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A mixed finite element–finite volume formulation of the black-oil model. (English) Zbl 0959.76039
The authors apply a hybrid finite element-finite volume technique to numerical study of the so-called black-oil model, where three compressible independent components (oil, gas and water) form three phases (liquid, vapor and water) presented in the reservoir. The Brezzi-Douglas-Marini elements of degree one are used for the calculation of Darcy velocity governed by the parabolic pressure equation, whereas the system of mass conservation laws is solved by a finite volume method. More precisely, first pressure and velocity are calculated by an implicit Euler scheme assuming that mass concentrations are known. Then, an explicit predictor-corrector Godunov-type scheme extended to triangular unstructured grid is used to solve hyperbolic mass conservation equations. To verify the method, the authors present numerical results for some practical problems (mixed saturated-unsaturated flows in one dimension, and a quarter of five spots in two dimensions).

76M10 Finite element methods applied to problems in fluid mechanics
76M12 Finite volume methods applied to problems in fluid mechanics
76T30 Three or more component flows
76S05 Flows in porous media; filtration; seepage
86A60 Geological problems
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