European Numerical Mathematics and
Advanced Applications Conference 2019
30th sep - 4th okt 2019, Egmond aan Zee, The Netherlands
10:40   MS46: Numerical methods for coupled flow, reactive transport and deformation in heterogeneous porous media (Part 2)
Chair: Iuliu Sorin Pop
25 mins
Mixed-dimensional multigrid methods for single-phase flow problems in fractured porous media
Andres Arraras, Francisco J. Gaspar, Laura Portero, Carmen Rodrigo
Abstract: This work presents a monolithic multigrid method for the efficient numerical solution of single-phase flow problems in factured porous media. Specifically, we consider a reduced model which couples Darcy flow in a two-dimensional porous matrix with either Darcy or Forchheimer flow within a set of vertical and/or horizontal possibly intersecting one-dimensional fractures. The proposed multigrid solver takes into account the mixed-dimensional nature of the problem by using mixed-dimensional smoothing and inter-grid transfer operators. Numerical experiments illustrate the robustness of the method with respect to the permeability of the fractures, the Forchheimer coefficient, the grid size and the number of fractures in the network.
25 mins
On The Convergence of Flow and Mechanics Iterative Coupling Schemes in Fractured Heterogeneous Poro-Elastic Media
Tameem Almani, Kundan Kumar, Abdulrahman Manea
Abstract: Nowadays, solving the coupled flow and geomechanics problem is of high importance in modeling upstream petroleum operations. This is due to the fact that different multiscale and multi-physical processes cannot be modeled correctly without incorporating an underlying geomechanical model. Such a model enables upstream engineers to simulate different natural and induced physical phenomena including reservoir deformation, pore collapse, wellbore stability, fault activation, and hydraulic fracturing [3]. Since fractures have significant effects on reservoir flow profiles, incorporating an accurate model for fractures when studying the coupled geomechanics and flow problem is a must. The main objective of this work is to establish the convergence of an adaptation of the fixed-stress split coupling scheme [2] in fractured heterogeneous poro-elastic media. In our approach, fractures are modeled as possibly non-planar interfaces, and the flow in the fracture is described by a lubrication type system [3]. The flow in the reservoir matrix and in the fracture are coupled to the geomechanics model through a fixed-stress split iteration [1, 2], in which mass balance equations (for both flow in the matrix, and in the fracture) are augmented with fixed-stress split regularization terms. The convergence proof we provide shall determine the appropriate localized values of these regularization terms, which ensure the convergence of the coupling scheme in heterogeneous media. Banach fixed-point contraction results will be derived for this flow-mechanics coupled system by studying the equations satisfied by the difference of iterates, which will lead to the desired Banach contraction argument. Geometric convergence to the unique solution of the system follows immediately as the sequence of iterates represents a convergent Cauchy sequence. To the best of our knowledge, this is the first time in literature a rigorous convergence analysis is established for coupling flow with geomechanics in fractured heterogeneous poroelastic media. References: [1] A. Mikelic, and M. F. Wheeler, “Convergence of Iterative Coupling for Coupled Flow and Geomchanics”. Computational Geomechanics, 17(3), 455-461, 2013. [2] V. Girault, K. Kumar, and M. F. Wheeler, “Convergence of iterative coupling of geomechanics with flow in a fractured poroelastic medium”. Computational Geosciences, 20 (5), 997-101, 2016. [3] V. Girault, M. F. Wheeler, B. Ganis, and M. E. Mear: A Lubrication Fracture Model in a Poro-elastic Medium. Mathematical Models and Methods in Applied Sciences, 25 (04), 587-645, 2015.
25 mins
A fully-mixed flow formulation for the coupling of the Navier-Stokes and Biot models
Ivan Yotov, Sergio Caucao
Abstract: We develop and analyze a mathematical model for the coupling of the Navier-Stokes and Biot equations based on a fully mixed flow formulation. The Navier-Stokes formulation is based on weakly symmetric deviatoric stress, velocity, and vorticity. The porous media flow formulation is based on Darcy velocity and pressure. The analysis is performed in non-Hilbert space setting in order to deal with the low regularity of the fluid velocity in the mixed Navier-Stokes formulation. Using the theory of evolutionary saddle point problems and a fixed point theorem, we show that the model has a unique solution under a small data assumption. We further analyze the stability and accuracy of the finite element approximation of the model and present numerical experiments.
25 mins
Iterative Methods for Coupled Flow and Mechanics in Fractured Porous Media
Kundan Kumar, Tameem Almani, Vivette Girault, Maarten de Hoop, Mary Wheeler, Ruichao Ye
Abstract: The coupled flow and geomechanical effects in fractured porous media are important. Activities related to energy extraction or injection operations can cause significant changes of fluid-pressure in subsurface formations leading to a change in the in-situ stress conditions. Fractures have strong influence on the flow behaviour and at the same time are the vulnerable regions for mechanical failures as the faults or fractures become prone to slipping. We consider fractures that are big enough and explicitly described by lower dimensional objects (Discrete Fracture Networks). The resulting model is a single phase quasi-static Biot model in the porous matrix coupled to a Darcy flow model on the fractures. We report here some of the developments in suitable iterative schemes for such models and their extensions. Our work progressively builds on the three components: 1. Developing suitable iterative schemes by decoupling flow in fractured media and the mechanics, 2. Developing multirate schemes by exploiting the different time scales of mechanics and flow solve by taking coarser time step for mechanics and smaller time steps for flow, 3. Considering advanced models for rate- and state- dependent friction laws for the fracture slippage for the mechanics equation. We analyse these iterative multirate schemes and rigorously analyse the convergence and stability properties of these schemes. References 1. V. Girault, M. F. Wheeler, K. Kumar, G. Singh, Mixed formulation of a linearized lubrication fracture model in a poro-elastic medium, Springer-ECCOMAS series Computational Methods in Applied Sciences, 171--219, 47, Springer, Cham. 2. K. Kumar, V. Girault, M. F. Wheeler, Convergence of iterative coupling of geomechanics with flow in a fractured poroelastic medium, Computational Geosciences, 2016, 20, 997--1011. 3. T. Almani, K. Kumar, A. Dogru, G. Singh, M. F. Wheeler, Convergence Analysis of Multirate Fixed-Stress Split Iterative Schemes for Coupling Flow with Geomechanics, Computer Methods for Applied Mechanics and Engineering, 311, 2016, 180--207. 4. R. Ye, K. Kumar, M. V. de Hoop, M. Campillo, A multi-rate iterative coupling scheme for dynamic ruptures and seismic waves generation in the self-gravitating Earth: the discontinuous Galerkin method, 2018, submitted.