Sequential Methods for Coupled Geomechanics and Multiphase Flow

Chapter 4
Convergence of the Drained and
Undrained Splits
4.1 Error Estimation
Sequential implicit solution schemes for coupled geomechanics and flow entail solutions of
two implicit problems, one for mechanics and one for flow, in sequence. In such schemes, one
often performs a single pass or a few iterations, and it is crucial to assess the convergence
behavior of the sequential coupling strategy. In this Chapter, two sequential schemes are
investigated: the drained and undrained splits. The backward Euler time discretization is
used, and the linear coupled problem is considered for simplicity. We employ matrix and
spectral methods to analyze the behavior of the error as a function of iteration.
4.1.1 Error estimation of the drained split by matrix algebra
For a sequential solution strategy, we can write
e n+1,niter
ts
≤ x n+1
t − xn+1
f +
= O(∆t) +
x n+1
f
81
x n+1
f
− xn+1,niter
s
− xn+1,niter s
, (4.1)