An implicit local time-stepping method based on cell reordering for multiphase flow in porous media

We discuss how to introduce local time-step refinements in a sequential implicit method for multiphase flow in porous media. Our approach relies heavily on causality-based optimal ordering, which implies that cells can be ordered according to total fluxes after the pressure field has been computed, leaving the transport problem as a sequence of ordinary differential equations, which can be solved cell-by-cell or block-by-block. The method is suitable for arbitrary local time steps and grids, is mass-conservative, and reduces to the standard implicit upwind finite-volume method in the case of equal time steps in adjacent cells. The method is validated by a series of numerical simulations. We discuss various strategies for selecting local time steps and demonstrate the efficiency of the method and several of these strategies by through a series of numerical examples.publishedVersio

Sequential Fully Implicit Methods for Multiscale Modeling of Compositional Flows

The fully implicit (FI) method is widely used for numerical modeling of multiphase flow and transport in porous media. It entails iterative linearization and solution of fully-coupled linear systems with mixed elliptic/hyperbolic character. However, in methods that treat the near-elliptic (flow) and hyperbolic (transport) parts separately, such as multiscale formulations (Jenny et al., JCP 2003, Møyner and Lie, JCP 2016), sequential solution strategies are used to couple the flow (pressures and velocities) and the transport (saturations / compositions). SFI schemes solve the fully coupled system in two steps: (1) Construct and solve the pressure equation (flow problem). (2) Solve the coupled species transport equations for the phase saturations and phase compositions. In SFI, each outer iteration involves this two-step sequence. Here, we propose a new SFI variant based on a nonlinear overall-volume balance equation. The first step consists of forming and solving a nonlinear pressure equation, which is a weighted sum of all the component mass conservation equations. The resulting pressure field is used to compute the total-velocity. The second step of the new SFI scheme entails introducing the overall-mass density as a degree-of-freedom, and solving the full set of component conservation equations cast in the natural-variables form (i.e., saturations and phase compositions). During the second step, the pressure and the total-velocity fields are fixed. The SFI scheme with a nonlinear pressure extends the SFI approach of Jenny et al. (JCP 2006) to multi-component compositional processes with interphase mass transfer. We analyze the `splitting errors' associated with the compositional SFI scheme, and we show how to control these errors in order to converge to the same solution as the FI method. We also show that phase-potential upwinding is incompatible with the total-velocity formulation of the fluxes, which is common in SFI schemes. We observe that in cases with strong capillary pressure or gravity, it is possible to have flow reversals. These reversals can strongly affect the convergence rate of SFI methods. We employ phase upwinding (PU) as well as a new hybrid upwinding (HU) scheme. HU determines the upwinding direction differently for the viscous, capillary pressure and buoyancy terms in the phase velocity expression. The use of HU leads to a consistent SFI scheme in terms of both pressure and compositions, and it improves the SFI convergence significantly in settings with strong capillarity and/or buoyancy. Finally, we use the multiscale restriction-smoothed basis (MsRSB) method (Møyner and Lie, JCP 2016) for the parabolic pressure operator. This sequential scheme then allows the design of robust numerical methods that are optimized for the sub-problems of flow and transport. Thus, we strongly recommend using this SFI method for sequential formulations in general, and multiscale formulations in particular

Nearwell local space and time refinement in reservoir simulation

International audienceIn reservoir simulations, nearwell regions usually require finer space and time scales compared with the remaining of the reservoir domain. We present a domain decomposition algorithm for a two phase Darcy flow model coupling nearwell regions locally refined in space and time with a coarser reservoir discretization. The algorithm is based on an optimized Schwarz method using a full overlap at the coarse level. The main advantage of this approach is to apply to fully implicit discretizations of general multiphase flow models and to allow a simple optimization of the interface conditions based on a single phase flow equation

Nearwell Local Space and Time Refinement for Multi-phase Porous Media Flows

International audienceNearwell regions in reservoir simulations usually require fine space and time scales due to several physical processes such as higher Darcy velocities, the coupling of the stationary well model with the transient reservoir model, high non linearities due to phase appearance (typically gas), complex physics such as formation damage models. If Local Grid Refinement is commonly used in reservoir simulations in the nearwell regions, current commercial simulators still make use of a single time stepping on the whole reservoir domain. It results that the time step is globally constrained both by the nearwell small refined cells and by the high Darcy velocities and high non linearities in the nearwell region. A Local Time Stepping with a small time step in the nearwell regions and a larger time step in the reservoir region is clearly a promising field of investigation in order to save CPU time. It is a difficult topic in the context of reservoir simulation due to the implicit time integration, and to the coupling between a mainly elliptic or parabolic unknown, the pressure, and mainly hyperbolic unknowns, the saturations and compositions. Our proposed approach is based on a Schwarz Domain Decomposition (DDM) Robin-Neumann algorithm using a full overlap at the coarse level to speed up the convergence of the iterative process. The matching conditions at the nearwell reservoir interfaces use optimized Robin conditions for the pressure and Dirichlet conditions for the saturations and compositions. At the well interfaces, a Neumann condition is imposed for the pressure (assuming to fix ideas that the well condition is a fixed pressure) and input Dirichlet conditions are imposed for saturations and compositions. The optimization of the Robin coefficients can be done on a pressure equation only using existing theory for elliptic/parabolic equations while the algorithm is applied on fully implicit discretization of multi-phase Darcy flows. Numerical experiments on 3D test cases including gas injection and gas-condensate reservoir exhibit the efficiency of the method both in terms of improved accuracy compared with the classical sequential windowing algorithm, and in terms of convergence of the DDM algorithm using Robin coefficients optimized once and for all on the single phase flow equation only