Kouli_etal_2008_Groundwater modelling_BOOK.pdf - Pantelis ...

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Nigel J. Cassidy
(2000). In general, the models correlate well to the experimental data for simple materials
(sands, rocks, low-clay content soils, etc) and, more significantly, have the capability of
including the complex permittivity into their formulations. This is particularly important for
detailed contaminant studies where the loss effects of the contaminating fluids results in
subtle spectral changes associated with the loss or imaginary component of the permittivity
(Cassidy, 2007).
Of all the models, the Complex Refractive Index Model (CRIM) is one of the most
popular for hydrological/contaminant based applications as it is simple to apply, robust for
most materials and accurate over the GPR frequency range (Ajo-Franklin et al., 2004;
Darayan et al., 1998; Endres and Knight, 1992). Strictly a one-dimensional, layered medium
model, CRIM has been shown to be effective for medium-to-coarse grained, multi-phase
mixtures involving simple granular materials (e.g., semi-spherical sand grains, etc) and
moderate-to-low viscosity fluids. It has the advantage of being a volumetric model that
requires only a knowledge of a material’s permittivities and their fractional volume
percentages. The general CRIM formula is
2
ε
⎞
i ⎟
⎠
⎛
= ⎜ N
e
ε ∑ f
(14)
mix i
⎝ i=
1
where ε e mix is the bulk effective permittivity of the mixture, f i is the volume fraction of each i
th component and ε i the permittivity of each i th component. Any number of phases can be
included but, in most cases, a three-phase model is appropriate with ε w , ε g , and ε m representing
the measured effective permittivities of water, gas (air) and the matrix respectively. As such,
the CRIM formula becomes
ε
e
mix
[( φ S ε ) + (( −φ)
ε ) + ( φ ( 1−
S ) ε ) ] 2
= (15)
w
w
1
m
w g
where φ is the porosity, S w the water or fluid saturation (i.e., percentage of pore space filled
with fluid) ε e mix the effective permittivity of the mixture and ε w , ε g , ε m the permittivities of the
water/fluid, gas and matrix phases respectively. For contaminated materials, an extra term can
be added to the CRIM model that represents the contaminant fraction, which effectively
becomes a component part of fluid phase. A representative example of CRIM’s use is shown
in Figure 9 were the complex effective permittivity spectrum of contaminant saturated sand is
compared to a groundwater saturated sand over the whole GPR frequency range (the
contaminant is a hydrocarbon-based, light non-aqueous phase liquid or LNAPL).
To develop the model, the complex permittivity of the matrix material (medium-to-coarse
grained, aeolian quartz-rich sands), groundwater and the LNAPL contaminant was measured
over the full frequency range using a vector network analyser technique (Cassidy, 2007). The
porosity of the sands was determined using standard laboratory methods (~40% by volume)
and a range of CRIM models produced for different contaminant and water saturations
(Cassidy, 2007). In comparison to measured values (Figure 9), the mixing model performed
well over the GPR frequency range of 10 MHz – 1GHz, with an excellent fit to the real
component of the permittivity and a good fit to the imaginary component.