear sorption) nonequilibrium
processes in soil (Brusseau et al., 1997), determination of nonreactive
solute exchange between the matrix porosity and preferential flow paths in
fractured shale (Reedy et al., 1996), quantifying the effect of aggregate
radius on diffusive timescales in dual porosity media (Cote et al., 1999),
numerical modelling of aqueous contaminant release in nonequilibrium flow
conditions (Wehrer and Totsche, 2003), empirical modelling of the release of
dissolved organic species (Guimont et al., 2005; Ma and Selim, 1996; Totsche
et al., 2006; Wehrer and Totsche, 2005, 2009) and heavy metals (Buczko et
al., 2004), increasing the efficiency of solute leaching (Cote et al., 2000),
empirical modelling of conservative tracer transport in a laminated sandstone
core sample (Bashar and Tellam, 2006), and characterising in situ aquifer
heterogeneity (Gong et al., 2010). One area where comparatively few studies
exist, however, is in characterising the hydraulic properties of aquitards
(e.g. clay-dominated soils and sediments, shales, and mudstones). Such
research is of particular interest because preferential flow paths, by their
intrinsic nature, can significantly compromise the integrity of aquitard
units as local and regional barriers to the movement of groundwater
contaminants. There are significant technical difficulties at present,
however, in characterising such features at appropriate scales (Cuthbert et
al., 2010). For example, it is well known that the K of glacial till is
scale dependent, with laboratory permeability measurements often yielding
values lower than field-based measurements and modelling (Cuthbert et
al., 2010). As a consequence, a key requirement of laboratory-scale aquitard
characterisation is that the core sample must be of sufficient volume in
order to incorporate the key dual porosity features which govern the overall
formation. A second technical challenge is that laboratory testing typically
requires generation of flow through the sample whilst maintaining relevant
in situ hydro-geotechnical conditions. One method which has been demonstrated
as effective for this purpose is centrifugation, which is increasingly being
used for hydraulic and geotechnical testing of low K materials (Hensley and
Schofield, 1991; Nimmo and Mello, 1991; Timms et al., 2009; Timms and Hendry,
2008). Moreover, experiments using geotechnical centrifuges with payload
capacities exceeding several kilograms can provide the additional benefit of
being able to use core samples of representative scale for the overall
formation. Here we present, for the first time, an interrupted-flow
methodology using a centrifuge permeameter (CP) to characterise possible dual
porosity behaviour of low permeability porous media. A novel dual domain
model is also described which has been used to guide physical interpretation
of the experimental tracer breakthrough curves.
Experimental methods
Core and groundwater sampling methodology
The clay core (101.6 mm in diameter, Treifus core barrel, nonstandard
C size) and groundwater were sourced from a 40 m thick, semi-consolidated,
clay-rich alluvium deposit located approximately 100 km south of Gunnedah,
New South Wales, Australia (31∘31′9′′ S,
150∘28′7′′ E). Equipment and procedures for
obtaining minimally disturbed cores were compliant with ASTM (2012). See
Timms et al. (2014) for a review of the procedure. Groundwater samples were
taken from piezometers using standard groundwater quality sampling techniques
(Sundaram et al., 2009). A 240 V electric submersible pump (GRUNDFOS MP1)
and a surface flow cell were used to obtain representative samples after
purging stagnant water to achieve constant field measurements of electrical
conductivity (EC), pH, dissolved oxygen (DO) and reduction potential
(Eh).
Centrifuge permeameter theory
During centrifugation, increased centrifugal force generates a body force
which accelerates both solid and fluid phases within the sample. Centrifugal
acceleration at any point within a centrifuge sample is calculated as
follows:
a=ω2r,
where a is the centrifugal acceleration (ms-2), ω is the
angular velocity (rads-1), and r is the radius from the axis
of rotation (m). The g level is the scaling factor (a/g) for accelerated
gravity, where g is gravity at Earth's surface.
Vertical hydraulic conductivity, Kv (ms-1), is
calculated using ASTM (2000) (Eq. 2), where Q is the steady-state fluid
flux (mLh-1), A is the sample flow area (cm2),
rm is the radial distance at the midpoint of the core sample
(cm), and RPM is revolutions per minute.
Kv=0.248QArm(RPM)2
The estimated in situ stress applied at the base of the core samples was
calculated according to Eq. (3) and assumes that the overlaying formations
were fully saturated and of a similar density to the core samples.
