The effect of variability in rock thermal conductivity on exhumation rate estimates from thermochronological data
Introduction
Thermochronology is an essential tool for estimating the rate of rock exhumation on geological time scales (Braun et al., 2006, Brown, 1991). A cooling age corresponds to the time in the past when a rock cooled through a given temperature (called by Dodson (1973) the “closure temperature” or Tc). A first-order, mean exhumation rate can be obtained by dividing the depth, zc, of the closure temperature isotherm (at the time the rock cooled through it) by the cooling age. Estimating zc is not trivial as it depends on the geothermal gradient (the rate of change of temperature with depth) which can be affected by a number of things including the rate of exhumation itself (Batt and Braun, 1997). Even when the exhumation rate is low and does not perturb the shape of the geotherm, the geotherm is controlled by the rate of heat loss through the Earth's surface (the sum of the heat coming from the mantle and the vertically integrated crustal heat sources) and the conductivity of rocks, which is quite variable in the Earth's continental crust. For example, the conductivity of sedimentary rocks ranges between 0.5 and 2.5 W/m/K, that of volcanic and plutonic rocks between 1.5 and 3.5 W/m/K, and that of metamorphic rocks between 2 and 7 depending on their quartz content, mostly (Clauser and Huenges, 1995).
To help the reader appreciate how variable the conductivity of rocks is in the continental crust, we show in Fig. 1a observed and simulated conductivity profiles as a function of depth and temperature. The solid black line represents the theoretical variation of conductivity with temperature and pressure as proposed by Chapman (1986), assuming a uniform “average” granitic upper crust and a geothermal gradient of 28 ° C/km. More recently, Whittington et al. (2009) suggested that the temperature-dependence of conductivity is actually stronger and proposed a relationship shown as the dashed black line in Fig. 1a. We note that both curves are characterized by a very similar gradient and that, over the upper crustal depth-range, conductivity decreases by a factor of 2. Both authors agree that the temperature dependence dominates over the pressure dependence. Another important source of variability in conductivity comes from variability in lithology. This is shown by the light grey boxes which represent the range of conductivity values measured in the laboratory (at room temperature) by Clauser and Huenges (1995) for a range of rock types which we have associated, rather arbitrarily, with a depth range in the upper crust. This is meant to illustrate the point that rocks deposited in sedimentary basins will have a much lower conductivity than rocks in the underlying granitic basement, which, in turn, have lower conductivity than that of deeply buried metamorphic rocks. In general conductivity decreases with increasing porosity, and increases with the quartz content of the rock (Clauser and Huenges, 1995). We see that the overall lithological effect is to cause a strong increase in conductivity with depth over a range that is similar (but inverse) to that related to the temperature effect. To further illustrate this point, we show conductivity measurements (range and mean) from the KTB deep borehole from Clauser et al. (1997). We see that in the uppermost crust the temperature effect is dominant while lithological variations (transition from gneiss to metabasalt in this case) take over as the factor controlling conductivity values at depths larger than 5 km.
For a given rate of conductive heat loss through the crust, the product of conductivity by the geothermal gradient is a constant, implying that any spatial variability in thermal conductivity results in a similar spatial variability in the geothermal gradient. This means that variability in conductivity must be taken into account when interpreting cooling ages in terms of apparent exhumation rate. Interestingly, this has not been done so far. Ehlers, 2005, Braun et al., 2006 discussed the importance of incorporating variability in rock conductivity when interpreting thermochronological data, but do not assess its importance quantitatively. Dempster and Persano (2006) concluded from the lack of correlation between observed ages and thermal properties of rocks (conductivity and heat production) that these only play a minor role in controlling the thermal structure of the shallowest part of the crust. They suggested that fluid flow must homogenize the near-surface temperature field. Glotzbach et al. (2009) incorporated laterally variable thermal conductivities in a 2D model that they used to interpret thermochronological data form the central Alps. They showed that it affected the geometry of the closure temperature isotherm. However, no attempt has been made so far at quantifying in a rigorous and methodic way the importance of thermal conductivity variations when interpreting thermochronological data.
This is what we propose to do in this paper. First we consider how a linear variation of the conductivity of rocks with depth affects the apparent exhumation rate derived from cooling ages. For this we use an analytical, steady-state solution of the 1D heat equation obtained under the assumption that conductivity is a “static” function of depth (Fig. 1b). By this we mean to represent the effect of exhumation-driven near-surface fracturing and alteration, or the presence of water which could decrease the effective thermal conductivity close to the surface (Clauser and Huenges, 1995); fluid convection in the more permeable upper crust can also be parameterised as an apparent increase in heat conductivity near the surface (Russell, 1935); the well-known dependence of conductivity on temperature and, to a lesser degree, pressure (Whittington et al., 2009) can also be regarded as a “static” variation in conductivity with depth, keeping in mind that the temperature profile may depend on erosion rate, in which case conductivity is unlikely to vary as a simple linear function of depth.
In a second step, we use a numerical solution of the heat equation based on the method developed by Braun (2003) to show how predicted surface ages are affected by an initial vertical gradient in conductivity that is advected with the deformation; we say, in that case, that the conductivity is a “dynamic” function of depth (Fig. 1b), which represents conductivity variability associated with variability in rock type. As shown in Fig. 1, different rocks have different conductivity, mostly due to their different chemical composition or mineralogy (such as quartz content which appears to be the main control in continental rocks) and physical state, such as previously acquired permeability, fracture density or weathering intensity. In each case, we compute synthetic ages that we use, in turn, to compute apparent exhumation rates. In this way we are able to quantify the error made by neglecting the effect of variable thermal conductivity on estimates of exhumation (or erosion) rates from thermochronological data.
Section snippets
The 1D heat equation
Vertical heat transport by conduction and advection is governed by the following partial differential equation: where T is temperature, z is depth (positive downwards), t is time, ρ is rock density, c is heat capacity, is exhumation (or erosion) rate, and k is thermal conductivity. We wish to investigate first the effect of a depth-dependent thermal conductivity on the vertical distribution of temperature and its effect on apparent exhumation rate obtained from
Method and setup
We now investigate the effect of conductivity variability associated with variability in rock type. This means that, contrary to what was done earlier, any spatial variation in conductivity is advected with the rocks as they are exhumed through the upper surface. To do this, we use a modified version of the Pecubesoftware (Braun, 2003, Braun et al., 2012) that solves the complete three dimensional heat transport equation: where v is a full three-dimensional velocity
How important is conductivity variability for estimating exhumation rate?
We have shown that variability in thermal conductivity of rocks has a measurable effect on predicted cooling ages and the exhumation rate estimates that are deduced from them. There is no easy way to remove that dependence, as all methods of interpretation of cooling ages are affected by the depth dependence of thermal conductivity (single age, age–elevation relationship and time lags derived from detrital age distributions). However, two cases must be considered. If, on the one hand, the
Conclusions
We have quantified the error made by assuming a uniform rock conductivity in estimating exhumation rate and exhumation rate histories from thermochronological data. Using simple analytical and more sophisticated numerical solutions to the heat equation, we have shown that the effect of neglecting depth variations in conductivity is relatively minor when the conductivity is assumed to be a “dynamic” function of depth (i.e., it is advected by rock exhumation). In such a case, whatever the
Acknowledgments
Financial support for C.S.’s visit to J.B. in Grenoble is fromDeutsche Forschungsgemeinschaft (DFG)grant182/18-1 to U.G. The latest version of Pecube, including the variable conductivity option, can be obtained by contacting J.B. by email. We thank D. Whipp and C. Glotzbach for their useful and constructive reviews.
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