# A Relaxation Projection Analytical–Numerical Approach in Hysteretic Two-Phase Flows in Porous Media

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## Abstract

Hysteresis phenomenon plays an important role in fluid flow through porous media and exhibits convoluted behavior that are often poorly understood and that is lacking of rigorous mathematical analysis. We propose a twofold approach, by analysis and computing to deal with hysteretic, two-phase flows in porous media. First, we introduce a new analytical projection method for construction of the wave sequence in the Riemann problem for the system of equations for a prototype two-phase flow model via relaxation. Second, a new computational method is formally developed to corroborate our analysis along with a representative set of numerical experiments to improve the understanding of the fundamental relaxation modeling of hysteresis for two-phase flows. Using the projection method we show the existence by analytical construction of the solution. The proposed computational method is based on combining locally conservative hybrid finite element method and finite volume discretizations within an operator splitting formulation to address effectively the stiff relaxation hysteretic system modeling fundamental two-phase flows in porous media.

## Keywords

Hyperbolic conservation laws Riemann problem Projection method Relaxation Hysteretic two-phase flow Finite volume/element## Mathematics Subject Classification

76S05 76M10 76M20## Notes

## Supplementary material

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