A non-local two-phase flow model for immiscible displacement in highly heterogeneous porous media and its parametrization

In this paper, we present an upscaled model for horizontal immiscible displacement in highly heterogeneous media. This type of heterogeneity can be found, for instance, in fractured rock, which consists of two flow domains, a mobile fracture and a virtually immobile matrix. We derive an upscaled double-continuum model capable of predicting flow in mobile-immobile domains using homogenization theory. The model consists of a flow equation for the saturation of displacing fluid in the fracture domain, and a capillary flow equation for saturation in the matrix, which are coupled via a source term. By linearizing capillary counter current flow in the matrix domain, we combine this system of equations into a non-local single-equation model for the fracture saturation, which can be interpreted as a multi-rate mass-transfer (MRMT) model for immiscible displacement. We discuss this simplification and the parametrization of the upscaled model equation from local hydraulic parameters obtained from rock samples and from knowledge of the average flow properties of the fracture network. We demonstrate its performance for predicting two-phase flow by considering a single fracture with imbibition into a rectangular matrix domain. The upscaled model is parameterized directly from geometry and hydraulic parameters of matrix and fracture of the reference model, which means that no parameters need to be fitted. We compare the detailed and upscaled models in terms of breakthrough curves for the displaced fluid at a control plane within the medium. Both the detailed numerical simulations and the upscaled model show a preasymptotic ^{t - 1 / 2} scaling and a breakoff at the characteristic time scale for filling the matrix by counter current flow.