σi=ρsdg,
where σi is the in situ stress (kPa), ρs is
saturated density of core (kgm-3), d is the depth to the base
of the core sample (m below ground level (b.g.l.)); and
g is the gravitational acceleration (ms-2). The applied stress
at the base of the core (σg, kPa) during the centrifuge experiments
was calculated according to Eq. (4) (Timms et al., 2014).
σg=[(ρbLc)+ρw(Lc+hw)]ab,
where ρb is the core bulk density (kgm-3),
Lc is the length of the CP core specimen (mm), ρw is
the influent density (kgm-3), hw is the height of
influent water above the CP core specimen (mm), and ab is the
centrifugal acceleration at the base of the CP core specimen.
Centrifuge permeameter sample preparation
A Broadbent geotechnical centrifuge (GMT GT 18/0.7 F) with a custom-built
permeameter module (Timms et al., 2014) was used for this study. Prior to
mounting into the CP, the outer 5 mm of the clay cores were trimmed
and the trimmed cores were then inserted into Teflon cylindrical core holders
(100 mm internal diameter, 220 mm length) using a custom-built mechanical
cutting and loading device. The cores were trimmed in order to remove any
physical and chemical disturbance associated with the core extraction
(drilling) process. A 5 mm thick A14 Geofabrics Bidim geofabric filter
(100 micron, K=33 ms-1) was placed above and below the
sample in order to prevent clogging of the effluent drainage plate with
colloid material from the sample. The geofabric filter was held in position
above the sample using a plastic clamp.
The core holders (with the core sample held within) were placed into 3000 mL
glass beakers containing 1000 mL of groundwater derived from the
piezometer at the closest depth to the core sample (see Table 1) and allowed
to saturate from the base upwards. In total three core samples were analysed,
which were taken from depths of 5.03, 9.52, and 21.75 m b.g.l.
Saturation was performed by immersing the core holder into a reservoir of
groundwater with the level of the water 5 cm higher than the top of
the core sample. The mass of each core was then monitored every 24 h
until no further increase in mass was recorded, saturation was then assumed
to have occurred. The core holders (containing the saturated core samples)
were mounted to the CP system via double O-ring seals. An influent head was
added to all samples (see Table 1), which was maintained during
centrifugation by a custom-built automated influent level monitoring and
pumping system. The system comprises a carbon fibre EC electrode array which is connected via a fibre optic rotary
joint to a peristaltic pump that supplies influent from an external 100 mL
burette. Effluent samples were
collected in an effluent reservoir and extracted using a 50 mL syringe. All
experiments were conducted under steady-state flow, which is defined as a
< 10 % difference between influent and effluent flow rates. The
influent volume was determined by manual measurements of the water level in
the external burette and effluent volumes were measured by multiplying their
mass by their density.
Interrupted-flow experiment methodology
The idea of interrupting the flow during a breakthrough experiment is to
differentiate between advection and diffusion processes. The method comprises
a minimum of three phases.
Flow is induced at a constant centrifugal force for a fixed time period
with effluent samples collected at multiple periodic intervals. The g level
and influent reservoir height are selected so that the maximum total stress
on the core approached the estimated in situ stress of the material at the
given depth in the formation (Eqs. 3, 4). The time period between each
effluent sampling interval is selected in order to gain sufficient effluent
volume (namely > 1 mL) for accurate volume and nonreactive tracer
concentration measurement.
Flow is interrupted (stopped) for a fixed time period during which time
the permeameters are disconnected from the centrifuge module and positioned
upright, the influent reservoir is also removed to limit any downward
migration of solutes. A relatively long interrupted-flow period
(> 12 h) is selected so that slow mass transfer processes can be
identified.
Phase 1 is then repeated.
All phases can be repeated multiple times in order to record sufficient
nonreactive tracer breakthrough which enables the mass transport behaviour
to be accurately characterised. Deuterium oxide (D2O) (Acros
Organics, 99.8 % concentration) was used as a nonreactive tracer. A
concentration of 3.12 mLL-1 was used, which raised the
concentration of D2O to approximately 200 %. This was selected as
sufficiently high in order to result in accurately measurable mass transfer
changes. Effluent samples were filtered using a 0.2 µm cellulose
acetate filter, stored at 4 ∘C and analysed for δD within
7 days of testing. δD was determined by measuring the
1H/2H ratio to an accuracy of 0.1 % using a Los Gatos
DLT100 isotope analyser.
