The effect of capillary forces on immiscible two-phase flow in heterogeneous porous media

Duijn, C.J. van; Molenaar, J.; Neef, Matthias

doi:10.1007/bf00615335

Cited by 143 publications

(76 citation statements)

References 16 publications

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“…Therefore, a crucial question is the existence of traveling waves, i.e., waves with a constant wave profile. In the case τ = 0, it is well known that traveling waves have a monotonous profile [19,20]. In the dynamic case, the existence and behavior of traveling wave solutions was studied in a recent paper ( [21], see also [3]).…”

Section: Discussion Of the Behaviormentioning

confidence: 98%

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Dynamic capillary effects in heterogeneous porous media

2007

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In standard multi-phase flow models on porous media, a capillary pressure saturation relationship developed under static conditions is assumed. Recent experiments have shown that this static relationship cannot explain dynamic effects as seen for example in outflow experiments. In this paper, we use a static capillary pressure model and a dynamic capillary pressure model based on the concept of Hassanizadeh and Gray and examine the behavior with respect to material interfaces. We introduce a new numerical scheme for the one-dimensional case using a Lagrange multiplier approach and develop a suitable interface condition. The behavior at the interface is discussed and verified by various numerical simulations.

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Section: Discussion Of the Behaviormentioning

confidence: 98%

“…The continuity of the capillary pressure at the interface is derived using a regularization technique, see [19].…”

Section: Materials Interfacesmentioning

confidence: 99%

Dynamic capillary effects in heterogeneous porous media

2007

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“…The partial differential equation (3) has to be completed with suitable initial conditions for t = 0 and boundary conditions on ∂Ω.…”

Section: Porous Media: Two-phase Flow With Entry Pressurementioning

confidence: 99%

Semismooth Newton methods for variational problems with inequality constraints

Hager

Wohlmuth

2010

GAMM-Mitteilungen

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Inequality constraints occur in many different fields of application, e.g., in structural mechanics, flow processes in porous media or mathematical finance. In this paper, we make use of the mathematical structure of these conditions in order to obtain an abstract computational framework for problems with inequality conditions. The constraints are enforced locally by means of Lagrange multipliers which are defined with respect to dual basis functions. The reformulation of the inequality conditions in terms of nonlinear complementarity functions leads to a system of semismooth nonlinear equations that is solved by a generalized version of Newton's method for semismooth problems. By this, both nonlinearities in the pde model and inequality constraints are treated within a single Newton iteration which converges locally superlinear. The scheme can efficiently be implemented in terms of an active set strategy with local static condensation of the non-essential variables. Numerical examples from different fields of application illustrate the generality and the robustness of the method.

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“…One of the other current challenging questions in the modeling of two-phase flow processes is the approximation of the porous media interfaces [10,18,29]. The heterogeneous material leads to a possibly discontinuous saturation solution at porous media interfaces.…”

Section: Motivationmentioning

confidence: 99%

Variational inequalities for modeling flow in heterogeneous porous media with entry pressure

2009

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One of the driving forces in porous media flow is the capillary pressure. In standard models, it is given depending on the saturation. However, recent experiments have shown disagreement between measurements and numerical solutions using such simple models. Hence, we consider in this paper two extensions to standard capillary pressure relationships. Firstly, to correct the nonphysical behavior, we use a recently established saturation-dependent retardation term. Secondly, in the case of heterogeneous porous media, we apply a model with a capillary threshold pressure that controls the penetration process. Mathematically, we rewrite this model as inequality constraint at the interfaces, which allows discontinuities in the saturation and pressure. For the standard model, often finite-volume schemes resulting in a nonlinear system for the saturation are applied. To handle the enhanced model at the interfaces correctly, we apply a mortar discretization method on nonmatching meshes. Introducing the flux as a new variable allows us to solve the inequality constraint efficiently. This method can be applied to both the standard and the enhanced capillary model. As nonlinear solver, we use an active set strategy combined with a Newton method. Several numerical examples demonstrate This work was supported in part by IRTG NUPUS. the efficiency and flexibility of the new algorithm in 2D and 3D and show the influence of the retardation term.Keywords Porous media · Entry pressure · Variational inequality · Mortar · Active set strategy MotivationFlow processes in porous media involving two immiscible fluids need to be understood and predicted when dealing with subsurface hydrosystems or industrial applications. For example, in the unsaturated zone, the spatial distribution of the water and air phase, as well as their fluxes, serves as a basis for modeling transport of contaminants, such as pesticides or heavy metals (e.g., [8]). As examples for industrial applications in two-phase flow, the movement of fluids through a filter or the infiltration of ink into paper (see [24]) can be considered. All these applications have in common that the porous media structure is, in general, highly heterogeneous. The challenges are to combine the complex multiphase flow processes with the heterogeneity distribution of the porous media properties.The physical-mathematical model underlying simulations of two-phase flow on the Darcy scale usually requires a constitutive relationship between (wetting phase) saturation S w and the capillary pressure p c . Traditionally, one assumes that this relationship is determined under quasistatic or steady-state conditions but can also be applied to any transient flow processes fulfilling the Reynolds number criterion. However, recently, some works have questioned this assumption (see, e.g., [25]). The authors were able to improve numerical simulation results by applying a model

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The effect of capillary forces on immiscible two-phase flow in heterogeneous porous media

Cited by 143 publications

References 16 publications

Dynamic capillary effects in heterogeneous porous media

Dynamic capillary effects in heterogeneous porous media

Semismooth Newton methods for variational problems with inequality constraints

Variational inequalities for modeling flow in heterogeneous porous media with entry pressure

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