A POD-DEIM Reduced Model for Compressible Gas Reservoir Flow Based on the Peng-Robinson Equation of State

The efficient simulation of gas flow in porous media is highly required in petroleum engineering, CO2 sequestration, etc. However, it still remains a great challenge attributing to the gas compressibility compared with incompressible fluid flows. The commonly-used equation of states (EOS) of the gas are cubic equation and need to be updated in each iteration, and it also leads to the nonlinearity in flow equations, thus the computational cost mainly stems from the solution of gas EOS. In this paper, a hybrid reduced order model (ROM) coupling the proper orthogonal decomposition (POD) and the discrete empirical interpolation method (DEIM) is presented to accelerate the calculation of compressible single-phase gas reservoir flow, in which the Peng-Robinson EOS (P-R EOS) is considered to describe the gas states. To this end, in the hybrid ROM framework POD is applied to solve the flow equation and DEIM is used to solve the P-R EOS, respectively. The selection of POD modes and DEIM interpolation points, which plays a crucial role in the hybrid ROM, is discussed and carried out carefully. Performances of the proposed POD-DEIM-ROM are evaluated and demonstrated by two numerical cases. Simulation results illustrate that the proposed hybrid ROM displays a satisfactory computational speed-up (two orders of magnitudes faster) without sacrificing numerical accuracy significantly compared with the standard finite difference method. In addition, DEIM shows excellent acceleration and it is a perfect choice for solving the cubic gas EOS.