Convergence of multirate fixed stress split iterative schemes for a fractured Biot model

Abstract

This paper considers the convergence analysis of a coupled mixed dimensional flow and mechanics problem in a fractured poro-elastic medium. In this mixed dimensional type system, the flow equation on a $d$ dimensional porous matrix is coupled to the flow equation on a $d-1$ dimensional fracture surface. The fracture geometry is treated as a possibly non-planar interface, and the fracture is assumed to remain open (i.e., ignoring contact mechanics boundary conditions). The main contribution is developing a multirate scheme in which flow takes multiple fine time steps within one coarse mechanics time step using the standard fixed stress splitting approach. In this coupled system, the linear quasi-static Biot model is assumed for the porous matrix, and the lubrication-type system (Girault et al., 2015) is assumed for the fracture. More specifically, two variations (i.e., algorithms) of the multirate scheme are formulated and their convergence analyses are established. The convergence analysis is based on proving a Banach fixed-point contraction argument which establishes the geometric convergence, and hence, the uniqueness of the obtained solution for both algorithms. The theoretical investigations are supplemented by preliminary numerical results validating the theoretical findings.