Abstract
AbstractWe introduce a new cell‐centered finite volume discretization for elasticity with weakly enforced symmetry of the stress tensor. The method is motivated by the need for robust discretization methods for deformation and flow in porous media and falls in the category of multi‐point stress approximations (MPSAs). By enforcing symmetry weakly, the resulting method has flexibility beyond previous MPSA methods. This allows for a construction of a method that is applicable to simplexes, quadrilaterals, and most planar‐faced polyhedral grids in both 2D and 3D, and in particular, the method amends a convergence failure in previous MPSA methods for certain simplex grids. We prove convergence of the new method for a wide range of problems, with conditions that can be verified at the time of discretization. We present the first set of comprehensive numerical tests for the MPSA methods in three dimensions, covering Cartesian and simplex grids, with both heterogeneous and nearly incompressible media. The tests show that the new method consistently is second order convergent in displacement, despite being the lowest order, with a rate that mostly is between 1 and 2 for stresses. The results further show that the new method is more robust and computationally cheaper than previous MPSA methods. Copyright © 2017 The Authors. International Journal for Numerical Methods in Engineering Published by John Wiley & Sons Ltd.
Highlights
Fluid flow in porous materials is intrinsically coupled to mechanical stresses induced on the skeleton
We have recently proved that the multi-point stress approximations (MPSAs)-O method is a convergent discretization for mechanics [33]
We have not carried out a similar comparison here, but the experiments in this paper indicate that the MPSA-W method will compare favorably with the finite element methods (FEMs)
Summary
Fluid flow in porous materials is intrinsically coupled to mechanical stresses induced on the skeleton. In many geological subsurface applications, such as CO2 storage [1], geothermal energy extraction, for example, [2], and petroleum extraction [3], engineering design calls for high flow rates, and the induced mechanical stresses become significant. In this context, simulation of coupled flow and deformation in geological porous media is becoming increasingly important. It is customary to discretize the mechanical equations of equilibrium using finite element methods (FEMs), for example, [5] This situation has the disadvantage that finite volume and FEMs inherently use different data structures and are best adapted to different grid types. This makes the construction of efficient simulation codes difficult, and as a consequence, fixed-point iteration between disparate computational tools has become industry standard [6]
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