Scientific computing is at the core of many High-Performance Computing applications, including computational flow dynamics. Because of the utmost importance to simulate increasingly larger computational models, hardware acceleration is receiving increased attention due to its potential to maximize the performance of scientific computing. Field-Programmable Gate Arrays could accelerate scientific computing because of the possibility to fully customize the memory hierarchy important in irregular applications such as iterative linear solvers. In this article, we study the potential of using Field-Programmable Gate Arrays in High-Performance Computing because of the rapid advances in reconfigurable hardware, such as the increase in on-chip memory size, increasing number of logic cells, and the integration of High-Bandwidth Memories on board. To perform this study, we propose a novel Sparse Matrix-Vector multiplication unit and an ILU0 preconditioner tightly integrated with a BiCGStab solver kernel. We integrate the developed preconditioned iterative solver in Flow from the Open Porous Media project, a state-of-the-art open source reservoir simulator. Finally, we perform a thorough evaluation of the FPGA solver kernel in both stand-alone mode and integrated in the reservoir simulator, using the NORNE field, a real-world case reservoir model using a grid with more than 105 cells and using three unknowns per cell.