Convergence analysis of single rate and multirate fixed stress split iterative coupling schemes in heterogeneous poroelastic media

Abstract

Recently, the accurate modeling of flow-structure interactions has gained more attention and importance for both petroleum and environmental engineering applications. Of particular interest is the coupling between subsurface flow and reservoir geomechanics. Different single rate and multirate iterative and explicit coupling schemes have been proposed and analyzed in the past. In addition, Banach fixed point contraction results were obtained for iterative coupling schemes, and conditionally stable results were obtained for explicit coupling schemes. In this work, we will consider the mathematical analysis of the single rate and multirate fixed stress split iterative coupling schemes for spatially heterogeneous poroelastic media. We will re-establish the contractivity for both schemes in the localized case, and we will show that heterogeneities come at the expense of imposing more restricted conditions on the number of fine flow time steps that can be taken within one coarse mechanics time step in the multirate case. Our mathematical analysis is supplemented by numerical simulations validating our derived upper bounds. To the best of our knowledge, this is the first rigorous mathematical analysis of the multirate fixed-stress split iterative coupling scheme in heterogeneous poroelastic media.

Sequential formulation of all‐way coupled finite strain thermoporomechanics for largely deformable gas hydrate deposits

“…Let us introduce the following equations from Equation (35) to describe the extended fixed stress method.…”

“…30 In time discretization, we take the fixed stress sequential implicit method, where individual problems of flow and geomechanics are solved implicitly and sequentially. Specifically, the fixed stress method solves the flow problem first by fixing the total stress rate locally and the geomechanics problem is solved at the next step [31][32][33][34][35][36] In this sequential approach, one may iterate this sequential procedure at each time step until convergence, where the solution is then identical to that from the monolithic approach, or perform a fixed number of iterations per time step. It is common to employ a staggered method (one iteration per time step), similar to the fractional step method.…”

Robust block diagonal preconditioners for poroelastic problems with strongly heterogeneous material

“…The space discretization leads to a system of algebraic equations with a block structure where the particular blocks correspond either to the physical fields or to their mutual couplings. The related numerical solution methods can be divided into block‐splitting methods with Uzawa‐type algorithms 22–25 and preconditioned Krylov methods, applied on the original monolithic algebraic system 26,27 . The related preconditioners reflect the block structure and are based on Schur complements 16,28–31 .…”

Process understanding of induced seismicity during stimulation of enhanced geothermal systems

“…These steps may be repeated in an iterative way. Therefore, several iterations might be necessary for each time step until all the equations reach convergence and provide the same solution as that of the fully coupled approach (Almani et al, 2023(Almani et al, , 2016Castelletto et al, 2015;Kim et al, 2011;Preisig and Prévost, 2011). The number of iterations may be large, such that the calculation time is similar to the fully coupled method.…”