deletions | additions       diff --git a/discussion_thermal constraints.tex b/discussion_thermal constraints.tex  index 6040ca2..1dcf8af 100644  --- a/discussion_thermal constraints.tex  +++ b/discussion_thermal constraints.tex 
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Hydrothermal fluid at an initial temperature $\Theta$_{i} $\Theta_{i}$  enters the fault at depth \textit{x}_{1} $\textit{x}_{1}$  and flows up along the fault to a depth of \textit{x}_{2} $\textit{x}_{2}$  where it has cooled to a temperature $\Theta$_{1}. $\Theta_{1}$.  We assume that the heat lost from the fluid (i) heats the immediate fault zone to the temperature $\Theta$_1, $\Theta_1$,  and; (ii) is lost through thermal diffusion perpendicular to the fault walls. For a parallel plate slab we can express this energy balance per metre length of fault as: \[V \cdot \left( {{\Theta _i} - {\Theta _1}} \right) \cdot C_p^w \cdot {\rho _w} = \left[ {({x_1} - {x_2}) \cdot h \cdot \left( {\frac{{{\Theta _1} - {\Theta _2}}}{2}} \right) + \left( {{\Theta _1} - {\Theta _2}} \right) \cdot \left( {{x_1} - {x_2}} \right) \cdot {{\left( {\frac{{kt}}{\pi }} \right)}^{0.5}}} \right] \cdot C_p^r \cdot {\rho _r}\]  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} $\rho_{w}$  and $\rho$_{r} $\rho_{r}$  the densities of water and rock, and \textit{k} $\textit{k}$  the thermal diffusivity of rock. Noting that $C_{p}^{w}$ $\cdot$ $\rho_{w}$ $\approx$ 2 $\cdot$ $C_{p}^{r}$ $\cdot$ $\rho_{r}$ and selecting values of 120$^{\circ}$, 100$^{\circ}$ and 40$^{\circ}$C respectively for $\Theta$_{i}, $\Theta$_{1} $\Theta_{i}$, $\Theta_{1}$  and $\Theta$_{2}, $\Theta_{2}$,  3000m for \textit{x}_{1}, $\textit{x}_{1}$,  1000m for \textit{x}_{2} $\textit{x}_{2}$  and an effective fault width of 1m we find that the maximum duration (s) for an individual fluid pulse is: \[t = \frac{{\pi \cdot \left( {3.3 \times {{10}^{ - 4}} \cdot V - 0.5} \right)^2}}{{1 \times {{10}^{ - 6}}}}\]  Table 2 lists values of t for different values of V. The values of V were chosen corresponding to the volumes of fluid expelled from an overpressured 40km x 2kmm 2km  thick sediment sequence at depth with incremental changes in porosity of 0.1, 1 and 10\% on dewatering. These approximate to the width and thickness of shale in the Carboniferous Edale basin to the north-east of the Derbyshire platform. The porosity changes range from the smallest incremental changes calculated for individual dewatering pulses of overpressured sediments to the integrated maximum volume of fluid that might be expelled \citep{Cathles:1983tj}. The corresponding values of t are 67, 6912 and 694000 years respectively. These correspond to mean effusive fluxes of 136, 13.2 and 1.3 litres.hr^{-1} litres.hr$^{-1}$  for every metre length of the fault. Such flow rates are not geologically unrealistic. The highest rates associated with the smallest fluid pulses are on the order of the rates of effusion from springs that have been monitored for periods of several years following moderate earthquakes e.g (add refs by Nur and Tsuneishi et al., 1970). One can legitimately question the model details and parameter estimates but the point of this somewhat heuristic approach is not to be an accurate model. Rather, it is to give an indication of the flow rates that are needed to sustain the maximum observed thermal anomaly within the Dirtlow Rake and to assess whether these are reasonable. That the model may only be accurate to within a factor of 5 to 10 does not invalidate the central result that if fluid release occurs in a pulsed and episodic manner then each event must be of short duration to sustain the necessary thermal anomaly. Moreover the duration of these events and maximum flow rates associated with them are commensurate with fluid fluxes associated with modern observations of fluid behaviour following earthquake rupture.  These observations about the pulsed and rapid nature of fluid flow in MVT systems are convergent with ideas expressed by several researchers e.g. Cathles and Smith (1986), \citet{Bodnar:2009hc, Wilkinson06022009}. They also pose serious questions about our understanding concerning the transport and precipitation of carbonate minerals in fracture systems. For example modelling and experimental data on vein formation suggest that an extremely large volume of fluid is required to produce calcite veins in nature with fluid:calcite volume ratios of 10^{5} 10$^{5}$  to 10^{6}. 10$^{6}$.  These same models also predict that the flow needs to be sustained for thousands to millions of years \citep{Lee_1996, Lee_1999}.