Partitioned time stepping method with different time scales for a dual-porosity-Stokes model

Abstract

In this report, we consider confined flow in a dual-porosity media coupled with free flow in embedded macrofractures and conduits. More specifically, the flow in dual-porosity media, which consists of both matrix and microfractures, is described by a dual-porosity model of both matrix pressure and microfracture pressure; the flow in the conduits/macrofractures is governed by the Stokes model. There are four physically valid interface conditions to couple the two models on the interface, including a no-exchange condition, a mass balance condition, a force balance condition, and the Beavers-Joseph condition. To decouple the complex model into three simple sub-problems and solve them separately, we propose a partitioned time stepping method. It solves one, uncoupled matrix pressure equation, microfracture pressure equation and Stokes equation per time step. Most importantly, considering the flows in two sub-regions act with different speed, our scheme allows different time scales to discretize different sub-problems. Under a modest time step restriction, we prove the stability of the method. We also derive its optimal error estimates. Numerical tests verify the theoretical results.