M3 version 3.0: Verification and validation

Hydrochemical evaluation is a complex type of work that is carried out by specialists. The outcome of this work is generally presented as qualitative models and process descriptions of a site. To support and help to quantify the processes in an objective way, a multivariate mathematical tool entitled M3 (Multivariate Mixing and Mass balance calculations) has been constructed. The computer code can be used to trace the origin of the groundwater, and to calculate the mixing proportions and mass balances from groundwater data. The M3 code is a groundwater response model, which means that changes in the groundwater chemistry in terms of sources and sinks are traced in relation to an ideal mixing model. The complexity of the measured groundwater data determines the configuration of the ideal mixing model. Deviations from the ideal mixing model are interpreted as being due to reactions. Assumptions concerning important mineral phases altering the groundwater or uncertainties associated with thermodynamic constants do not affect the modelling because the calculations are solely based on the measured groundwater composition. M3 uses the opposite approach to that of many standard hydrochemical models. In M3, mixing is evaluated and calculated first. The constituents that cannot be described by mixing are described by reactions. The M3 model consists of three steps: the first is a standard principal component analysis, followed by mixing and finally mass balance calculations. The measured groundwater composition can be described in terms of mixing proportions (%), while the sinks and sources of an element associated with reactions are reported in mg/L. This report contains a set of verification and validation exercises with the intention of building confidence in the use of the M3 methodology. At the same time, clear answers are given to questions related to the accuracy and the precision of the results, including the inherent uncertainties and the errors that can be made when using M3 outside its realm of applicability. The verification exercises are designed to test the correct functioning of each part of the M3 code (Principal Components Analysis, mixing routines, mass balance routines, End-member Selection Module, and End-member Variability Module). Each test focuses on a particular algorithm or module. Synthetic datasets have been used in many tests as this is the best way to verify the results when dealing with mixing proportions and mass balance calculations. All the verification tests have been passed by M3, except the one dealing with the two-principal component mixing routine. This way of computing mixing proportions only gives consistent results with three end-members, and should not be used for cases with four or more end-members. The validation exercises go one step forward and test the ability of M3 methodology to solve mixing and reaction problems. Some tests focus on the uncertainties in the mixing proportions and others on the uncertainties in the calculated mass balances. Many validation exercises use a number of synthetic water samples inserted in a real groundwater dataset from the Laxemar- Simpevarp area in Sweden in order to assess the accuracy of the computed mixing proportions and deviations, to determine the limits of M3 applicability. In this respect, several validation exercises give clear indications that an incorrect use of M3 (i.e. for systems in which mixing is not the dominant process controlling the chemistry of the waters) can give rise to erroneous results. Most tests insist on the need for an independent assessment of the validity of the results given by M3 using, for example, expert judgment, other geochemical codes, or several lines of reasoning. In general, M3 deals successfully with most validation tests, although several clearly indicate where the limits of applicability are. This is particularly evident when chemical reactions significantly change the composition of the resulting mixed water. When reactions are more important than mixing, the computed mixing proportions may differ significantly from the real ones. However, checking the deviations between computed and real concentrations for the water conservative elements is the simplest way of assessing the quality of the computed mixing proportions. M3 is not the only code that can be used for mixing calculations with several (more than three) end-members. The performance of M3 against several other mixing codes has also been tested, and in all cases the accuracy of M3 has been as good as that achieved by the other codes. This is an important confidence assessment that supports the capabilities of M3.