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The g-function model is nowadays the most popular approach for the modeling of the thermal response of geothermal heat exchangers. By assuming the mean azimuthal borehole wall temperature to be uniform along the boreholes, it achieves a convenient uncoupling of the heat transfer problem in the borehole and its surroundings from the heat conduction problem in the ground, which significantly reduces the computational cost. This assumption, however, renders the g-function model ill-posed when the buried depth D of the boreholes is zero. This issue has passed unreported for the last 30 years, most probably due to the need of numerical discretizations with extremely fine meshes to observe the divergent behavior of the problem. In the present work a mathematical explanation for this ill-posedness is given. Also the well-posedness for all values of D of the alternative definition for the g-function model found in the literature is shown.