It is well known that the construction of traditional reservoir simulation models can be very time and resources consuming. Particularly in the case of mature fields with long history and large number of wells where such models can be extremely difficult and long to history match. In this case data driven models can represent a cost-effective alternative, or they can provide complementary analysis to classical reservoir modelling. Due to data scarcity full machine learning approaches are also usually doomed to fail. In this work we develop a new Physics-Constrained Deep Learning approach that combined neural networks with a reduced physics approach: Capacitance Resistive Model (CRM). CRM are data-driven methods that are based on a simple material balance approximation, that can provide very useful reservoir insight. CRM can be used to analyze the underlying connections between producer wells and injector wells that can then be used to better allocate water injection. Such analysis can usually require very long tracer tests or very expensive 4D seismic acquisition and interpretation. CRM can provide directly these wells connection information using only available production and pressure data. The problem with CRM approaches, based on classical optimizers, is that they often detect spurious correlations and can be not very robust and reliable. Our physics-constrained deep learning approach called Deep-CRM performs production data regularization via the neural network approximation that helps to provide a better CRM parameter identification also with the use of robust gradient descent optimization methods developed and widely used by the large deep learning community. We show first on a synthetic and then in real reservoir case that Deep-CRM was able to identify most of the injector-producer connections with higher accuracy with respect to traditional CRM. Deep-CRM produced also better liquid production forecasts on the performed blind tests.