Sequential Methods for Coupled Geomechanics and Multiphase Flow

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208 CHAPTER 7. CONCLUSIONS 7.2 Coupled Mechanics and Single-Phase flow In Chapter 3–5, we limited our analysis to single-phase flow, but we accounted for both elastic and elasto-plastic material behaviors. The four sequential methods analyzed fall in two categories: those that solve the mechanical problem first (drained and undrained splits), and those that solve the flow problem first (fixed-strain and fixed-stress splits). The drained and fixed-strain splits are the obvious splits, as they freeze one of the state variables (pressure or displacement, respectively) in the sequential solution strategy. As we demonstrate quite clearly in this thesis, these obvious splits are not a good choice. 7.2.1 Stability We have performed a thorough stability analysis using the Von Neumann and energy methods with the generalized midpoint rule for time integration. From the Von Neumann stability analysis, the coupling strength τ emerges as the key parameter. When we apply backward Euler time discretization (i.e., α = 1.0), the drained and fixed-strain splits yield conditional stability, and this limit is independent of time step size and depends only on the coupling strength. This implies that stability of the drained and fixed-strain splits cannot be achieved by simply tuning the time step size. Therefore, physical problems with high coupling strength cannot be solved by the drained or fixed-strain splits, regardless of time step size. The stability criterion in one dimension is τ ≤ 1. This criterion can be applied, however, to multi-dimensional problems. From the Von Neumann stability analysis we also determine that the drained and fixed-strain splits can have negative amplification factors, which explain their oscillatory behaviors, even when they are stable. When the midpoint rule α = 0.5 is used, the drained split is unconditionally unstable, whereas the fixed-strain split is conditionally stable, where the stability condition is τ ≤ 1. But when we use a mixed time discretization, where α = 1.0 for mechanics and α = 0.5 for flow, the drained split yields the same stability behavior as the backward Euler time discretization. On the other hand, the undrained and fixed-stress splits show unconditional stability for
7.2. COUPLED MECHANICS AND SINGLE-PHASE FLOW 209 both elasticity and elasto-plasticity. Using the Von Neumann method, the undrained and fixed-stress splits are unconditionally stable for α ≥ 0.5. Moreover, their error amplification factors are always positive for the backward Euler time discretization scheme; as a result, the undrained and fixed-stress numerical solutions do not exhibit oscillations in time. In particular, the fixed-stress split has the same amplification factors as those associated with the fully coupled method. The undrained and fixed-stress splits can be applied safely to general poro-mechanical behaviors, such as compressible solid grains and plasticity with hardening. For operator splitting, the undrained and fixed-stress splits are contractive relative to the Helmholtz free-energy and the complementary Helmholtz free-energy norms, leading to well-posed problems. For algorithmic (discrete) stability, the undrained and fixed-stress splits yield B-stability relative to the two energy norms, when α ≥ 0.5. 7.2.2 Convergence For convergence analysis, we estimated the error propagation for the sequential methods based on the backward Euler time discretization using matrix and spectral methods. One of the key results of this thesis is that the drained and fixed-strain splits with a fixed number of iterations are not convergent. That is, the drained and fixed-strain splits (with, say, a single iteration per time step) converge to a different (wrong) solution as ∆t → 0, even though they are numerically stable. This result is particularly relevant because the converged solution, even if incorrect, can appear to be physically plausible. We have illustrated this behavior with several test cases, including the Terzaghi problem. Clearly, for the drained and fixed- strain methods, taking more iterations per time step is a better strategy than cutting the time step size. On the other hand, the undrained and fixed-stress methods are convergent for a compressible fluid with first-order accuracy in time, even if a staggered (one-pass) method is used. In the undrained and fixed-stress splits, reducing the time step size is a better strategy than taking more iterations.
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SEQUENTIAL METHODS FOR COUPLED GEOM
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20 CHAPTER 2. FORMULATION coupling
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Page 270 and 271: 244 BIBLIOGRAPHY Mech Eng 122: 145-
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Sequential Methods for Coupled Geomechanics and Multiphase Flow

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