Mathematical formulation of the diffusion phenomenon might be described through a differential equation, which takes into account complementary and different effects with respect to the physical processes simulated with the support of the Fick´s equation, which is usually adopted to represent the diffusion process. In particular, diffusion applied to spatio-temporal retention problems with bimodal mass transmission are highlighted. To better understand this physical phenomenon, the proper use of the analytical Green function (GF) or the steady-state fundamental solution was investigated. In this case, we use the Boundary Element Method formulation is presented for the solution of the anomalous diffusion equation for one-dimensional problems. The formulation employs the steady-state fundamental solution. Besides the basic integral equation, another one is required, due to the fourth-order differential operator in the differential equation of the problem. The domain discretization employs linear cells. The first order time derivative is approximated by a backward finite difference scheme. Two examples are presented. Numerical results are compared with analytical solutions, showing good agreement between them.

Here, we develop and test an artificial intelligence (AI)-based approach to monitor major Brazilian aquifers. The approach combines Gravity Recovery and Climate Experiment (GRACE) data and ground-based hydrogeological measurements from Brazil’s Integrated Groundwater Monitoring Network at hundreds of wells distributed in twelve aquifers across the country. We use a model ensemble composed of four different AI models: Extreme Gradient Boost, Light Gradient Boosting Model, CatBoost and Multilayer Perceptron. The approach is further boosted with wavelet and seasonal decomposition processes applied to GRACE data. To determine the sensitivity of the AI approach to data availability, we propose four experiments combining hydrogeological measurements from different aquifers. Groundwater storage estimates from the Global Land Data Assimilation System (GLDAS) are used as the benchmark. The AI approach successfully reproduces groundwater storage estimates ​​at all Brazilian aquifers. Results show that the proposed approach outperforms GLDAS in all experiments, with an average Nash-Sutcliff efficiency of 0.91 and an average RMSE of 0.43cm for the experiment that covers all monitored wells in Brazil. GLDAS resulted in -1.311 and 5.84cm, respectively. This study demonstrates that combining satellite data and AI can be a cost-effective alternative to monitor poorly equipped aquifers at the continental scale.