Stability & Convergence of Sequentially Coupled Flow-Deformation Models in Porous Media

1. Stability & Convergence of Sequentially Coupled Flow-Deformation Models in Porous Media By Paul Delgado

2. Outline • Motivation • Flow-Deformation Equations • Discretization • Operator Splitting • Multiphysics Coupling • Fixed State Splitting • Other Splitting • Conclusions

3. Motivation • (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? Flow Mechanics σ = Total Stress Tensor f = body forces per unit area mf = variation in mass flux relative to solid wf = mass flux relative to solid Sf = mass source term Courtesy: Houston Tomorrow

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

5. Multiphysics Solvers • Iteration between physics models within a single time step • computationally cheap • Enables code reuse • Easier to achieve higher order accuracy • Variable convergence properties • 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 • We will examine the strategies for sequential coupling and their convergence properties • We summarize the work of Kim (2009, 2010) illustrating iterative coupling strategies.

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

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

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

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

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

11. Classification Courtesy: Kim (2010)

12. Numerical Analysis • 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

13. More Splitting • 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. t+k t t+1 t+k-1 t+2 Flow + Deform Flow Flow + Deform Flow … Flow

14. Future Direction • MicroscalePoroelasticity • 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)

15. References Kim J. et al. (2009) Stability, Accuracy, and Efficiency of Sequential Methods for Coupled Flow and Geomechanics, SPE Reservoir Simulation Symposium Feb. 2009. 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. 2581-2589