We present a method that combines a physics-informed deep neural network and Stokes' second problem to estimate the porosity and the permeability of a porous medium. Particularly, we investigate the accuracy of physics-informed deep neural networks in predicting the hidden quantities of interest, such as velocity and unknown parameters, including permeability and porosity, by employing different network architectures and different sizes of input data sets. The employed neural network is jointly trained to match the essential class of physical laws governing fluid motion in porous media (Darcy's law and mass conservation) and the fluid velocities in the domain or region of interest. Therefore, the described approach allows the estimation of hidden quantities of interest. This strategy conditions the neural network to honor physical principles. Thus, the model adapts to fit best the data provided while striving to respect the governing physical laws. Results show that the proposed approach achieves significant accuracy in estimating the velocity, permeability, and porosity of the media, even when the neural network is trained by a relatively small input data-set. Also, results demonstrate that using the optimal neural network architecture is indispensable to increase the porosity and permeability prediction accuracy.