Abstract: Viscous fingering (VF)in the random Sierpinski carpet is investigated by means of the successive over-relaxation technique and under the assumption that bond radii are of Rayleigh distribution. In the random Sierpinski network, the VF pattern of porous media in the limit $M\to \infty$ ($M$ is the viscosity ratio and equals $\eta_2$/$\eta_1$ where $\eta_1$ and $\eta_2$ are the viscosities of the injected and displaced fluids, respectively) is found to be similar to the diffusion-limited aggregation (DLA) pattern. The interior of the cluster of the displacing fluid is compact on long length scales when $M=1$, and the pores in the interior of the cluster have been completely swept by the displacing fluid. For finite values of $M$, such as $M\geq10$,the pores in the interior of the cluster have been only partly swept by the displacing fluid on short length scales. But for values of $M$ in $1<M \leq5$, the pores in the interior of the cluster have been completely swept by the displacing fluid on short length scales. The symmetry of the growth of VF is broken by randomizing the positions of the holes. The fractal dimension for VF in fractal space is calculated. However,the sweep efficiency of the displacement processes mainly depends upon the length of the network system and also on the viscosity ratio $M$. The fractal dimension $D$ can be reasonably regarded as a useful parameter to evaluate the sweep efficiencies. The topology and geometry of the porous media have a strong effect on the structure of VF and the displacement process. The distribution of velocities normal to the interface has been studied by means of multifractal theory.Results show that the distribution is consistent with the hypothesis that, for a system of size $L$, $L^{f(\alpha)}$ sites have velocities scaling as $L^{-\alpha}$; and the scaling function $f(\alpha)$ is measured and its variation with $M$ is found. 

Key words: viscous fingering, fractal construction, Sierpinski carpet