6.4 Mass Balance

The principle of mass balance can be applied widely across many disciplines, such as meteorology, oceanography and geochemistry. In essence it applies when a set of inputs merge perfectly and completely into a set of outputs. Quite often a system is simplified into only 2 or 3 inputs and a single output. For example, precipitation, groundwater and surface water could be the inputs to a river, which is the output. Assuming conservation of mass, in other words, no losses along the way, then the river quantity and quality must comprise the sum of the quantity multiplied by the quality, of all of the inputs.

For most calculations, the quantities can be flow rates in liters or cubic meters per unit time, or something similar. The use of moles is necessary when dealing with different compounds, but this is beyond the scope of this text and is seldom needed when dealing with water only. For isotope compositions, it is generally fine to represent them as delta values, but they are most accurately calculated using isotope ratios (Hayes, 2004). The errors introduced by using delta values are caused by rounding errors and mole fraction mixing, both of which are worse when there are large differences in delta values (personal communication from Robert Kalin 2022). These discrepancies are mostly below analytical or other errors, and therefore probably only need to be considered when using high precision data (Steur et al., 2020).

Mass balance can be presented as an equation, here using stable isotopes ( values) as the quality parameter in Equation 14.

where:

It is important that the stable isotope compositions differ enough in the end-members for the method to work.

The simplest application of mass balance in stable isotope hydrology is to evaluate the groundwater contribution to surface water during precipitation events, often called storms. In this simple case, the mass balance equation is expressed by Equation 15.

where:

The isotopic composition of groundwater and pre-storm-event stream flow is assumed to be the same, as the river is assumed to be totally fed by baseflow, in other words, δ_g = pre-storm-δ_s. Note that these assumptions may often be close to the truth, but may be quite incorrect in other cases. The H and O isotope compositions for the precipitation (δ_p), groundwater (d_g) and storm flow (δ_s) are measured, as is the total river flow during the storm or flood event (Q_s). This leaves 2 unknowns, Q_g and Q_p, but these are related because they both add up to the streamflow. In equation form, this is as shown in Equation 16.

Then, by substituting into the original mass balance equation (Equation 15) and rearranging the terms, results in Equation 17.

In this way, with only one flow measurement (stream discharge), and three isotope values (precipitation, pre-storm streamflow and storm streamflow), we can calculate the baseflow and precipitation contributions to storm flow in a stream. In reality, due to the variability of isotope values in both precipitation and streamflow, multiple samples produce a more reliable result. For precipitation, instead of multiple samples, a single cumulative sample of all the precipitation during the storm can be taken, but this would most likely miss out on a lot of the finer detail and perhaps some critical changes in the contributions to flow that are known to occur (Xie et al., 2016).

This technique is suitable for small catchments with simple geology. Where groundwater contributions from the vadose and phreatic zones may differ, or where there is a large upstream catchment and the river flow may be expected to change stable isotope values during the storm due to varied isotope composition coming downstream, this method will need adjustment and more measurements will be required (Das et al., 2020).

Mass balance on a part of the Gariep River, the second largest river in southern Africa, after the Zambezi, was demonstrated by Diamond and Jack (2018) using only surface waters, as groundwater input was insignificant over short reaches. The stable isotopes were measured upstream on two tributaries and downstream after being well mixed at the location of a flow-gauging station. The contributions from the two tributaries were then calculated using the above equations.