In this article, we study theoretically and numerically the heat equation coupled with Darcy’s law by a nonlinear viscosity depending on the temperature. We establish existence of a solution by using a Galerkin method and we prove uniqueness. We propose and analyze two numerical schemes based on finite element methods. An optimal a priori error estimate is then derived for each numerical scheme. Numerical experiments are presented that confirm the theoretical accuracy of the discretization. ------------------------- This paper has been published in Numer. Math., 139, (2018), pp. 315-348, doi 10.1007/s00211-017-0938-y