A modification of the invasion percolation model with trapping is proposed by introduction of special extraction sites on the lattice with given concentration. It is studied as an idealised model of the forced quasi-static water invasion in artificial hydrophobic soils with plants at reduced gravity. The special sites model effectively the presence of plants in the porous media. These special sites, when invaded by the water, can block with certain probability (called extraction probability) the further spreading of the water to neighbour sites in the network. Through Monte Carlo simulations, we study how the presence of the extraction sites modifies the amount of the residual air (or, respectively, the amount of the water) on the lattice and its distribution. We also study how the spreading process itself is affected, i.e. we study the influence of the concentration of special sites and the extraction probability on the rate at which the water invades the medium. Our results show that for the finite systems considered, the presence of the extraction sites leads to an increase of the water fraction on the lattice at low concentrations but further increase of the concentration, as expected, eventually blocks the spreading process. The phase diagram of the model system in the (concentration /extraction probability) plane is obtained.