A stochastic model to quantify the steady-state crustal geotherms subject to uncertainties in thermal conductivity

In this communication the 1-D steady-state heat conduction problem is solved in a stochastic framework incorporating uncertainties in the depth-dependent thermal conductivity. For this purpose, a new approach to the perturbation method, an expansion series method, which allows for the incorporation of a large variance in the controlling parameters, has been used. This method helps in avoiding assumptions on the probability distribution of the parameter and instead uses information pertaining to the mean and spatial correlation structure. This information is easily available in most geological situations and hence the thermal conductivity is assumed to have a Gaussian coloured noise correlation structure. With this information the stochastic heat conduction equation in equilibrium is solved and analytical expressions for the first two moments, that is, the mean and variance of the temperature field, are obtained. The expression for variance shows that it is highly dependent on the coefficient of variability of thermal conductivity, on the correlation length scale and on the depth. The methodology developed has been applied to quantify the steady-state geotherms, along with their associated error bounds, on a standard crustal model.