Perturbation analysis for the stochastic Darcy problem

We study the single-phase flow in a saturated, bounded randomly heterogeneous porous medium. We model the permeability tensor as a lognormal random field. We perform a perturbation analysis, expanding the solution in Taylor series. We give a theoretical upper bound for the norm of the residual of the Taylor series which predicts the divergence of the series as the polynomial degree goes to infinity. We then numerically determine the optimal order for the Taylor polynomial and the maximum achievable accuracy of the perturbation approach.