Complementary Constrains on Component based Multiphase Flow Problems, Should It Be Implemented Locally or Globally?

Abstract:

Multiphase flow problems are numerically difficult to solve, as it often contains nonlinear Phase transition phenomena A conventional technique is to introduce the complementarity constraints where fluid properties such as liquid saturations are confined within a physically reasonable range. Based on such constraints, the mathematical model can be reformulated into a system of nonlinear partial differential equations coupled with variational inequalities. They can be then numerically handled by optimization algorithms.

In this work, two different approaches utilizing the complementarity constraints based on persistent primary variables formulation^[4] are implemented and investigated. The first approach proposed by Marchand et.al^[1] is using “local complementary constraints”, i.e. coupling the constraints with the local constitutive equations. The second approach^[2],[3] , namely the “global complementary constrains”, applies the constraints globally with the mass conservation equation. We will discuss how these two approaches are applied to solve non-isothermal componential multiphase flow problem with the phase change phenomenon. Several benchmarks will be presented for investigating the overall numerical performance of different approaches. The advantages and disadvantages of different models will also be concluded.

References

[1] E.Marchand, T.Mueller and P.Knabner. Fully coupled generalized hybrid-mixed finite element approximation of two-phase two-component flow in porous media. Part I: formulation and properties of the mathematical model, Computational Geosciences 17(2): 431-442, (2013).

[2] A. Lauser, C. Hager, R. Helmig, B. Wohlmuth.  A new approach for phase transitions in miscible multi-phase flow in porous media. Water Resour., 34,(2011), 957–966.

[3] J. Jaffré, and A. Sboui. Henry’s Law and Gas Phase Disappearance. Transp. Porous Media. 82, (2010), 521–526.

[4] A. Bourgeat, M. Jurak and F. Smaï. Two-phase partially miscible flow and transport modeling in porous media : application to gas migration in a nuclear waste repository, Comp.Geosciences. (2009), Volume 13, Number 1, 29–42.