Inverse modeling of tracer flow via a mass conservative generalized multiscale finite volume/element method and stochastic collocation

In this paper we consider a stochastic inverse flow problem with a high-contrast and stochastic permeability coefficient. This problem is solved within the context of the Markov chain Monte Carlo method in which a number of forward simulations are needed to construct the posterior distribution. Due to the large number of forward solves, a constrained version of the generalized multiscale finite element method (GMsFEM) is used to reduce the global dimension of the pressure solutions. Furthermore, since the permeability coefficient exhibits a large variation in scales, GMsFEM allows us to construct coarse solutions that are able to accurately capture the underlying fine effects of the system. To ensure a local mass conservation property, Lagrange multipliers are used in the problem formulation and discretized solution technique. Since the construction of GMsFEM spaces involves solving a number of localized eigenvalue problems that can become burdensome in the context of random sampling, we employ the use of a sparse-grid collocation technique for local stochastic reduction. In particular, rather than recomputing the eigenfunctions and basis functions for each realization, a fixed set of collocation points is used for eigenfunction pre-computations. Then, for each realization a sparse interpolant is used in the place of the direct computations for the coarse space constructions. In doing so, we are able to incorporate a reduced stochastic model (that combines GMsFEM-FV and sparse-grid collocation) in order to present a suitable alternative for determining input permeability distributions conditioned to available tracer cut data.