A domain decomposition method to couple nonisothermal compositional gas liquid Darcy and free gas flows

Abstract

A domain decomposition algorithm is introduced to couple nonisothermal compositional gas liquid Darcy and free gas flow and transport. At each time step, our algorithm solves iteratively the nonlinear system coupling the nonisothermal compositional Darcy flow in the porous medium, the RANS gas flow in the free-flow domain, and the transport of the species and of energy in the free-flow domain. In order to speed up the convergence of the algorithm, the transmission conditions at the interface are replaced by Robin type boundary conditions. The Robin coefficients are obtained from a diagonal approximation of the Dirichlet to Neumann operator related to a simplified model in the neighbouring subdomain. The efficiency of our domain decomposition algorithm is assessed on several test cases focusing on the modeling of the mass and energy exchanges at the interface between the geological formation and the ventilation galleries of geological radioactive waste disposal.

Domain decomposition methods to model heat exchanges between a well and a rock mass

“…The analytical solution of this min max problem seems difficult to compute analytically, and a numerical computation is not convenient to implement. In the spirit of [2], we propose below a suboptimal, but simple choice. Let us choose for β rock the so called Taylor 0th order approximation of DtN rock which is exact for the lower frequency k = 0 ie…”

“…Figure 7 considers a 3D radial mesh of size 50 × 50 × 200 of the rock mass domain (r w , R w ) × (0, 2π) × (0, L), R w = 10 m, L = 1500 m, exponentially refined at the well boundary. The thermal conductivity is taken randomly in the interval [2,5] W.K −1 .m −1 at each cell of the mesh. For such heterogeneous test case, the usual strategy is simply to use, at each face σ of the well boundary, the following value of the thermal conductivity at the face σ:…”

“…Heterogeneous domain decomposition methods have previously been used to couple different physical models at the interface between subdomains. For example, let us refer to [7], [4], [8] for the coupling of a Darcy flow with a Stokes or Navier-Stokes free flow, to [2] for the coupling of non-isothermal liquid gas Darcy and free gas flows, and to [14] for the coupling of Helmholtz and Laplace Equations. Let us also refer to [1] and [23] for Robin-Robin domain decomposition methods developped for the nonlinear Richards equation modelling partially saturated flow in porous media.…”

An efficient multigrid-DEIM semi-reduced-order model for simulation of single-phase compressible flow in porous media

“…In general, there are two kinds of popular acceleration methods for solving the flow equation, one is full-order model, and the other is reduced-order model. The full-order model mainly includes parallel computing (Schrefler et al 1999; Edited by Xiu-Qiu Peng Akhmetzyanov et al 2012;Fung et al 2016;Jambunathan and Levin 2017;Tanaka et al 2018;Ho et al 2019), domain decomposition method (Popov and Power 1999;Stavroulakis and Papadrakakis 2009;Skogestad et al 2013;Birgle et al 2018), multigrid method (Gries et al 2014;Lovett et al 2015;Alzahabi et al 2016;la Cour et al 2016;Wang et al 2017a, b), etc. These methods use CPU/GPU parallel processing, regional partitioning technique and multilevel recursive calculation, respectively, to solve the flow equation efficiently in original full-order space to achieve the acceleration.…”

Robust monolithic solvers for the Stokes-Darcy problem with the Darcy equation in primal form

“…For completeness, we mention that there are also works that successfully apply DD techniques for problems with more complex physics, e.g. [12].…”