Capillary corrections to Buckley-Leverett flow
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Copyright 2015, Society of Petroleum Engineers. At the reservoir scale, multiphase fluid flow is well characterized by the Buckley-Leverett flow equations, neglecting capillarity. However, as we extend our studies in higher resolution using multiscale calculations, or evaluate tighter or higher contrast heterogeneous or fractured reservoirs, capillarity becomes increasingly important. To improve the understanding of these situations, we have extended the analytic solution of the Buckley-Leverett equations to include capillarity. Specifically we have solved the incompressible waterflood flow equations along a streamtube or streamline with arbitrary cross-section for a heterogeneous porous media with variable injection water rate, including capillarity. The methodology is an application of a singular perturbation expansion with matched asymptotic solutions. The outer solution is identical to the continuous portion of the Buckley-Leverett saturation profile while the inner solution is the steady state solution first noted by Terwilliger experimentally. The two solutions match at the Buckley-Leverett shock saturation and all solutions can be expressed in closed form. The result of this analytical solution is tested against high resolution flow simulation to verify its validity. This analysis is also applied to the calculation of capillary end effects in laboratory core floods, where the length scale of the saturation correction can be predicted. We also demonstrate, as the shape of the saturation profile near the front depends upon the capillary pressure function, that this analytical solution can used to interpret experimental data and derive and calibrate the capillary pressure function.
