An Efficient Fully-Implicit MFD-MUSCL Method Based on a Novel Multislope Limiting Procedure
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AbstractStandard reservoir simulation schemes use single-point upstream weighting for computing the convective fluxes when multi-phase/component fluids are present. These schemes are only first-order and may cause high viscosity effect. Second-order schemes provide a better resolution and reduce the smoothing near the shocks. In reservoir simulation practice, implicit discretisations capable of taking large time steps are preferred over explicit schemes. However, even for a simple first-order method, solving a large nonlinear system is often very expensive; Extra coupling and nonlinearity of the discretised equations can be introduced from a higher-order spatial discretisation. It has been shown that strong nonlinearity as well as lack of continuous differentiability in numerical flux function and flux limiter can cause serious nonlinear convergence problem.Cell-centered finite-volume (CCFV) discretizations may offer several attractive features, especially for fluid flow in heterogeneous or fractured porous media. The objectives of this work are to develop a fully-implicit CCFV framework that could achieve second-order spatial accuracy on smooth solutions, while at the same time maintain robustness and nonlinear convergence performance. We develop a novel multislope MUSCL method which interpolates the required values at the edge centroids in a simpler way by taking advantage of some geometric features of the triangular mesh. Through the edge centroids, the numerical diffusion caused by mesh skewness is significantly reduced, and optimal second-order accuracy can be achieved. An improved gradually-switching piecewise-linear flux-limiter is introduced according to mesh non-uniformity in order to prevent spurious oscillations. The smooth flux-limiter can achieve high accuracy without degrading nonlinear convergence behavior.For the discretization of pressure and Darcy velocities, a mimetic finite difference method that provides flux- continuity and an accurate total velocity field is used. The fully-coupled MFD-MUSCL framework is adapted to accommodate a lower-dimensional discrete fracture-matrix (DFM) model. Several numerical tests with discrete fractured system are carried out to demonstrate the efficiency and robustness of the numerical model. The results show that the high-order MUSCL method effectively reduces numerical diffusion, leading to improved resolution of saturation fronts compared with the first-order method. In addition, it is shown that the developed multislope method and adaptive flux limiter exhibit superior nonlinear convergence compared with other alternatives.
