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An asymptotic solution for two‐phase flow in the presence of capillary forces
Author(s) -
Vasco D. W.
Publication year - 2004
Publication title -
water resources research
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.863
H-Index - 217
eISSN - 1944-7973
pISSN - 0043-1397
DOI - 10.1029/2003wr002587
Subject(s) - saturation (graph theory) , amplitude , flow (mathematics) , mechanics , computation , capillary action , trajectory , mathematics , phase (matter) , mathematical analysis , physics , thermodynamics , algorithm , quantum mechanics , combinatorics , astronomy
Under the assumption of smoothly varying background properties I derive an asymptotic solution for two‐phase flow. This formulation partitions the modeling of two‐phase flow into two subproblems: an arrival time calculation and a saturation amplitude computation. The asymptotic solution itself is defined along a trajectory though the model. If gravitational forces are not important and the flow field is independent of changes in the background saturation, the trajectory may be identified with a streamline, and the asymptotic approach provides a mathematical basis for streamline simulation. The evolution of the saturation amplitude is governed by a generalization of Burgers' equation, defined along the trajectory. If the capillary properties are uniform and gravitational forces are negligible, one can derive self‐similar solutions for the three limiting behaviors of a two‐phase front. Asymptotic solutions were found to deviate by less than 5% from saturations calculated using a numerical simulator. The numerical results indicate that changes in capillary properties can introduce significant variations in the arrival time of a specific aqueous fraction. A more robust definition of arrival time appears to be the time associated with the peak rate of change in aqueous phase fraction, the greatest slope of the breakthrough curve.