Intrinsic versus super-rough anomalous scaling in spontaneous imbibition

We study spontaneous imbibition using a phase field model in a two dimensional system with a dichotomic quenched noise. By imposing a constant pressure $\mu_{a}<0$ at the origin, we study the case when the interface advances at low velocities, obtaining the scaling exponents $z=3.0\pm 0.1$, $\alpha=1.50\pm 0.02$ and $\alpha_{loc}= 0.95\pm 0.03$ within the intrinsic anomalous scaling scenario. These results are in quite good agreement with experimental data recently published. Likewise, when we increase the interface velocity, the resulting scaling exponents are $z=4.0 \pm 0.1$, $\alpha=1.25\pm 0.02$ and $\alpha_{loc}= 0.95\pm 0.03$. Moreover, we observe that the local properties of the interface change from a super-rough to an intrinsic anomalous description when the contrast between the two values of the dichotomic noise is increased. From a linearized interface equation we can compute analytically the global scaling exponents which are comparable to the numerical results, introducing some properties of the quenched noise.