deletions | additions       diff --git a/untitled.tex b/untitled.tex  index 46608a0..e31a72d 100644  --- a/untitled.tex  +++ b/untitled.tex 
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The results of isotopic and clumped isotope analysis of all samples and standards are reported in Table 1 and plotted in Figure 5 and 6.  The spatial distribution of isotope values and temperature for DLR7 is plotted in Figure 5(a) and for DLR7i in Figure 5(b). Because the two sections were cut at different locations across the vein we don't expect a direct correlation between them. There are, however, similarities. DLR7 has a distinctive, repeating asymetric saw-tooth pattern along the growth direction that is most marked for $\delta$^{18}O and T($\Delta$_{47}). Four regions of section DLR7, separated by the gray bands in Figure 5(a), are identified. The bands mark step changes in T($\Delta$_{47}) between adjacent samples. Each section is characterized by a rising temperature from a minimum of 40$^{\circ}$ to a maximum of 90$^{\circ}$C. This pattern is mirrored by antithetic changes in $\delta$^{18}O, falling from a maximum value of -8‰ to a minimum of -10‰_{VPDB}. The pattern of change in $\delta$^{13}C is not so marked or consistent but is still apparent. For example between 10 and 25mm there is a positive covariation of $\delta$^{18}O and \delta^{13}C $\delta$^{13}C  with an inverse correlation between 30 and 50mm. In contrast, the asymmetric, sawtooth pattern observed in section DLR7 is not readily apparent in section DLR7i. None-the-less it is still possible to identify regions where there is a marked pattern of variation for both T($\Delta$_{47}) and $\delta$^{18}O, Figure 5(b). Between 10 and 20, 55 and 60, and 65 and 75mm T($\Delta$_{47}) rises whilst there is an inverse fall in the $\delta$^{18}O value. Between 25 and 50mm in section DLR7i there is an apparent breakdown in the inverse relationship between T($\Delta$_{47}) and $\delta$^{18}O. In this region temperatures for the most part are greater than 70$^{\circ}$C whilst $\delta$^{18}O remains constant and close to -9.8‰_{VPDB}. As with section DLR7 the carbon isotope data does not show a marked covariation with either temperature or oxygen isotope composition. The exception is in the section between 10 and 20mm where there is a marked inverse relationship between $\delta$^{13}C and $\delta$^{18}O with a 2‰ positive trend in carbon isotope values matched by a 2‰ negative trend in the oxygen isotope values. This trend, however, is heavily levered by a single point with $\delta$^{13}C = 1.5‰_{VPDB} and $\delta$^{18}O = -7.5‰_{VPDB}. 
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\[V \cdot \left( {{\Theta _i} - {\Theta _1}} \right) \cdot C_p^w \cdot {\rho _w} = \left[ {({x_1} - {x_2}) \cdot h \cdot \left( {\frac{{{\Theta _1} - {\Theta _2}}}{2}} \right) + \left( {{\Theta _1} - {\Theta _2}} \right) \cdot \left( {{x_1} - {x_2}} \right) \cdot {{\left( {\frac{{kt}}{\pi }} \right)}^{0.5}}} \right] \cdot C_p^r \cdot {\rho _r}\]  where V is the volume of fluid expelled along the fault, h is the effective width of the fault, C_{p}^{w} and C_{p}^{r} are the specific heat capacities of water and rock respectively, $\rho$_{w} and $\rho$_{r} the densities of water and rock, and \textit{k} the thermal diffusivity of rock. Noting that and selecting values of 120$^{\circ}$, 100$^{\circ}$ and 40$^{\circ}$C respectively for $\Theta$_{i}, $\Theta$_{1} and $\Theta$_{2}, 3000m for \textit{x}_{1}, 1000m for \textit{x}_{2} and an effective fault width of 1m we find that the maximum duration for an individual fluid pulse is a function