Two-phase porous media flows with dynamic capillary effects and hysteresis: Uniqueness of weak solutions

Cao, Xiulei; Pop, Sorin

doi:10.1016/j.camwa.2015.02.009

Cited by 33 publications

(38 citation statements)

References 17 publications

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“…In general, one cannot expect that solutions exist in a classical sense. We refer to [5,6,12,13,14,25,28,27,41,42,43] for results concerning the existence and uniqueness of weak solutions for hysteresis models, dynamic capillarity models, or for models including both effects. In particular we refer to [41,42,13] where, as suggested in [3,4], (2.12) is used to express ∂ t S as a function of S and p. We rely on the same idea for the TW analysis below.…”

Section: Scaling and Assumptionsmentioning

confidence: 99%

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Travelling wave solutions for the Richards equation incorporating non-equilibrium effects in the capillarity pressure

Duijn

Mitra

Pop

2018

Nonlinear Analysis: Real World Applications

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The Richards equation is a mathematical model for the unsaturated flow through porous media. This paper considers an extension of the Richards equation, where non-equilibrium effects like hysteresis and dynamic capillarity are incorporated in the relationship that relates the water pressure and the saturation. The focus is on travelling wave solutions, for which the existence is investigated first for the model including hysteresis and subsequently for model including dynamic capillarity effect. In particular, such solutions may have non monotonic profiles, which are ruled out when considering standard, equilibrium type models, but have been observed experimentally. The paper ends with numerical experiments confirming the theoretical results. In this sense the original system of partial differential equations is solved by means of an implicit scheme. This relies on an equivalent, mixed formulation of the system. For solving the resulting nonlinear, time-discrete problems, a linear iterative scheme is proposed.

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Section: Scaling and Assumptionsmentioning

confidence: 99%

“…We define the vector-valued function 13) and denote its components by F S and F u respectively. A direct calculation gives…”

Section: Properties For Fixed ε >mentioning

confidence: 99%

Travelling wave solutions for the Richards equation incorporating non-equilibrium effects in the capillarity pressure

Duijn

Mitra

Pop

2018

Nonlinear Analysis: Real World Applications

Self Cite

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“…A similar choice of the primary variables is made in [6,24]. We refer to [24][25][26] for existence and uniqueness results for System (5).…”

Section: Primary Variablesmentioning

confidence: 99%

Analysis of a linearization scheme for an interior penalty discontinuous Galerkin method for two‐phase flow in porous media with dynamic capillarity effects

Karpinski

Pop

Radu

2017

Numerical Meth Engineering

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SUMMARYAbstract We present a linearization scheme for an interior penalty discontinuous Galerkin method for two-phase porous media flow model which includes dynamic effects in the capillary pressure. The fluids are assumed immiscible and incompressible, and the solid matrix non-deformable. The physical laws are approximated in their original form, without using the global or complementary pressures. The linearization scheme does not require any regularization step. Furthermore, in contrast with Newton or Picard methods, there is no computation of derivatives involved. We prove rigorously that the scheme is robust and linearly convergent. Numerical results, including examples in realistic, heterogeneous media are presented to sustain the applicability of the scheme.

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“…The system (1) -(5) is derived from (10) - (12) by proper non-dimensionalization [43]. The gravity term is not considered for the mathematical analysis but is included in the numerical section.…”

Section: Introductionmentioning

confidence: 99%

“…In [35,48], the authors consider numerical algorithms for unsaturated flow through highly heterogeneous media with dynamic effect. For the full two-phase flow model, the existence and uniqueness of the weak solutions are proved in [39,12], with the assumption that the equations are non-degenerate (i.e. all non-linearities are bounded away from 0 or +∞).…”

Section: Introductionmentioning

confidence: 99%

Error estimates for a mixed finite element discretization of a two-phase porous media flow model with dynamic capillarity

Cao

Mitra

2019

Journal of Computational and Applied Mathematics

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Two-phase porous media flows with dynamic capillary effects and hysteresis: Uniqueness of weak solutions

Cited by 33 publications

References 17 publications

Travelling wave solutions for the Richards equation incorporating non-equilibrium effects in the capillarity pressure

Travelling wave solutions for the Richards equation incorporating non-equilibrium effects in the capillarity pressure

Analysis of a linearization scheme for an interior penalty discontinuous Galerkin method for two‐phase flow in porous media with dynamic capillarity effects

Error estimates for a mixed finite element discretization of a two-phase porous media flow model with dynamic capillarity

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