Polythermal reaction paths

Polythermal reactions paths are those in which temperature varies as a function of reaction progress, . In the simplest case, the modeler prescribes the temperatures T0 and Tf at the beginning and end of the reaction path. The model then varies temperature linearly with reaction progress. This type of model is sometimes called a sliding temperature path. 

The calculation procedure for a sliding temperature path is straightforward. In taking a reaction step, the model evaluates the temperature to be attained at the step s end. Since varies from zero to one, temperature at any point in reaction progress is given by, 

A second type of polythermal path traces temperature as reactants mix into the 

In this equation, and are the values of reaction progress at the beginning and end of the step nj is the mass in kg of the fluid (equal to nw, the water mass, plus the mass of the solutes) nk is the mole number of each mineral nr is the reaction rate (moles) for each reactant Mwk is the mole weight (g mol-1) of each mineral, and Mwr is the mole weight for each reactant and T, jsp is the fraction of the fluid displaced over the reaction step in a flush model (Adlsp is zero if a flush model is not invoked). 

In an example of a sliding temperature path, we consider the effects of cooling from 300 °C to 25 °C a system in which a 1 molal NaCl solution is in equilibrium with the feldspars albite (NaAlSiaOs) and microcline (KAlSisOg), quartz (SiC 2), and muscovite [KAl3Si30io(OH)2]. To set up the calculation, we enter the commands 

A second type of polythermal path traces temperature as reactants mix into the equilibrium system. This case differs from a sliding temperature path only in the manner in which temperature is determined. The modeler assigns a temperature To to the initial system, as before, and a distinct temperature Tr to the reactants. By assuming that the heat capacities CP, CPk, and CPr of the fluid, minerals, and reactants are constant over the temperature range of interest, we can calculate temperature ( ) from energy balance and the temperature T(c ) at the onset of the step according to 

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Fig. 14.1. Mineralogical results of tracing a polythermal reaction path. In the calculation, a 1 molal NaCl solution in equilibrium with albite, microcline, muscovite, and quartz cools from 300 °C to 25 °C.

To model the problem, we take a packet of water in contact with the fracture walls over a polythermal reaction path. The fact that the packet moves relative to the walls is of no concern, since the fracture surface area exposed to the packet is approximately constant. Since the system contains 1 kg of water, we can show from geometry that the surface area 4s (in cm2) of the fracture lining is,... 

Reaction paths can be traced at steady or varying temperature the latter case is known as a polythermal path. Strictly speaking, heat transfer occurs even at constant temperature, albeit commonly in small amounts, to offset reaction enthalpies. For convenience, modelers generally define polythermal paths in terms of changes in temperature rather than heat fluxes. 

Polythermal reaction models (Section 14.1), however, are commonly applied to closed systems, as in studies of groundwater geothermometry (Chapter 23), and interpretations of laboratory experiments. In hydrothermal experiments, for example, researchers sample and analyze fluids from runs conducted at high temperature, but can determine pH only at room temperature (Fig. 2.2). To reconstruct the original pH (e.g., Reed and Spycher, 1984), assuming that gas did not escape from the fluid before it was analyzed, an experimentalist can calculate the equilibrium state at room temperature and follow a polythermal path to estimate the fluid chemistry at high temperature. 

To invoke our geothermometer, we need to recombine the vapor and fluid phases and then heat the mixture to determine saturation indices as functions of temperature. We could do this in two steps, first titrating the vapor phase into the liquid and then picking up the results as the starting point for a polythermal path. We will employ a small trick, however, to accomplish these steps in a single reaction path. The trick is to add the vapor phase quickly during the first part of the reaction path but use the cutoff option to prevent mass transfer over the remainder of the path. The commands to set the mass transfer are... 

As an example of how the dump option might be used, consider the problem of predicting whether scale will form in the wellbore as groundwater is produced from a well (Fig. 2.10). The fluid is in equilibrium with the minerals in the formation, so the initial system contains both fluid and minerals. The dump option simulates movement of a packet of fluid from the formation into the wellbore, since the minerals in the formation are no longer available to the packet. As the packet ascends the wellbore, it cools, perhaps exsolves gas as it moves toward lower pressure, and leaves behind any scale produced. The reaction model, then, is a polythermal, sliding-fugacity, and flow-through path combined with the dump option. 

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