European Numerical Mathematics and
Advanced Applications Conference 2019
30th sep - 4th okt 2019, Egmond aan Zee, The Netherlands
08:30   MS46: Numerical methods for coupled flow, reactive transport and deformation in heterogeneous porous media (Part 1)
Chair: Iuliu Sorin Pop
25 mins
Implicit Hybrid Upwinding for Non-linear Porous Media Flow and Transport
Sebastian Bosma
Abstract: We describe a method to compute numerical fluxes that improves Newton efficiency and robustness in fully implicit reservoir simulation. Specifically, we present a hybrid upwinding scheme for coupled porous media flow and transport with strong buoyancy and capillarity. Through a total velocity formulation, the scheme splits the flux computation with respect to different forces. The fluid properties in hyperbolic and parabolic terms can then be evaluated separately and consistent with the corresponding physics. In addition, the fluid properties used to obtain the total velocity can smoothly transition where phase velocities change direction. As a result, the presented hybrid upwinding scheme removes numerically induced non-linearities from the solution space. We show this reduces Newton iterations and the likelihood of diverging. Finally we discuss possibilities to enhance accuracy in the presence of capillary heterogeneity and the benefits beyond fully implicit simulations.
25 mins
An efficient numerical scheme for fully coupled flow and reactive transport in variably saturated porous media including dynamic capillary effects
Davide Illiano, Iuliu Sorin Pop, Florin Adrian Radu
Abstract: In this paper, we study a model for the transport of an external component, e.g., a surfactant, in variably saturated porous media. We discretize the model in time and space by combining a backward Euler method with the linear Galerkin finite elements. The Newton method and the L-Scheme are employed for the linearization and the performance of these schemes is studied numerically. A special focus is set on the effects of dynamic capillarity on the transport equation.
25 mins
Numerical simulation of a phase field model for reactive transport in porous media.
Manuela Bastidas, Carina Bringedal, Sorin Pop
Abstract: Precipitation and dissolution processes in porous media appear in many real-life applications, such as soil salinization and geothermal energy. Appearing at the scale of pores (the micro scale) such processes can affect the micro-scale structure of the medium. This impacts the behaviour of the system at a larger scale (macro-scale), which is of primary interest. When performing numerical simulations, one has to account for the occurrence of different scales, and with the fact that the micro-structure is changing in time in a manner that is not known a-priori. Here we consider a phase field model for precipitation and dissolution in porous media [1, 4]. Using periodic homogenization, we derive an upscaled (macro-scale) model. The resulting is a system of equations coupling parabolic and elliptic equations at the macro-scale. These involve effective parameters like diffusivity, porosity and permeability, which are obtained by solving locally coupled systems again. We proposed a combination of numerical techniques to solve the upscaled models. At each time step, the macro-scale system receives information from the micro-scale problems, which are updated as the phase field evolves. To deal with the nonlinearity of the model, we use an L-type linearization scheme [2, 3]. This is combined with mesh refinement at the micro-scale, improving both the accuracy and the efficiency of the simulations. References [1] C. Bringedal, L. von Wolff, and I.S. Pop, Phase field modeling of precipitation and dissolution processes in porous media: Upscaling and numerical experiments, CMAT Report UP-19-01 (2019), Hasselt University. [2] F. List and F.A. Radu, A study on iterative methods for solving Richards’ equation, Comput. Geosci. 20 (2016), pp. 341–353. [3] I.S. Pop, F.A. Radu and P. Knabner, Mixed finite elements for the Richards' equation: linearization procedure, J. Comput. Appl. Math. 168 (2004), pp. 365-373. [4] M. Redeker, C. Rohde and I.S. Pop, Upscaling of a tri-phase phase-field model for precipitation in porous media, IMA J. Appl. Math. 81 (2016), pp. 898-939.
25 mins
Nodal Discretization of Two-Phase Darcy flows in Highly Heterogeneous Media
Konstantin Brenner, Julian Hennicker, Roland Masson
Abstract: This talk discusses the use of nodal discretizations to simulate two-phase Darcy flows in heterogeneous media. Our objective is to improve the usual transport Two-Point upwind scheme at interfaces between highly heterogeneous rocktypes. To fix ideas, we consider the case of a two-phase Discrete Fracture Matrix Model with fractures acting as drains immersed in a much lower permeable matrix. Our approach is based on the Vertex Approximate Gradient (VAG) scheme using cell, fracture face and node unknowns. The VAG scheme is combined with an adaptation of the porous volume definition at matrix fracture interfaces as well as with a Multi-Point upwind scheme for the transport. This allows to avoid both the drain enlargement classical effect of nodal discretizations as well as phase fluxes going out on the wrong side of the fractures due to dual control volume effects at matrix fracture interfaces. 2D and 3D numerical simulations illustrate the efficiency of our approach compared with usual Control Volume Finite Element discretizations.