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 solve d in a stochastic framework incorporating uncertainties in the depth-depend ent thermal conductivity. For this purpose, a new approach to the perturbat ion method, an expansion series method, which allows for the incorporation of a large variance in the controlling parameters, has been used. This meth od helps in avoiding assumptions on the probability distribution of the par ameter and instead uses information pertaining to the mean and spatial corr elation 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 expressio ns for the first two moments, that is, the mean and variance of the tempera ture field, are obtained. The expression for variance shows that it is high ly dependent on the coefficient of variability of thermal conductivity, on the correlation length scale and on the depth. The methodology developed ha s been applied to quantify the steady-state geotherms, along with their ass ociated error bounds, on a standard crustal model.