4.1 Diffusion and Dispersion
Diffusion causes dissolved constituents to spread and decrease in concentration. Diffusion is caused by the random motion of molecules, known as Fickian motion, in which the system seeks maximum disorder, so that solute or isotopes in one location will spread to decrease that “order”. The magnitude of diffusion in porous media is expressed as a coefficient, D*, in units of length squared over time (e.g., m2/s) and depends on the nature of the dissolved species (size is a major factor) and the character of the porous medium (its porosity, size of the pores and tortuosity of the pore spaces are influencing factors). Dispersion also spreads dissolved constituents, but this spreading is caused by variations in velocities in heterogeneous materials. It is also expressed in units of length squared over time and includes the diffusion coefficient. Together, diffusion and dispersion cause dissolved constituents to spread in a groundwater system.
The previous section discussed how tracer ages are affected by mixing of discrete water samples. Groundwater flow systems are typically complex and heterogeneous, so rarely produce mixtures of parcels of water of discrete ages, because dispersion will cause mixing of water in one zone with water in adjacent zones. This usually means mixing water with other water that is slightly younger (above and/or upgradient) and slightly older (below and/or downgradient). The effect on bomb tracers such as 3H and 36Cl is that concentrations measured on groundwater samples from the 1950s and 1960s are always lower than the maximum concentrations that occurred in rainfall. Nevertheless, if samples are collected along a flow line, or as vertical profiles, the location of highest concentration in groundwater will indicate the period of highest concentration in rainfall. Furthermore, because the effect of dispersion can be large, the shape of the tracer profile can be used to estimate the dispersivity of the aquifer materials (Robertson and Cherry, 1989).
For tracers whose concentration varies more smoothly with time, the effect of dispersion on tracer concentrations will be less but can still be important. In the simplest case of one-dimensional flow in a homogeneous aquifer, longitudinal dispersion will cause 14C ages to be less than hydraulic ages, although the effect will only be significant for very large dispersivity values and low groundwater velocities. For a water velocity of 1 m/y and dispersivity of 1000 m, the hydraulic age will exceed the tracer age by less than 5 percent, although the difference increases to 30 percent for a velocity of 0.1 m/y and the same dispersivity (Castro and Goblet, 2005). For event markers, apparent tracer ages can either overestimate or underestimate hydraulic ages, depending on how the atmospheric input curve has changed with time. For CFCs, a water velocity of 1 m/y and dispersivity of 2 m results in tracer ages that differ from hydraulic ages by up to 10 percent for water with estimated recharge dates between the early 1970s and early 1990s, but the effect is greater (up to 30 percent) for water with estimated recharge dates from the early 1950s. Similar age discrepancies arise for 85Kr, but errors are larger on groundwater that was recharged prior to the mid-1960s (Ekwurzel et al., 1994). 3H/3He represents a special case, because the groundwater age is calculated from the ratio of two tracers, and these tracers can be transported at different rates. Errors associated with 3H/3He ages can be large for groundwaters recharged prior to the 1970s, as concentration gradients are greatest for groundwaters recharged immediately before and after the bomb peak (Schlosser et al., 1989).
Calculations of the extent of mixing due to dispersion in uniform, homogeneous flow fields can potentially underestimate the extent of mixing within real aquifer systems. The effect of mixing in heterogeneous aquifers on CFC concentrations and ages was examined by Weissmann and others (2002), who simulated solute transport in a small aquifer system consisting of five different interlayered materials with hydraulic conductivity values that varied over five orders-of-magnitude. The model found that variations in the travel paths of groundwater intercepted by wells with 1.5 m long intake zones resulted in relatively broad age distributions for groundwater samples. Apparent CFC-11 and CFC-12 ages of these mixtures would have underestimated mean hydraulic ages by a factor of approximately two. The extent of this bias in mean age caused by mixing within the aquifer will depend upon the degree of aquifer heterogeneity, the timing and magnitude of tracer input and the diffusion coefficient of the tracer (McCallum et al., 2014). Thus, the effect will differ between tracers. It is still unclear how significant this effect might be in different aquifers. However, similar apparent ages have been obtained using tracers with very different atmospheric input curves (e.g., CFC-12 and 3H/3He) in some aquifer systems (e.g., Ekwurzel et al., 1994), suggesting that this problem will be aquifer specific.