Shriram Srinivasan

Multiscale direction-splitting algorithms for parabolic equations with highly heterogeneous coefficients

In this paper we discuss two methods for upscaling of highly heterogeneous data for parabolic problems in the context of a direction splitting time approximation. The first method is a direct application of the idea of Jenny etźal. (2003) in the context of the direction splitting approach. The second method devises the approximation from the Schur complement corresponding to the interface unknowns of the coarse grid, by applying a proper L 2 projection operator to it. The spatial discretization employed in this paper is based on a MAC finite volume stencil but the same approach can be implemented within a proper finite element discretization. A key feature of the present approach is that it can extend to 3D problems with very little computational overhead. The properties of the resulting approximations are demonstrated numerically on some benchmark coefficient data available in the literature.

A Generalized Darcy–Dupuit–Forchheimer Model with Pressure-Dependent Drag Coefficient for Flow Through Porous Media Under Large Pressure Gradients

Prior works have discussed the appearance of a ceiling flux for the generalized Darcy model, but there has been no such discussion on the Forchheimer model. Moreover, the earlier results were obtained through numerical computations. Here we employ a semi-inverse solution to get analytical expressions for the solution and the volumetric flux, and we demonstrate, by direct comparison, that for the flux computation, the semi-inverse approximation is as accurate as a numerical solution to the problem. It is shown that neither model (Darcy or Forchheimer) has the desirable property of attaining a limiting flux for large driving pressures. However, when the pressure-dependence of viscosity is incorporated suitably to generalize the classical models, the generalized Darcy and Forchheimer models show the desirable behaviour of a ceiling flux for large pressures. Moreover, the volumetric flux obtained from the generalized models is always lesser than that obtained from the classical models. Such generalized models are most appropriate to model the flow in applications involving large pressure gradients, like enhanced oil recovery and carbon sequestration.