Core and influent properties, experimental parameters and
Kv results for the interrupted-flow experiments. Calculations are
based on Eq. (2) for Kv, Eq. (3) for estimated in situ total
stress and Eq. (4) for total stress at the base of the core specimen during
centrifugation.
Core depth
Estimated
Influent
Influent EC
g level
Core
Height of
Kv
Total stress
(m b.g.l.)
in situ
groundwater
(µScm-1)
applied
length,
influent water
(ms-1)
at base of
total stress, σi
depth
Lc
above core, hw
core during
(kPa)
(m b.g.l.)
(mm)
(mm)
centrifugation,
σg (kPa)
5.03
89
10
18 470
20
36
61
1.4×10-8
75
9.52
177
10
18 470
20
47
81
3.9×10-9
127
21.75
383
20
13 160
80
54
48
2.7×10-9
373
Dual domain transport modelling
Dual porosity models were created using COMSOL Multiphysics v. 4.4
(http://www.comsol.com) modified from well-known formulations
described, for example, by Coats and Smith (1964) and Bear and Bachmat
(1990). The purpose of the modelling was to aid physical interpretation of
the tracer breakthrough curves and validate the hypothesis that the step
changes in tracer concentrations observed during no-flow periods could be
explained by the presence of dual porosity in the samples. The models
comprised a classical advection–dispersion equation for a mobile zone
(subscript m) representing preferential flow pathways with a source/sink term
representing exchange of solute with an immobile zone (subscript im). Solute
transport in the immobile zone was by diffusion only. The exchanged flux
between the immobile and mobile zones was modelled as being proportional to
the concentration difference between the zones. The governing equations are
as follows:
∂Cm∂t=Dm∂2Cm∂z2-q(t)∅m∂Cm∂z-γ∅m(Cm-Cim),∂Cim∂t=μ∂2Cim∂z2+γ∅im(Cm-Cim),Dm=∝q(t)∅m+μ,
where C is the δD isotope ratio (1), t is time (T), z is
distance along the column (L), q is fluid flux (LT-1), α
is hydrodynamic dispersivity (L), μ is the coefficient of molecular
diffusion (L2T-1). The porosity, ∅, of the mobile and
immobile domain is defined as
∅m=Vp,mVT,∅im=Vp,imVT,
where Vp,m is the pore volume of the mobile domain (L),
Vp,im is the pore volume of the immobile domain (L) and
VT is the total volume of the saturated core (L). The mass
transfer coefficient, γ (T-1), is defined as
γ=β∅mμa2,
where β is the dimensionless geometry coefficient, which typically
ranges from 3 for rectangular slabs to 15 for spherical aggregates, and a
is the characteristic half width of the matrix block (L) (Gerke and van
Genuchten, 1993).
The initial concentration conditions were set to zero for both domains for
all model runs. During centrifugation periods, a variable solute flux upper
boundary condition was used for the mobile domain and varied according to the product of the
measured fluid flux and input concentration (C0) during each experiment as
follows:
qt∅mC0=qt∅mCm+Dm∂Cm∂z.
A Dirichlet (constant concentration) upper boundary condition was used for
the immobile domain during times of centrifugation. A novel aspect of the
models, facilitated by the flexibility of model structure variations possible
in COMSOL Multiphysics, was that the upstream transport boundary for both
domains was switched to a zero flux condition during the interrupted-flow
phases. The downstream transport boundary conditions for both domains were
given by
∂Cm,im∂z=0,
at z=Lb, where Lb was sufficiently large to ensure
the results at the column outlet distance (at z=Lc,
Lc≪Lb) were not sensitive to the position of the
boundary. The total mass flux at the distance from the upstream boundary
corresponding with the length of the experimental column was output from the
models and integrated over the sampling periods for comparisons to the
observed breakthrough curves. μ was calculated as 3.43×10-5 m2d-1 which is the diffusion coefficient of
D2O in H2O at 25.0 ∘C (Orr and Butler, 1935)
multiplied by the average tortuosity of 0.15 reported by Barnes and Allison
(1988) for clay bearing media. Model output was fitted to the observed data
by varying the unconstrained parameters: α and γ. Note that
∅m and ∅im were also considered
unconstrained parameters but their sum was constrained to equal total
∅ measured for each sample by oven drying at 105 ∘C for
24 h. In order to quantify the deviation between the recorded data
and the dual porosity model, the normalised root mean square error (NRMSE) and
the Nash–Sutcliffe model efficiency coefficient (NSMEC) were calculated
(Nash and Sutcliffe, 1970). The mesh size and model tolerance were set
sufficiently small so that the results were no longer sensitive to further
reduction, to ensure the accuracy of the model output. The models runs
presented were all executed using an extra fine mesh size and a relative
tolerance of 0.00001.
