## Q Qo Q

and for the chemical constituent

where Q is flow, C is concentration, the subscript o indicates old or pre-event water, and the subscript n represents new or event water. These are two simultaneous equations in two unknowns (Qo and Qn) and can therefore be solved easily when it can be assumed that the concentrations for the two sources are constant in time and distinctive. Substituting (11.1) into (11.2) and rearranging gives the result:

The need for a difference in the old and new water concentrations is then obvious. If the two waters cannot be differentiated, the denominator in (11.3) is zero and the calculation is indeterminate. Given a distinctive difference in the source concentrations, however, (11.3) can then be applied at every time step during a hydrograph to effect the separation between old and new water. An example using isotope concentrations is shown in Fig. 11.5. In effect, the constant concentration assumption is equivalent to assuming that the sources of pre-event water are well mixed before the event and are displaced from storage by the storm inputs.

Example: Oxygen isotope (18O) concentration data are to be used to carry out a two-component hydrograph separation for a storm on 26 July 1995 in the 160-ha catchment of Shelter Creek in the Catskill Mountains of New York State (adapted from Brown et al., 1999). The mean concentration of 18O in the rainfall recorded for the event was -4.83 S18O (oxygen isotope concentrations are expressed in S units, as differences from an international reference standard). The concentration in the stream prior to the event, taken as indicative of the old water component was -9.85 S18O. Concentrations in the stream water were measured on grab samples taken during the storm.

At the peak discharge of 0.59 mm h-1, the 18O concentration in the stream was -7.06S18O. Applying equation (11.3) gives an old water contribution to the total discharge of:

Qo = 0.59 x (-7.06 + 4.83)/(-9.85 + 5.83) = 0.33 mm h-1

This is 55 per cent of the total flow at the hydrograph peak, leaving only 45 per cent estimated as being supplied by event water. The 'old' water is being displaced from storage by the infiltration of event water into the soils. This calculation can be repeated at each time step for which an isotope concentration is available. The full results for this storm are shown in Fig. 11.5.

Brown et al. (1999) also carried out a three-component mixing (see next section) which suggested that there was an additional component of pre-event soil water contributing to the hydrograph.

A particularly interesting application of two-component mixing has been in the investigation of incremental lateral subsurface inputs into a stream channel. For each reach of the channel, the two components are the flow from upstream and the subsurface lateral input (which again have to have different concentrations). This can also be considered a form of successive dilution gauging (see Section 7.5). An early application by Huff et al. (1982) using an artificial tracer added to the Walker Branch stream

Fig. 11.5 Results of a two-component hydrograph separation using l8O as an environmental tracer for the event of 26 July 1995 in the Shelter Creek catchment, Catskills, New York State. Solid line is total discharge (mm h-1), dotted line is pre-event or old water component (mm h-1). (Data taken from Brown et al., 1999, with kind permission of Elsevier.)

Fig. 11.5 Results of a two-component hydrograph separation using l8O as an environmental tracer for the event of 26 July 1995 in the Shelter Creek catchment, Catskills, New York State. Solid line is total discharge (mm h-1), dotted line is pre-event or old water component (mm h-1). (Data taken from Brown et al., 1999, with kind permission of Elsevier.)

in Tennessee revealed that there were strong spatial differences in subsurface inputs to the stream related to upward dipping bedrock structures. These results were later confirmed at different discharges by Genereux et al. (1993; Fig. 11.6). They also investigated the changing patterns of lateral inflows under different discharge conditions. A modern variation on this is to use the spatial pattern of temperatures in a stream to indicate mixing between surface and subsurface sources. This is a much more complicated problem, however, and requires modelling a number of different energy balance components in each reach of a river (Westhoff et al., 2007).

### 11.3.2 End-member mixing analysis

More complex mixing models are also possible and are referred to under the general name of end-member mixing analysis (EMMA). Modifications to the two-component model have been made to allow for temporal variability in the input and end-member concentrations and to include more components. If we wish to separate out more components, then more chemical constituents will be needed. Three components will result in three unknown variables and, assuming that mass balance holds, will require two chemical constituents that have different concentrations for different water sources. For three-component mixing this can be visualised in the form of a mixing diagram such as that of Fig. 11.7. This shows how, for this small catchment in Switzerland, measured concentrations of calcium and silica in rainwater are clearly different from those in soil water and those in deeper groundwater. This is a result of the particular conditions in this catchment, in which the soils are relatively acid, favouring high-silica concentrations, but waters that penetrate to the Tertiary period

 Qweir = 395 L min-1 I-1 —i , u Ur- i—

100 200 Meters upstream of weir

### 100 200 Meters upstream of weir

Fig. 11.6 Lateral inflows to the West Fork of Walker Branch, Oak Ridge (TN) determined by a two-component mixing model using salt as a tracer. (Reproduced from Genereux et al., 1993, with kind permission of Elsevier.)

Acid soil [AS]

/ Direct precipitation (DP)

500 1000 1500 2000 Calcium [^eq L-1]

2500

3000

Fig. 11.7 Mixing diagram for separation of sources of stream discharge in the Haute Menthue catchment, Switzerland, using silica and calcium concentration observations (after lorgulescu et al., 2005, with kind permission of John Wiley & Sons).

Qweir = 1148 L min-1

Molasse bedrock are alkaline and high in calcium. Thus, in this case, the mixing equations are as follows for water

for the silica

for the calcium

The solution to these equations, again under the assumptions that the concentrations stay constant during an event, is shown in Fig. 11.8. Such an assumption is less tenable if we wish to consider a whole sequence of events, since in a second event the input of rainwater from the first event might expect to have an effect on the concentrations in different storages in a catchment. Iorgulescu et al. (2005) have attempted to model this, under different hypotheses about mixing of different sources over time for this same catchment. The results suggest that the proportion of soil water sources in the hydrograph increases significantly over time (Fig. 11.9). Fig. 11.7 also shows that the cd p

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