Pressure-Transient Behavior of Vertical Wells Considering Dynamic Water Hammer and Dynamic Induced Fracture: Theory and Case Studies

Water injection can result in the creation of induced fracture by connecting natural fractures. The induced fracture penetrates the entire reservoir, leading to the interconnection of injection and production wells and ultimately resulting in water breakthrough and the abandonment of production wells. During well work, the induced fracture exhibits dynamic behavior characterized by extension and closure, known as the dynamic-induced fracture phenomena. During the shut-in process, fracture closure phenomena are often accompanied by water hammer phenomena, which can be detected bottom hole pressure data. However, numerical simulation methods are difficult to describe their dynamic processes. Therefore, we urgently need a mathematical model to fill this gap. In this work, we developed a waterflooding-induced dynamic fracture (WIDF) model. Dynamic water hammer phenomena, multi-dynamic closure phenomena, and fracture storage phenomena are introduced into the WIDF model to describe the induced fracture dynamic behavior. Field cases demonstrate that the WIDF model can improve the accuracy of interpreted parameters and correct incorrect model selection. Green function and Newman product method are used to characterize single-phase water flows within tight reservoirs. Then, the induced fracture is discrete into N segments through the boundary element method. Discrete-induced fractures are divided into n parts. Based on the induced fracture bending property, conductivity variation is independent in each part. The feature line method is used to solve the pressure response of the dynamic water hammer. The Duhamel principle is applied to couple the storage effects and reservoir pressure response. The Laplace transform method transforms the model into Laplace space so that it can be solved easily, and the Stehfest numerical inversion method transforms the model into real space to obtain the final solution. Our results show that the dynamic water hammer flow (DWH) regime with an oscillation curve and the multi-dynamic closure flow (MDCF) regime with peaks exist on the type curve. Resistance coefficient (f) and inertia coefficient (I) control the DWH regime. Fracture storage coefficient (C_f), dynamic fracture closure rates (Delpat_m), and induced fracture closure half-lengths (x_dfm) control the MDCF regime. It is worth noting that the peak position can be adjusted by the Delpat_m and x_dfm parameters, achieving a more accurate match to the field cases. The C_f is identified, which corrected for the amplified wellbore storage coefficient. Obtained permeability is also in a reasonable range of permeability for tight reservoirs. In summary, a series of mathematical methods are used to develop and solve an injection well model to identify the dynamic behavior of the induced fracture. Numerical simulation methods are used to verify the accuracy of the WIDF model. Two field cases from X Oilfield demonstrate the practicability of the WIDF model. Dynamic identification of the induced fracture would help engineers develop appropriate injection schemes to prevent water breakthroughs.