Dual domain model sensitivity testing
Sensitivity analysis of the dual domain model (for the core taken from
5.03 m) was conducted in order to determine how sensitive the model
was to changes in the constrained (Lc, ∅ and μ) and
unconstrained (∅m, α and γ) parameters.
Sensitivity factors for constrained parameters were determined according to
the estimated percentage error associated with each parameter, whilst
±50 % was selected for the unconstrained parameters in order to
determine their influence on the NSMEC. The percentage error for
Lc was calculated to be ±2.78 % due to the core length
being 36 mm, and the error associated with measurement at each end was
±0.5 mm. The percentage error for ∅ was calculated to
be ±2.79 %, which comprises the Lc measurement error plus
0.0026 % which is the calculated error associated with the two mass
measurements. The percentage error for μ was determined to be
±50 % due to the range in tortuosity of 0.1–0.2 documented by Barnes
and Allison (1988) and references therein.
Results and discussion
D2O breakthrough
D2O breakthrough data and best-fit dual porosity model output for the
interrupted-flow experiments conducted using core samples taken from 5.03,
9.52, and 21.75 m b.g.l. are displayed in Fig. 1. A close fit was
achieved between the dual porosity model output and the original data, with a
NSMEC of 0.97, 0.99 and 0.97 and a NRMSE of 5, 3, and 5 % recorded for
D2O breakthrough data from core samples taken from 5.03, 9.52, and
21.75 m b.g.l., respectively. The D2O breakthrough curves for
all core samples exhibited a relatively elongated shape, with 100 %
breakthrough not recorded for any of the timescales tested. This was expected
given that a “long tailing” is a common feature of dual (or multi-)porosity
materials, i.e. systems where the mobile domain is coupled to a less mobile,
or immobile, domain. In such instances the dominant solute transport
mechanism during imposed flow in the mobile domain(s) is typically advection;
however, solute exchange also occurs in parallel with the immobile domain(s),
typically via molecular diffusion. Following each interrupted-flow (no-flow)
period a decrease in δD was recorded for all samples, and attributed
to the diffusion of D2O from the preferential flow domain(s) into the
low-flow, or immobile-flow, domain(s). The shape of the D2O
breakthrough curves and the magnitude of the δD decrease following
the interrupted-flow periods are different for all samples, with a 42.6,
18.5, and 28.4 % decrease recorded for the core samples taken from 5.03,
9.52, and 21.75 m b.g.l., respectively, after the first interrupted-flow
period. In addition, the Kv of each sample was recorded as
different (Fig. 2), with average values of 1.4×10-8, 3.9×10-9, and 2.7×10-9 ms-1 for the core samples
taken from 5.03, 9.52, and 21.75 m b.g.l., respectively. The
Kv was recorded to decrease during the initial stages of each
centrifugation period, attributed to the partial consolidation of the
clay due to the stress applied by the centrifugal force. Following this
initial consolidation period a more constant Kv as a function of
time was recorded for all cores, indicating that relative equilibrium had
been achieved between stress applied by the centrifugal force and the
compaction state of the core.
Dual domain model
The close model fits confirm that preferential flow through a dual porosity
structure is a plausible hypothesis to explain the shape of the observed
breakthrough curves. The unconstrained (∅m, α and
γ) parameters that yielded the best dual domain model output fit to
the D2O breakthrough data are displayed in Table 2. It is noted that
the pore volume of the mobile domain per total volume of the core,
∅m, was modelled to be 0.06, 0.04, and 0.08 for core taken
from 5.03, 9.52, and 21.75 m b.g.l., respectively. With total porosity,
∅, measured as 0.44, 0.47, and 0.43, this equates to 13.6, 8.5, and
18.6 % of the total pore volume, respectively, suggesting that
preferential flow features comprise a relatively large proportion of the
total pore porosity in each sample. Hydrodynamic dispersivity, α, for
best-fit model output for all core samples was Lc/2, which is
larger than typically reported for laboratory-scale column experiments (e.g.
Shukla et al., 2003). It can be noted that all of the core samples were
assumed to have remained saturated throughout the breakthrough experiments
because all influent and effluent flow rates were recorded at steady state.
