We report experiments on spontaneous imbibition of a viscous fluid by a model porous medium in the absence of gravity. The average position of the interface satisfies Washburn’s law. Scaling of the interface fluctuations suggests a dynamic exponent z ≃ 3 , indicative of global dynamics driven by capillary forces. The complete set of exponents clearly shows that interfaces are not self-affine, exhibiting distinct local and global scaling, both for time ( β = 0.64 ± 0.02 , β ∗ = 0.33 ± 0.03 ) and space ( α = 1.94 ± 0.20 , α loc = 0.94 ± 0.10 ). These values are compatible with an intrinsic anomalous scaling scenario.