Inverse relationship between hmod and tmod
This new approach has been applied to a large dataset of tracer tests mostly carried out in surface waters and karst or fractured groundwater systems. No information was available at a large scale for porous aquifers to apply this approach. Results from the MADE tracer tests (Adams and Gelhar
1992) have been used to validate the inverse relationship between mode and modal time of the RTD in a heterogeneous porous aquifer. In addition, factors ranging from 0.041 to 0.256 (0.1 on average) between macro-dispersivity and scale were found by Pickens and Grisak (
1981) in a stratified granular aquifer. Following Eq. (
3), these values correspond to a Pe ranging from 4 to 24 (10 on average), which is an intermediate value between fractured media and karst aquifers (Fig.
3).
Using regression analysis, Jobson (
1996) and Morales et al. (
2007) found power coefficients
b = −0.89 and
b = −0.85, respectively, to describe the evolution of the mode of the RTD with time in karst systems. These results were interpreted by the authors as evidence of non-Fickian behavior. Accordingly, Fig.
3 could also be used to fit power-law relationships relating
hmod to
tmod. Primarily, this interpretation prevents the assessment of the Peclet number using Eq. (
7), and another theory should be used to characterize transport processes. This would also mean for instance that all tracing systems performed in karst aquifers could be characterized by the same power-law relationship. There is however a bias in this interpretation when various tracing systems are brought together: the more advection dominates the solute transport processes, the easier it is to perform tracer tests over very long distances, and therefore with relatively long residence times. This sampling bias causes a positive trend between Pe and
tmod, which can explain why the fit of a power-law relationship related to surface streams, karst or fractured media in Fig.
3 will result in a power coefficient slightly lower than 1 (−0.88, −0.92, −0.85 respectively). This bias is not identified for a given tracing system (Fig.
1), which supports this interpretation.
Another explanation for the spreading of points in Fig.
3 comes from the use of a global characterization of hydrodispersive parameters of the tracing system. If the latter consists of an injection zone with distinct hydrodispersive parameters such as a thick infiltration zone, or a mixing in the water column in an injection well, the residence time distribution of the tracer within this first tracing subsystem should be taken into account. Considering the tracing system as a whole may, thus, introduce another bias in the analysis according to the relative importance of transport processes in each tracing subsystem encountered by the tracer cloud. This should however not be correlated to
tmod, and thus not introduce any trend in Fig.
3, but it can explain some noise in the relationships.
Finally, the interpretation of tracer tests performed in fractured media in Fig.
3 gives a Pe value close to 1, which means that hydrodynamic dispersion dominates the solute transport processes. For such low values, the interpretation of Pe based on the
hmod/
tmod diagnosis is very sensitive to the initial boundary conditions, i.e. the injection mode in resident or flux concentration, but also to the dilution procedure that is used for well injection, which is the most common procedure for tracer test performed in fractured media. Different protocols of tracer test could thus explain a greater variability in results for this type of media.
Worthington and Smart (
2003,
2011) used their own tracer database of 195 tracer tests to propose various empirical relationships that can be used to assess the mass to be injected. Among them, one significant relationship previously proposed by Dole (
1906) is of the form:
$$ M=A\times {\left({t}_{\mathrm{mod}}\times Q\times {C}_{\mathrm{peak}}\right)}^B $$
(8)
where
A and
B are two fitted parameters,
M (g) is the injected mass of the tracer,
Q (m
3/s) is the discharge and
Cpeak (g/m
3) is the peak of concentration at the detection point. According to Worthington and Smart (
2011), with a correlation coefficient of 0.96, Eq. (
8) gives
A = 0.84 and
B = 0.96. The latter being really close to 1, Eq. (
8) can consequently be re-written using
B = 1, which gives:
$$ \frac{Q\times {C}_{\mathrm{peak}}}{M}\cong \frac{1}{0.84\times {t}_{\mathrm{mod}}} $$
(9)
Assuming permanent flow conditions and neglecting the effect of partial mass recovery, the left side of Eq. (
9) is precisely the analytical expression of
hmod, which appears as inversely proportional to
tmod with a Pe close to 20 using Eq. (
7). These examples show how previous empirical relationships found between parameters of RTDs support the assumption of an inverse relationship between
hmod and
tmod. All these results suggest that the Peclet number can be seen as an intrinsic parameter of the dispersive media for large-scale contaminant transport studies, which leads to an inverse relationship between the mode and the modal time of the RTD.