To have a better knowledge of problems occurring with drilling fluids in complex wells, we carried out a detailed rheological analysis of a typical drilling mud at low shear rates using both conventional rheometry and MRI velocimetry. We show the existence of a viscosity bifurcation effect: Below a critical stress value, the mud tends to completely stop flowing, whereas beyond this critical stress, it reaches an apparent shear rate larger than a finite (critical) value, and no stable flows can be obtained between this critical shear rate value and zero. These results are confirmed by MRI velocity profiles, which exhibit a slope break at the interface between the solid and the liquid phases inside the Couette geometry. Moreover, this viscosity bifurcation is a transient phenomenon, the progressive development of which can be observed by MRI. A further examination of MRI data shows that, in the transient regime, the shear rate does not vary monotonously in the rheometer gap and is particularly large along the outer (rough) cylinder, which might be at the origin of the development of a region of constant shear rate in the apparent flow curve.