We studied seismic attenuation of P- and S-waves caused by the physical mechanism of wave-induced fluid flow at the mesoscopic scale. Stress relaxation experiments were numerically simulated by solving Biot’s equations for consolidation of 2D poroelastic media with finite-element modeling. The experiments yielded time-dependent stress-strain relations that were used to calculate the complex moduli from which frequency-dependent attenuation was determined. Our model consisted of periodically distributed circular or elliptical heterogeneities with much lower porosity and permeability than the background media, which contained 80% of the total pore space of the media. This model can represent a hydrocarbon reservoir, where the porous background is fully saturated with oil or gas and the low-porosity regions are always saturated with water. Three different saturation scenarios were considered: oil-saturated (80% oil, 20% water), gas-saturated (80% gas, 20% water), and fully water-saturated media. Varying the dry bulk and shear moduli in the background and in the heterogeneities, a consistent tendency was observed in the relative behavior of the S-wave attenuation among the different saturation scenarios. First, in the gas-saturated media the S-wave attenuation was very low and much lower than in the oil-saturated or in the fully water-saturated media. Second, at low frequencies the S-wave attenuation was significantly higher in the oil-saturated media than in the fully water-saturated media. The P-wave attenuation exhibited a more variable relative behavior among the different saturation degrees. Based on the mechanism of wave-induced fluid flow and on our numerical results, we suggest that S-wave attenuation could be used as an indicator of fluid content in a reservoir. Additionally, we observed that impermeable barriers in the background can cause a significant increase in S-wave attenuation. This suggests that S-wave attenuation could also be an indicator of permeability changes in a reservoir due to, for example, fracturing operations.