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Stability and convergence of sequential methods for coupled flow and geomechanics: drained and undrained splits. (English) Zbl 1228.74106

Summary: We perform a stability and convergence analysis of sequential methods for coupled flow and geomechanics, in which the mechanics sub-problem is solved first. We consider slow deformations, so that inertia is negligible and the mechanical problem is governed by an elliptic equation. We use Biot’s self-consistent theory to obtain the classical parabolic-type flow problem. We use a generalized midpoint rule (parameter \(\alpha \) between 0 and 1) time discretization, and consider two classical sequential methods: the drained and undrained splits.
The von Neumann method provides sharp stability estimates for the linear poroelasticity problem. The drained split with backward Euler time discretization \((\alpha = 1)\) is conditionally stable, and its stability depends only on the coupling strength, and it is independent of time step size. The drained split with the midpoint rule \((\alpha = 0.5)\) is unconditionally unstable. The mixed time discretization, with \(\alpha = 1.0\) for mechanics and \(\alpha = 0.5\) for flow, has the same stability properties as the backward Euler scheme. The von Neumann method indicates that the undrained split is unconditionally stable when \(\alpha\geq 0.5\). We extend the stability analysis to the nonlinear regime (poro-elastoplasticity) via the energy method. It is well known that the drained split does not inherit the contractivity property of the continuum problem, thereby precluding unconditional stability. For the undrained split we show that it is B-stable (therefore unconditionally stable at the algorithmic level) when \(\alpha\geq 0.5\).
We also analyze convergence of the drained and undrained splits, and derive the a priori error estimates from matrix algebra and spectral analysis. We show that the drained split with a fixed number of iterations is not convergent even when it is stable. The undrained split with a fixed number of iterations is convergent for a compressible system (i.e., finite Biot modulus). For a nearly-incompressible system (i.e., very large Biot modulus), the undrained split loses first-order accuracy, and becomes non-convergent in time.
We also study the rate of convergence of both splits when they are used in a fully-iterated sequential scheme. When the medium permeability is high or the time step size is large, which corresponds to a high diffusion of pressure, the error amplification of the drained split is lower and therefore converges faster than the undrained split. The situation is reversed in the case of low permeability and small time step size.
We provide numerical experiments supporting all the stability and convergence estimates of the drained and undrained splits, in the linear and nonlinear regimes. We also show that our spatial discretization (finite volumes for flow and finite elements for mechanics) removes the well-documented spurious instability in consolidation problems at early times.

MSC:

74S20 Finite difference methods applied to problems in solid mechanics
74S05 Finite element methods applied to problems in solid mechanics
76M12 Finite volume methods applied to problems in fluid mechanics
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
76S05 Flows in porous media; filtration; seepage
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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