Computational modeling technique for numerical simulation of immiscible two-phase flow problems involving flow and transport phenomena in porous media with hysteresis

Abreu, Eduardo; Lambert, Wanderson

doi:10.1063/1.4711166

“…In [11], examples of how hysteresis influences hyperbolic solutions in mechanics are found. The role of relative permeability hysteresis, which is not addressed in this study, in determining the hyperbolic solutions is examined in [1,4,26,27]. However, the relation between p and S is assumed to be a linear one in these articles.…”

Section: Introductionmentioning

confidence: 99%

Capillary hysteresis and gravity segregation in two phase flow through porous media

Mitra

¹

,

Duijn

²

2021

Comput Geosci

0

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We study the gravity driven flow of two fluid phases in a one dimensional homogeneous porous column when history dependence of the pressure difference between the phases (capillary pressure) is taken into account. In the hyperbolic limit, solutions of such systems satisfy the Buckley-Leverett equation with a non-monotone flux function. However, solutions for the hysteretic case do not converge to the classical solutions in the hyperbolic limit in a wide range of situations. In particular, with Riemann data as initial condition, stationary shocks become possible in addition to classical components such as shocks, rarefaction waves and constant states. We derive an admissibility criterion for the stationary shocks and outline all admissible shocks. Depending on the capillary pressure functions, flux function and the Riemann data, two cases are identified a priori for which the solution consists of a stationary shock. In the first case, the shock remains at the point where the initial condition is discontinuous. In the second case, the solution is frozen in time in at least one semi-infinite half. The predictions are verified using numerical results.

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“…In [11], examples of how hysteresis influences hyperbolic solutions in mechanics are found. The role of relative permeability hysteresis, which is not addressed in this study, in determining the hyperbolic solutions is examined in [1,4,25,26]. However, the relation between p and S is assumed to be a linear one in these articles.…”

Section: Introductionmentioning

confidence: 99%

Capillary Hysteresis and Gravity Segregation in Two Phase Flow Through Porous Media

Mitra¹,

Duijn²

2020

Preprint

0

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We study the gravity driven flow of two fluid phases in a one dimensional homogeneous porous column when history dependence of the pressure difference between the phases (capillary pressure) is taken into account. In the hyperbolic limit, solutions of such systems satisfy the Buckley-Leverett equation with a non-monotone flux function. However, solutions for the hysteretic case do not converge to the classical solutions in the hyperbolic limit in a wide range of situations. In particular, with Riemann data as initial condition, stationary shocks become possible in addition to classical components such as shocks, rarefaction waves and constant states. We derive an admissibility criterion for the stationary shocks and outline all admissible shocks. Depending on the capillary pressure functions, flux function and the Riemann data, two cases are identified a priori for which the solution consists of a stationary shock. In the first case, the shock remains at the point where the initial condition is discontinuous. In the second case, the solution is frozen in time in at least one semi-infinite half. The predictions are verified using numerical results.

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“…Hysteresis may be the most important phenomenon in multiphase porous media Ćow that is lacking rigorous mathematical analysis (see, e.g., [27,12] and references therein). Hysteresis has been identiĄed for instance in physical phenomenas as such pseudo-elasticity, plasticity, superconductivity, ferromagnetism, biology, thin-Ąlm transistors, ferroelectricity, phase transitions, porous media Ąltration, and in several other physical phenomena.…”

Section: Considerações ąNaismentioning

confidence: 99%

“…The Ćuid phases are water and oil, indicated by the subscripts w and o, respectively, and they occupy the whole pore space (i.e., 𝑆 𝑤 + 𝑆 𝑜 = 1). An additional state variable, the hysteresis parameter Þ = Þ(x, 𝑡) is introduced to account for the dependence of the oil permeability 𝑘 𝑜 (𝑆 𝑤 , Þ) on the Ćow history [27,12]. Then, we write the governing system of equations of two-phase Ćow with hysteresis and no gravity, in the so-called phase formulation [50]; see also [2,7]:…”

Section: Considerações ąNaismentioning

confidence: 99%

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Implementação de um algoritmo multi-escala para sistemas de equações lineares de grande porte mal condicionados provenientes da discretização de problemas elípticos em dinâmica de fluidos em meios porosos

Ferraz¹

0

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The focus of this work is the numerical approximation of differential problems involving partial differential equations (PDEŠs) of elliptic nature, in the context of modelling and simulation in Ćuid dynamics in porous media. The dissertation aims to contribute with an implementation of a multiscale multigrid algorithm, recently introduced in the literature, designed for solving ill-conditioned large linear systems of equations derived from those classes of PDEŠs, typically associated with Poisson problems of pressure-velocity with boundary conditions typical of Ćow in porous media. The Poisson problem discussed here is identiĄed from the coupled convectiondiffusion transport system counterpart of PDEŠs, with dominated convection, and by a linearization by means the use of an operator splitting approach based on the physics of the underlying model. The methodology used for the discretization of the Poisson elliptic problem is mixed hybrid Ąnite elements. The numerical solution of the resulting linear system will be addressed by a multiscale multigrid preconditioned conjugate gradient (PCG) method. We combine both methodologies in order to capture the distinct wavelengths associated with the different wavelengths from the assosiated self-adjoint Poisson operator, strongly inĆuenced by the heterogeneity of the geological properties of the porous media, in particular to the absolute permeability, which in turn might exhibit very large Ćuctuations of orders of magnitude. Numerical experiments in applications of Ćuid dynamics problems in porous media are presented and discussed for a veriĄcation of the results obtained by direct numerical simulations with the multiscale multigrid algorithm under consideration.

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Computational modeling technique for numerical simulation of immiscible two-phase flow problems involving flow and transport phenomena in porous media with hysteresis

Cited by 4 publications

References 17 publications

Capillary hysteresis and gravity segregation in two phase flow through porous media

Capillary hysteresis and gravity segregation in two phase flow through porous media

Capillary Hysteresis and Gravity Segregation in Two Phase Flow Through Porous Media

Implementação de um algoritmo multi-escala para sistemas de equações lineares de grande porte mal condicionados provenientes da discretização de problemas elípticos em dinâmica de fluidos em meios porosos

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