Two-Phase Flow in Fractured Media – Homogenized Model with Mixing and Upscaling by a Stream-Configuration Method

The paper deals with the displacement process of two fluids, immiscible at the molecular scale, through a porous medium. The macroscale mixing between the phases is caused by the medium heterogeneity determined by a fracture network whose scale is much greater than the pore size, but much smaller than the reservoir size. At the heterogeneity scale the flow is described by a two-phase flow model with a generalized Darcy law, classical phase permeabilities and capillary pressure. The macroscale flow model is obtained by the two-scale asymptotic homogenization method. To capture the effect of dynamic mixing, the first order model is derived. <br>The mixing is described by the dispersion effect and the convective velocity renormalization. The dispersion tensor, the effective phase permeabilities and the velocity renormalization are defined through the cell problems as the functions of saturation, viscosity ratio and flow velocity. <br>To solve the cell problems we have developed the two-phase version of the stream-configuration method proposed earlier for single-phase flow. The method is based on the following features of a system of fine fractures: a) the limit flow is locally one-dimensional; b) the saturation and heterogeneity can be factorized, so the cell problem becomes independent of saturation. At the same time, the limit solution is shown to be non-unique due to a loss of information about the stream configuration geometry in the nodes of fracture intersection. The regularization procedure is developed which proposes additional conditions describing the type of stream configuration in each node. The types of configuration are determined from the principle of local energy minimum or entropy maximum. For a periodic regular fracture network, the method allows obtaining analytical solution for the dispersion tensor. For disordered networks, the method reduces the cell problem to an algebraic system of a rank equal to the number of nodes in a unite cell. A version of method is developed, based on a combination between the stream-configuration and the bordering techniques. <br>A singular dispersion regime is revealed in which the dispersion tensor becomes unbounded due to arising of stagnant zones in several fracture segments and fluid trapping. The data on comparison of our results with those obtained using other methods are illustrated. <br>The industrial application of this research consists of a new numerical algorithm of upscaling two-phase flow in fractured media, which is much faster than other methods. <br>The research is supported by the Schlumberger Technology Center in Abingdon.<br>