Adaptive conservative time integration for transport in fractured porous media

Abstract

The time step of an explicit time integration scheme for solving time-dependent hyperbolic partial differential equations in a finite volume method (FVM) framework is restricted by the Courant–Friedrichs–Lewy (CFL) criterion. Conventional time stepping integrates all grid cells with the same time step. This causes unnecessary computational costs when wave speeds and/or grid spacing vary considerably throughout the domain, and a few critical cells dictate the global time step, although most cells could be advanced with a much larger time step. Adaptive time stepping overcomes this issue by allowing different local time step sizes for each grid cell. The adaptive conservative time integration (ACTI) scheme is a recently developed adaptive time stepping method which relies on local time steps that are equal to the largest time step divided by powers of two.This work extends the ACTI scheme to tracer transport in fractured porous media. When fluid velocity within highly permeable fractures is higher than in the rock matrix and local grid refinement is applied around fractures, the CFL restriction would require prohibitively small time steps in the vicinity of fractures. For two-dimensional discrete fracture and matrix models of fracture patterns we demonstrate that ACTI reduces the computational cost by orders of magnitude compared to global time stepping while retaining solution accuracy. Empirically, we show that ACTI is stable in combination with a first-order explicit flux discretization scheme. Since combination with a standard higher-order MUSCL scheme can lead to spurious oscillations in the solution, we propose a modified MUSCL scheme relying on advection of an inclined reconstruction (MUSCL-AIR).