The flow and heat transfer from a heated semi-infinite horizontal circular cylinder which is moving with a constant speed into a porous medium is considered. It is assumed that the Grashof and Reynolds numbers are large so that the governing equations are the three dimensional boundary-layer equations. A numerical procedure for solving these equations is described and the asymptotic solutions which are valid both near and distant from the leading edge of the cylinder are presented. The range of validity of these asymptotic solutions is discussed and the results are compared in detail with the full numerical solution. The problem is of practical importance, for example in the drilling of pipes into a geothermal reservoir.