Coping with nonuniqueness

The examples in the previous section demonstrate that nonunique solutions to the equilibrium problem can occur when the modeler constrains the calculation by assuming equilibrium between the fluid and a mineral or gas phase. In each example, the nonuniqueness arises from the nature of the multicomponent equilibrium problem and the variety of species distributions that can exist in an aqueous fluid. When more than one root exists, the iteration method and its starting point control which root the software locates. 

In each of the cases, the dual roots differ from each other in terms of pH, sulfide content, or ionic strength, so that in a modeling study the correct root could readily be selected. The danger of nonuniqueness is that a modeler, having reached an inappropriate root, might not realize that a separate, more meaningful root to the problem exists. 

Unfortunately, no software techniques exist currently to automatically search for additional roots. Instead, modelers must rely on their understanding of geochemistry to demonstrate uniqueness to their satisfaction. Activity-activity diagrams such as those presented in Figs. 10.1-10.3 are the most useful tools for identifying additional roots. 

The end point of the reaction path is the second root to the problem. At 300 °C, we find by trial and error that we need to add 0.089 92 moles of CaCl2 to reach the second root. The differences among the roots are summarized 

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