The reservoir characterization aims to provide the analysis and quantification of the injection-production relationship, which is the fundamental work for production management. The connectivity between injectors and producers is dominated by geological properties, especially permeability. However, the permeability parameters are very heterogenous in oil reservoirs, and expensive to collect by well logging. The commercial simulators enable to get accurate simulation but require sufficient geological properties and consume excessive computation resources. In contrast, the data-driven models (physical models and machine learning models) are developed on the observed dynamic data, such as the rate and pressure data of the injectors and producers, constructing the connectivity relationship and forecasting the productivity by a series of nonlinear mappings or the control of specific physical principles. While, due to the “black box” feature of machine learning approaches, and the constraints and assumptions of physical models, the data-driven methods often face the challenges of poor interpretability and generalizability and the limited application scopes. To solve these issues, integrating the physical principle of the waterflooding process (material balance equation) with an artificial neural network (ANN), a knowledge interaction neural network (KINN) is proposed. KINN consists of three transparent modules with explicit physical significance, and different modules are joined together via the material balance equation and work cooperatively to approximate the waterflooding process. In addition, a gate function is proposed to distinguish the dominant flowing channels from weak connecting ones by their sparsity, and thus the inter-well connectivity can be indicated directly by the model parameters. Combining the strong nonlinear mapping ability with the guidance of physical knowledge, the interpretability of KINN is fully enhanced, and the prediction accuracy on the well productivity is improved. The effectiveness of KINN is proved by comparing its performance with the canonical ANN, on the inter-well connectivity analysis and productivity forecast tasks of three synthetic reservoir experiments. Meanwhile, the robustness of KINN is revealed by the sensitivity analysis on measurement noises and wells shut-in cases.

Waterflooding reservoir interwell connectivity characterization is the fundamental work in oil development, aiming to inverse the vital connecting channels between injectors and producers. In this paper, we endow an artificial neural network (ANN) with strong interpretability through the ordinary differential equation (ODE) of the material balance equation, proposing a physical knowledge fusion neural network (PKFNN). In addition, the proposed model could inherit the knowledge learned from different injector–producer pairs, fully improving the training efficiency. In this way, PKFNN combines the merits of both physical and machine learning approaches. Firstly, based on the physical control law and the ODE of the material balance equation, we endow the model with highly transparent modular architectures in the framework of feedforward neural network. In this way, our work has both high interpretability and excellent approximation ability, combining the merits of the physical and machine learning approaches. The proposed model shows great performance on productivity forecast and interwell connectivity reflection in several reservoir experiments. PKFNN provides a novel way to enhance the interpretability and robustness of the data-driven-based interwell connectivity-analyzing models by integrating the physical knowledge of waterflooding reservoirs.