Adaptive implicit finite element methods for multicomponent compressible flow in heterogeneous and fractured porous media

Abstract

This work presents adaptive implicit first‐order and second‐order discontinuous Galerkin (DG) methods for the transport of multicomponent compressible fluids in heterogeneous and fractured porous media, discretized by triangular, quadrilateral, and hexahedral grids. The adaptive implicit method (AIM) combines the advantages of purely explicit or implicit methods (in time). In grid cells with high fluxes or low pore volumes, the transport update is done implicitly to alleviate the Courant‐Friedrichs‐Lewy (CFL) time step constraints of the conditionally stable explicit approach. Grid cells with a large CFL condition are updated explicitly. Combined, this allows higher efficiency than explicit methods, but it reduces the “penalty” of implicit methods, which exhibit high numerical dispersion and are more computationally and storage expensive per time step. The advantages of AIM are modest for uniform grids and rock properties. However, in heterogeneous or fractured reservoirs explicit methods may become impractical, while a fully implicit approach introduces unnecessary numerical dispersion and is overkill for low‐permeability layers and matrix blocks. In such applications, AIM is shown to be significantly more efficient and accurate. The division between explicit and implicit grid cells is made adaptively in space and time. This allows for a high level of explicitness and can also adapt to high fluxes caused by, e.g., viscous and gravitational flow instabilities. Numerical examples demonstrate the powerful features of AIM to model, e.g., solute transport, carbon sequestration in saline aquifers, and miscible gas injection in fractured oil and gas reservoirs.

A higher-order finite element reactive transport model for unstructured and fractured grids

“…( 16 )) and the implementation of the Nernst-Planck Eq. ( 23 ) for electromigration, the implementation is identical to prior (non-reactive) works 20 , 32 , 33 , and is thus not repeated here in further detail.…”

“…Explicit, implicit, and adaptive implicit Euler time-discretizations have been implemented, where the adaptive method uses an implicit update for grid cells that have a small Courant-Friedrichs-Lewy (CFL) time-step constraint 50 and an explicit update elsewhere 33 . The advantage of implicit methods is that they are unconditionally stable and thus allow for larger time-steps.…”

“…Their other main strength is to provide accurate velocity fields 24 , particularly for heterogeneous and fractured domains 25 , 26 . These features also allow for an efficient discrete fracture model based on cross-flow equilibrium 27 – 33 .…”

A fully mass-conservative iterative IMPEC method for multicomponent compressible flow in porous media

“…Most of the reservoir simulators solve the main governing equations of flow and transport equations in porous media by the Darcy's law, the conservation of mass and a pressure relation. The fully implicit methods can be directly applied to solve the coupled equations, and one can also use the improved sequential implicit technique [19,7] and the adaptive implicit methods [30,23]. It is known that the implicit methods exhibit greater numerical dispersion than explicit methods on the same grid [38], although the implicit methods deserve better stability than the explicit methods.…”

“…Thus, compared with the fully implicit methods, the IMPEC scheme has become a popular choice to simulate the multicomponent compressible flow in porous media (cf. [10,6,11,7,41,23] and the references therein).…”