Stability and convergence of sequential methods for coupled flow and geomechanics: Fixed-stress and fixed-strain splits

Abstract

We analyze stability and convergence of sequential implicit methods for coupled flow and geomechanics, in which the flow problem is solved first. We employ the von Neumann and energy methods for linear and nonlinear problems, respectively. We consider two sequential methods with the generalized midpoint rule for t n+ α , where α is the parameter of time discretization: namely, the fixed-strain and fixed-stress splits. The von Neumann method indicates that the fixed-strain split is only conditionally stable, and that its stability limit is a coupling strength less than unity if α ⩾ 0.5. On the other hand, the fixed-stress split is unconditionally stable when α ⩾ 0.5, the amplification factors of the fixed-stress split are different from those of the undrained split and are identical to the fully coupled method. Unconditional stability of the fixed-stress split is also obtained from the energy method for poroelastoplasticity. We show that the fixed-stress split is contractive and B-stable when α ⩾ 0.5. We also estimate the convergence behaviors for the two sequential methods by the matrix based and spectral analyses for the backward Euler method in time. From the estimates, the fixed-strain split may not be convergent with a fixed number of iterations particularly around the stability limit even though it is stable. The fixed-stress split, however, is convergent for a fixed number of iterations, showing better accuracy than the undrained split. Even when we cannot obtain the exact local bulk modulus (or exact rock compressibility) at the flow step a priori due to complex boundary conditions or the nonlinearity of the materials, the fixed-stress split can still provide stability and convergence by an appropriate estimation of the local bulk modulus, such as the dimension-based estimation, by which the employed local bulk modulus is less stiff than the exact local bulk modulus. We provide numerical examples supporting all the estimates of stability and convergence for the fixed-strain and fixed-stress splits.