By Paul Delgado

Motivation Flow-Deformation Equations Discretization Operator Splitting Multiphysics Coupling Fixed State Splitting Other Splitting Conclusions

(Quasi-Static) Poroelasticity Equations Using constitutive relations, we obtain a fully coupled system of equations in terms of pore pressure (p) and deformation (u) How hard could it be to solve these equations? Courtesy: Houston Tomorrow Mechanics Flow m f = variation in mass flux relative to solid w f = mass flux relative to solid S f = mass source term σ = Total Stress Tensor f = body forces per unit area

Deformation Strong form Weak form Flow Strong form Weak form Backward Euler Form If constitutive relations are non-linear, => Non linear system

Simultaneous coupling between flow & deformation at each time step Computationally expensive Code Intrusive high order approximations are difficult to achieve Strong numerical stability & consistency properties Iteration between physics models within a single time step computationally cheap Enables code reuse Easier to achieve higher order accuracy Variable convergence properties We will examine the strategies for sequential coupling and their convergence properties We summarize the work of Kim (2009, 2010) illustrating iterative coupling strategies.

Based on Kumar (2005) Newton-Raphson at time t Rewrite the Jacobian matrix as: Until convergence J dd = mechanical equation with fixed pressure J fd + J ff = flow equation with solution from J dd Rewrite Newton-Raphson as Fixed Point Iteration In operator splitting, we apply this technique to the discrete (linear) operators governing the continuous system of equations.

Algorithm: 1. Hold pressure constant 2. Solve deformation first 3. Solve flow second 4. Repeat until convergence Iteration Deformation Flow t t+1 If converged If not converged State variables are held constant alternately How else can we decompose the operator?

Algorithm: 1. Hold mass constant 2. Solve deformation first 3. Solve flow second 4. Repeat until convergence Iteration Deformation Flow t t+1 If converged If not converged Conservation variable are held constant alternately Deformation solution produces pressure adjustment before solving flow equations

Algorithm: 1. Hold strain constant 2. Solve flow first 3. Solve deformation second 4. Repeat until convergence Iteration Flow Deformation t t+1 If converged If not converged State variables are held constant alternately Flow solution produces strain adjustment before solving deformation equations

Algorithm: 1. Hold stress constant 2. Solve flow first 3. Solve deformation second 4. Repeat until convergence Iteration Flow Deformation t t+1 If converged If not converged Conservation variable are held constant alternately Flow solution produces strain adjustment before solving deformation equations

Fixed StateFixed Conservation Deform 1 st Drained SplitUndrained Split Flow 1 st Fixed Strain SplitFixed Stress Split Courtesy: Kim (2010)

Kim et al. (2009) Derived stability criteria for all four operator splitting schemes using Fourier Analysis for the linear systems. Kim (2010) Tested operator splitting strategies on a variety of 1D & 2D cases Fixed number of iterations per time step => fixed state methods are inconsistent! Fixed conservation methods => consistent even with a single iteration! Undrained split suffers from numerical stiffness more than fixed-stress. Fixed Stress method => fewer iterations for same accuracy compared to undrained Fixed Stress Method is highly recommended for Consistency Stability Efficiency

Loose Coupling Minkoff et al. (2003) Special case of sequential coupling Solid mechanics equations not updated every timestep. Extremely computationally efficient Linear elasticity & porosity-pressure dependency leads to good convergence. Approximate rock compressibility factor in flow equations to compensate for non-linear elasticity in staggered coupling Heuristics to determine when to update elasticity equations. Flow + Deform Flow … Flow + Deform tt+1 t+2 t+k-1 t+k

Microscale Poroelasticity Continuum scale models assume fluid and solid occupy same space, in different volume fractions. For microscale models: Non-overlaping flow-deformation domains Discrete conservation laws and constitutive equations Discrete flow-deformation coupling relations Fixed Stress Operator Splitting Method??? Wu et al. (2012)

Kim J. et al. (2009) Stability, Accuracy, and Efficiency of Sequential Methods for Coupled Flow and Geomechanics, SPE Reservoir Simulation Symposium Feb Kim, J. (2010) Sequential Formulation of Coupled Geomechanics and Multiphase Flow, PhD Dissertation, Stanford University Kumar, V. (2005) Advanced Computational Techniques for Incompressible/Compressible Fluid- Structure Interactions. PhD Disseration, Rice University Wu, R. et al. (2012) Impacts of mixed wettability on liquid water and reactant gas transport through the gas diffusion layer of proton exchange membrane fuel cells. International Journal of Heat and Mass Transfer 55 (9-10), p