When one fluid displaces another in a confined environment, some energy is dissipated in the fluid bulk and the rest is dissipated near the contact line. Here we study the relative strengths of these two sources of dissipation with a novel experimental setup: constant-rate spontaneous imbibition experiments, achieved by introducing a viscous oil slug in front of the invading fluid inside a capillary tube. We show that a large fraction of dissipation can take place near the contact line, and rationalize the observations by means of a theoretical analysis of the dynamic contact angles of the front and back menisci of the oil slug. Our results bear important implications for macroscopic descriptions of multiphase flows in microfluidic systems and porous media.