Whilst dispersion is known to increase substantially as moisture content
decreases from saturation (e.g. Wilson and Gelhar, 1981), it is therefore
unlikely that this could have been a factor. The mass transfer coefficient,
γ, was also modelled as different for each core sample with 0.65,
1.50, and 1.20 yielding the best model fit for the core samples taken from 5.03,
9.52, and 21.75 m b.g.l., respectively. Using Eq. (10), the half width
of the matrix block (using a β range of 3–15 (3 for parallel slabs and
15 for spherical aggregates after Gerke and van Genuchten, 1993)), a, is
calculated as within the range of 8.0–17.8, 5.4–12.1, and
5.5–12.3 mm for the core samples taken from 5.03, 9.52, and
21.75 m b.g.l., respectively. This suggests that the
preferential flow channels present are likely to be separated by distances in
the order of several millimetres from each other within the cores. With the dimensions
of the cores significantly greater than these values, the model output
therefore suggests that several preferential flow features are present in
each core sample.
Normalised D2O breakthrough data along with best-fit dual
porosity model output for the interrupted-flow experiments conducted using
core samples taken from 5.03 m (left), 9.52 m (middle), and
21.75 m b.g.l. (right). The data points represent the
concentration averaged over each sampling period and the dashed line for the
model output represents the raw model output time series. In the empirical
experiment it was therefore not possible to measure the concentration of the
effluent during the no-flow phase because there was no effluent to collect
for analysis. Thus, due to this averaging, in the rising limb of the
breakthrough curve, the first point obtained by measurement during each flow
phase can be observed as consistently greater than the “starting
concentration” for the raw model output.
Vertical hydraulic conductivity (ms-1), calculated using
Eq. (2), for the interrupted-flow experiments conducted using core samples
taken from 5.03 m (left), 9.52 m (middle), and 21.75 m b.g.l.
(right).
Constrained (D, Lc, ∅, μ) and
unconstrained (∅m, α and γ) model
parameters. a is calculated using Eq. (10).
Core depth
Core
Core
Total
Pore volume
Coefficient of
Hydrodynamic
Mass
Half
(m b.g.l.)
diameter,
length,
porosity,
of the mobile
molecular
dispersivity, α
transfer
width of
D
Lc
∅
domain per
diffusion, μ
(L)
coefficient, γ
the matrix
(mm)
(mm)
total core
(L2T-1)
(T-1)
block, a
volume, ∅m
(mm)
5.03
100
36
0.44
0.06
3.43×10-5
Lc/2
0.65
8.0–17.8
9.52
100
47
0.47
0.04
3.43×10-5
Lc/2
1.50
5.4–12.1
21.75
100
55
0.43
0.08
3.43×10-5
Lc/2
1.20
5.5–12.3
Model output for mobile (solid lines) and immobile (dashed lines)
domains for core samples taken from 5.03 m (left), 9.52 m
(middle), and 21.75 m b.g.l. (right). The black, dark grey, and
light grey lines comprise model output for the base, middle, and top of the
cores, respectively.
Model output for the mobile and immobile domains at the top, middle, and base
of the core samples is displayed in Fig. 3. It is noted that, for all core
samples, diffusion into the immobile domain during the induced-flow periods is
relatively significant, with δDim/δDm at the
end of the first centrifugation (induced-flow) period recorded as 0.16, 0.32,
and 0.34 for the base of the core samples taken from 5.03, 9.52, and 21.75 m b.g.l., respectively. With respective average flow rates recorded as
0.017, 0.007, and 0.015 md-1 this behaviour is not obviously
related to the variation in flow rates between the samples but more likely
to the intrinsic properties of the preferential flow domain (namely
∅m, γ, and α). It is also noted that for all
core samples full equilibration between the mobile and immobile domains
occurred (δDim=δDm) during each no-flow
period. For example, δDim and δDm were
modelled to be within ±1 % of each other after 7.0, 2.6, and
6.1 h during the first no-flow period for the core samples taken from
5.03, 9.52, and 21.75 m b.g.l., respectively.
