Numerical homogenization of well singularities in the flow transport through heterogeneous porous media

We consider the steady flow transport through highly heterogeneous porous media driven by extraction wells. We develop a new upscaling technique which lumps the small-scale details of the medium property into a few representative macroscopic parameters on a coarse scale that preserve the large-scale behavior of the medium and are more appropriate for simulations. The method is based on the recently introduced oversampling multiscale finite element method and the introduction of new base functions that locally resolve the well singularities. The modeling error which reduces the original problem having wells into problems with Dirac sources is carefully analyzed. We also provide a detailed multiscale convergence analysis of the method under the assumption that the oscillating coefficients are locally periodic. While such a simplifying assumption is not required by our method, it allows us to use homogenization theory to obtain the asymptotic structure of the solutions. New homogenization results for Green functions are derived. A new convergent algorithm approximating Green functions in the vicinity of singularities is proposed which extends the well-known Peaceman technique in the engineering literature. Numerical experiments are carried out for flow transport in both periodic and randomly generated log-normal permeabilities to demonstrate the efficiency and accuracy of the proposed method.