Inverse Mass Balance Modeling

Garrels and Mackenzie (1967) introduced inverse mass balance modeling into geochemistry. They showed that if the chemistry of the start and end solutions are known, possible mass transfer reactions that had produced the compositional differences and the extent to which these reactions had taken place could be deduced from the mass balance principle. 

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The mass balance concept in the inverse mass balance models is quite simple (Figure 2.6). If one fluid is evolved from another, the compositional differences can be accounted for by the minerals and gases that leave or enter that packet of water ... 

Figure 2.6. Schematic representation of the inverse mass balance model. If we know that the spring at the foothill is evolved from rain water, possible mass transfer reactions can be modeled from the mass balance principle.

Inverse mass balance modeling here only employs the mass balance principle thermodynamics and equilibrium are not considered. Inverse models are usually nonunique. A number of combinations of mass transfer reactions can produce the same observed concentration changes along the flow path. Mass transfer reactions here refer to the reactions that result in the mass transfer between two or more phases, such as the dissolution of solid and gas or precipitation of solids. Chapter 9 describes the details of the models and shows a few examples. 

On the other hand, many investigations have shown that Saturation Indices are a reasonably good guide to certain phases, such as some carbonate and sulfate minerals. Saturation Indices are also quite useful in reaction path modeling (Chapter 8) and inverse mass balance modeling (Chapter 9) in the sense that they indicate (within the accuracy of the data) which processes are possible (e.g., precipitation of a phase having an SI > 0), and which are impossible (e.g., dissolution of a phase having an SI > 0). 

The applicability of inverse mass balance modeling hinges on a number of assumptions, which are seldom examined in detail ... 

Hydrodynamic dispersion may however be significant in small, local hydrogeological problems, such as a point source contamination (Plummer et al., 1992). Another instance where diffusion may play an important role in water chemistry is the diffusion from permeable to less permeable parts of the aquifer, or matrix diffusion. This process appears to be important in fractured aquifers (Maloszewski and Zuber, 1991 Neretnieks, 1981), volcanic rock aquifers, aquifers adjacent to confining units (Sudicky and Frind, 1981), and sand layers inter-stratified with confining clay layers (Sanford, 1997). In systems in which a chemical steady state (see below) has not been reached, matrix diffusion effects may severely limit the applicability of inverse mass balance modeling to those systems. 

From this assumption, inverse mass balance modeling is more likely to be applicable to large regional aquifer systems, but less applicable to aquifers with point source contamination. For example, in the acid mine drainage impacted aquifer described in Chapter 6, the chemistry at all points is changing with time. Applicability of inverse mass balance modeling to this type of system depends on the spatial locations of the flow path and the time-frame of interest. 

When the necessary condition that the chemistry at point A does not change with time is met, inverse mass balance models are still applicable when the chemistry at point B changes with time. An example is a laboratory column study, in which the chemistry of influent is maintained in the experiments while the effluent chemistry continues to change. In this case, we are assured that the effluent is chemically evolved from the influent. The variation of chemistry with time in the effluent does not violate the steady-state assumption. Another example is field injection of reactive tracers, during which the injectate chemistry is constant. Actually, laboratory titration experiments would also fit into this category because we know the initial solution chemistry from which the final solution evolves. Inverse mass balance modeling should find applications in these situations. 

These spreadsheet calculations are very intuitive they helped in the understanding of inverse mass balance modeling. We strongly recommend readers to practice using a... 

One can also use phreeqc to perform inverse mass balance modeling for the same calculations, phreeqc is capable of taking into account the analytical uncertainties. Here, we assume the analyses of snow and groundwater samples carry 6% errors. The input file is listed in Table 9.3. The readers are referred to the phreeqc manual for a detailed explanation of the input file (Parkhurst and Appelo, 1999). 

Of course, Mg can also come from dissolution of biotite or amphiboles. Similarly, K can come from the dissolution of orthoclase, micas, and amphiboles. The models can become very complicated, and we never can be sure what exactly are the mass transfer reactions that produced the groundwater chemistry. It is an ambitious proposition that we can know the groundwater genesis. Rather, inverse mass balance modeling is a useful tool that can help us to understand it. Sometimes, the results can be used quantitatively, and are sufficient for the purpose of our studies. 

Seven tentative inverse mass balance models were produced by phreeqc with the mass balance and phase constraints given (Table 9.6). A mixing fraction of 0.258 for well 504 water (hence 0.742 for well 403 water) was determined based on the conservative chemical Cl. Among the seven models, most are essentially combinations of proportions of different minerals. With different input constraints, more inverse models can be produced, which also produce the compositional differences between the initial and final wells. 

Chloride is usually a major constituent in groundwater and is widely considered a conservative tracer. In the N aquifer, Cl- concentrations are considerably higher in Holocene water than in late Pleistocene water (Figure 9.3). The groundwater ages indicated on the horizontal axis are results from inverse mass balance modeling and age corrections (Zhu, 2000). These age data are also supported by the SD and S lsO data of the same samples. It is generally known that Pleistocene water has depleted H and O stable isotope values with respect to recent water because of a cooler and more humid climate in the late Pleistocene (Merlivat and Jouzel, 1979). 

Note that the carbon isotopes are solely constrained by the DIC and < 13C values. Therefore, the two inverse mass balance models give the same results with respect to calculated travel time. Precipitation of illite, quartz, or kaolinite, the dissolution of gypsum and plagioclase, and Ca-Na exchange are needed to interpret the increases of Na+, SO2-, and SiC>2(aq), and decreases of Ca2+ and K+ in downgradient wells. 

The actual calculations were performed using balance, an earlier inverse mass balance modeling code that is now superseded by netpath and phreeqc. Chapelle and Lovley (1990) then calculated the time interval in these segments from flow velocities calculated by a numerical groundwater flow model and the length of the flow path. The total CO2 production rate from oxidation of organic matter is... 

Interpretation of Chemical Composition of Ground Water in Terms of Inverse Mass Balance Model... 

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