Sensitivity analysis
Sensitivity analysis plots for a ±50 % change in unconstrained
parameters (α, γ, and ∅m) for the core
sample taken from 5.03 m b.g.l. are displayed in Fig. 4, with
corresponding NSMEC data displayed in Table 3. The model fitting efficiency
is relatively insensitive to all three unconstrained parameters in the range
tested, with a less than 12 % change in the NSMEC compared to the NSMEC
recorded for the best fit (Table 3). Sensitivity for the estimated percentage error
associated with constrained parameters (∅, Lc, and
μ) are displayed in Fig. 5, with corresponding NSMEC data displayed in
Table 3. The model fitting efficiency is also relatively insensitive, with a
less than 1 % change in the NSMEC compared to the NSMEC recorded for the
best fit (Table 3). For the data presented, the relatively low sensitivity to
the parameters indicates that further testing, such as by dye tracing or
geophysical tomography, is necessary to resolve more precisely the nature of
the preferential flow paths. Nevertheless, the modelling has supported the
preferential flow conceptual model we have used to explain the step changes
in concentration observed after resting periods. It has also provided a
first-order approximation of the likely geometry of the flow paths.
Sensitivity of the dual domain model for the core sample taken from
5.03 m b.g.l. due to ±50 % change in unconstrained
parameters: ∅m (LHS), γ (middle), and α
(RHS).
Sensitivity of the dual domain model for the core sample taken from
5.03 m b.g.l. for the calculated error associated with the
constrained parameters: ∅ (LHS); Lc (middle), and μ
(RHS).
Comparison of dual and single domain modelling
In order to further demonstrate the practicality of the interrupted-flow
methodology, a numerical experiment was carried out using the dual domain
model developed above. Using the best-fit parameters from the core from
9.52 m b.g.l., an equivalent simulation to the laboratory experiment
described above was run but without interrupted-flow phases. The
breakthrough curve produced was then fit to the Ogata–Banks equation (Ogata
and Banks, 1961) on the assumption that flow was occurring only through a
single domain. The resulting fit was good (NRMSE = 3 %) with just
one fitting parameter being the dispersion term which yielded a reasonable
value of 1.27×10-8 m2s-1. This illustrates that,
without the use of interrupted-flow phases to reveal the disequilibrium
between two or more flow domains, a false assumption could easily be made
with regard to the structure and associated transport properties of the core
on the basis of a simple 1-D analytical model. This could have very
significant consequences for the prediction and management of solute
migration through such deposits.
NSMEC for the core sample taken from 5.03 m b.g.l. due to
changes in constrained (Lc, ∅, μ) and unconstrained
(∅m, α and γ) model parameters. Changes in
constrained parameters comprised the estimated percentage error per each
parameter, which was 2.78, 2.79, and 50 % for Lc,
∅,
and μ, respectively. Changes in unconstrained parameters were
±50 %. The NSMEC for the best fit was 0.972.
Model
Pore volume of the mobile
Mass
Hydrodynamic
Total
Core
Coefficient
parameter
domain per total
transfer
dispersivity,
porosity,
length,
of molecular
pore volume, ∅m
coefficient, γ
α
∅
Lc
diffusion, μ
NSMEC (+ change)
0.925
0.926
0.952
0.974
0.965
0.964
NSMEC (- change)
0.952
0.862
0.964
0.968
0.971
0.975
An additional numerical experiment was also undertaken to attempt to match
the observed data to a single domain model which included resting phases,
since no analytical solution is known for such a simulation. This was
accomplished using COMSOL Multiphysics with identical settings to the dual
domain models described above but with a disabled immobile domain.
Calibrating to the δD breakthrough data recorded for the core from
9.52 m b.g.l. by just varying dispersivity, but using the
measured porosity, we were unable to achieve a better fit than a NRMSE of
46 %, even with an unrealistically high dispersivity. A better fit is
possible (NRMSE = 9 %, NSMEC = 0.9) if porosity is decreased to
0.1 but, again, only with an unrealistically high value for dispersivity of
1000Lc (see Fig. 6). While such a model may be useful to suggest
that the effective porosity of the core through which solute is moving is
much less than the total porosity, it is only possible to fit the early time
data (e.g. only the first flow stage) very accurately at the expense of the
later time data. Perhaps more importantly than the lower NSMEC (or higher
NRMSE) compared to the dual domain models, the single domain model also
misses a key feature of the observed breakthrough curves: the decrease in
concentration during resting phases. Instead, modelled concentrations
increase during resting phases as would be expected in a single domain model
due to redistribution of the solute along the core by diffusion. This
additional numerical experiment thus strengthens the conclusions of the
study, which are that dual domain behaviour is indicated by our interrupted-flow
experiment observations, and that single domain models are inappropriate as a
means of analysis.
Comparison of single and dual domain interrupted-flow transport
model best-fit simulations for the core taken from 9.52 m b.g.l.