A very high order Flux Reconstruction (FR) method for the numerical simulation of 1-D compositional fluid flow model in petroleum reservoirs

Abstract

Compositional reservoir simulation is an extremely important tool for modeling the fluid flow in complex petroleum reservoirs, as in cases where the reservoir fluid is volatile and composed of several pseudo-components with distinct characteristics, or reservoirs that require the use of enhanced oil recovery process. For these cases, simpler black-oil models are not suitable. The compositional model involves the solution of a large system of partial differential equations, that comprises the mass conservation, the Darcy’s law and the fugacity constraints. However, the high number of equations and constraints render the compositional problems extremely complex and computationally demanding solutions. To improve accuracy and reduce the high computational costs, one can use higher than first or even second-order methods to approximate the advective flux terms in the hyperbolic conservation laws that describe the multicomponent transport in the reservoir. Those very high-order schemes can replace low-order approaches, such as the First Order Upwind (FOU) method, which is traditionally used in commercial petroleum reservoir simulators (CMG, 2019; ECLIPSE, 2022), obtaining more accurate solutions with reduced computational cost. In this work, we consider a three-phase and isothermal fluid flow of water, oil and gas and assume that there is no mass transfer between the water and the hydrocarbon phases. Physical dispersion and capillary effects are neglected. To solve the system of Partial Differential Equations (PDEs), we used the classical IMPEC (Implicit Pressure Explicit Composition) approach and, to model the complex phase behavior, we used the Peng–Robinson equation of state (PR-EOS). The classical Two Point Flux Approximation (TPFA) method is used to spatially discretize the diffusion terms in the pressure equation. For the first time in literature the very high order Flux Reconstruction (FR) method is adapted for the numerical simulation of the compositional model in petroleum reservoirs. The FR method is a numerical scheme employed to discretize the hyperbolic conservation laws using general grids. To avoid spurious oscillations in the vicinity of solution discontinuities, the h-MLP (hierarchical Multi-dimensional Limiting Process) is applied in the reconstruction stage. For the time integration, we have used the third-order Runge–Kutta approach. To appraise the robustness of our formulation, we have solved some one-dimensional benchmark problems found in literature and compared the FR solution with the FOU method and with the second-order Monotonic Upstream-Centered Scheme for Conservation Laws (MUSCL) method. From our results, the FR method has shown improved accuracy, efficiency and capability of detecting sharper shocks when compared to the other methods that we have considered for the numerical modeling of the compositional flow in the present paper.