A block operator splitting method for heterogeneous multiscale poroelasticity
Traditional models of poroelastic deformation in porous media assume relatively homogeneous material properties such that macroscopic constitutive relations lead to accurate results. Many realistic applications involve heterogeneous material properties whose oscillatory nature require multiscale methods to balance accuracy and efficiency in computation. ^ The current study develops a multiscale method for poroelastic deformation based on a fixed point iteration based operator splitting method and a heterogeneous multiscale method using finite volume and direct stiffness methods. To characterize the convergence of the operator splitting method, we use a numerical root finding algorithm to determine a threshold surface in a non-dimensional parameter space β separating convergent & divergent problems. We also use the method of manufactured solutions to verify the proposed multiscale algorithm. ^ Results suggest that non-dimensional parameter values β above the threshold surface ensure convergence, with increasing rate of convergence as β → ∞. For a given spatial discretization Δ x, convergence can be ensured by choosing larger time stepsizes Δ t. ^ The proposed multiscale algorithm converges for the decoupled solid deformation PDE with analogous behaviors as observed in Chu et al. (2012). We observed divergence in our multiscale algorithm for the decoupled fluid equation in the heterogeneous case and attribute it to lack of monotonicity preservation induced by the reaction term in the microscale model. An alternative cross-sectional flux estimator is proposed to improve convergence.^
Applied Mathematics|Geophysics|Engineering, Mechanical
Delgado, Paul M, "A block operator splitting method for heterogeneous multiscale poroelasticity" (2013). ETD Collection for University of Texas, El Paso. AAI1545157.