Adaptive Implicit Higher-Order Finite Element Methods for Multicomponent Transport in Heterogeneous and Fractured Porous Media

Modeling flow and transport in naturally fractured reservoirs is one of the most challenging problems in hydrogeology and hydrocarbon reservoir engineering, due to the orders of magnitude range in rock properties (e.g., permeability), flow rates, and spatial dimensions (from fracture apertures of < mm to km3-scale reservoirs). Porosity and permeability heterogeneity, e.g. layering, may pose similar challenges. Numerical approximations to the transport equations that are explicit in time are conditionally stable and suffer from stringent constraints on the maximum time-step sizes. Implicit methods, on the other hand, allow for large time-steps but also introduce higher numerical dispersion. In this work, we compare explicit and new fully-implicit and adaptive-implicit methods for a discrete fracture model of multicomponent compressible flow. Both lowest-order finite volume (FV) and second-order discontinuous Galerkin (DG) methods are developed and coupled to a mixed hybrid finite element method for the pressure and velocity fields. The adaptive implicit methods (AIM) are adaptive in both space and time. For fractured reservoirs, fractures and near-well regions are typically updated implicitly, while most of the grid blocks in the matrix can be updated explicitly. This results in a scheme that is more efficient in terms of both CPU time and storage requirements than a fully implicit approximation, while exhibiting lower numerical dispersion. We also demonstrate the strength of AIM in modeling flow instabilities in relatively uniform porous media for which our AIM approach automatically tracks the front of viscous and gravitational fingers. We present a range of numerical examples that demonstrate gains in CPU efficiency by 1 - 3 orders of magnitude by using DG-AIM. This allows one to accurately model large-scale 3D flow and transport problems in heterogeneous and fractured reservoirs that would be prohibitively expensive with methods that have high numerical dispersion and thus would require mesh refinement for comparable accuracy.