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Numerical and dimensional investigation of two-phase countercurrent imbibition in porous media. (English) Zbl 1343.76066
Summary: In this paper, we introduce a numerical solution of the problem of two-phase immiscible flow in porous media. In the first part of this work, we present the general conservation laws for multiphase flows in porous media as outlined in the literature for the sake of completion, where we emphasize the difficulties associated with these equations in their primitive form and the fact that they are, generally, unclosed. The second part concerns the 1D computation for dimensional and non-dimensional cases and a theoretical analysis of the problem under consideration. A time-scale based on the characteristic velocity is used to transform the macroscopic governing equations into a non-dimensional form. The resulting dimensionless governing equations involve some important dimensionless physical parameters such as Bond number $$Bo$$, capillary number $$Ca$$ and Darcy number $$Da$$. Numerical experiments on the Bond number effect is performed for two cases, gravity opposing and assisting. The theoretical analysis illustrates that for common formulations of the time-scale forces the coefficient $$Da^{1/2}/Ca$$ is equal to one. The formulation of dimensionless time based on a characteristic velocity allows the capillary and Darcy numbers to appear in the dimensionless governing equation, which leads to a wide range of scales and physical properties of fluids and rocks. The results indicate that the buoyancy effects due to gravity force take place depending on the location of the open boundary.

##### MSC:
 76S05 Flows in porous media; filtration; seepage 86A05 Hydrology, hydrography, oceanography
##### Keywords:
oil recovery; gravity effect; Galerkin method; conservation law
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