The Active Region 8151 (AR 8151) observed in February 1998 is the site of an eruptive event associated with a filament and a S-shaped structure, and producing a slow Coronal Mass Ejection (CME). In order to determine how the CME occurs, we compute the 3D coronal magnetic field and we derive some relevant parameters such as the free magnetic energy and the relative magnetic helicity. The 3D magnetic configuration is reconstructed from photospheric magnetic magnetograms (IVM, Mees Solar Observatory) in the case of a non-constant- force-free (nlff) field model. The reconstruction method is divided into three main steps: the analysis of vector magnetograms (transverse fields, vertical density of electric current, ambiguity of 180 degrees), the numerical scheme for the nlff magnetic field, the interpretation of the computed magnetic field with respect to the observations. For AR 8151, the nlff field matches the coronal observations from EIT/SOHO and from SXT/Yohkoh. In particular, three characteristic flux tubes are shown: a highly twisted flux tube, a long twisted flux tube and a quasi-potential flux tube. The maximum energy budget is estimated to 2.6 10^31 erg and the relative magnetic helicity to 4.7 10^34 G^2 cm^4. From the simple photospheric magnetic distribution and the evidence of highly twisted flux tubes, we argue that the flux rope model is the most likely to describe the initiation mechanism of the eruptive event associated with AR 8151.