Neural networks provide an attractive framework to monitor the water table level and the volume of stored water in porous media from seismic data in an automated, fast and cost-efficient manner. In this work, a subsurface reservoir is modeled as a coupled three-dimensional poroviscoelastic-viscoelastic medium. The wave propagation from source to receiver(s) is numerically simulated using a nodal discontinuous Galerkin method coupled with an Adams-Bashforth time-stepping scheme on a graphics processing unit cluster. The wave field solver is used to generate databases for the neural network model to estimate the water table level and actual volume of water. We use a deconvolution-based approach to normalize the effect from the source wavelet. The results demonstrate the capacity of the fully connected neural network for estimating both the water table level and the volume of stored water in the porous storage reservoir from both synthetic and measured data.