Haut PDF Flux calculation at the interface between two rock types for two-phase flow in porous media

Flux calculation at the interface between two rock types for two-phase flow in porous media

Flux calculation at the interface between two rock types for two-phase flow in porous media

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Numerical simulation of two-phase inertial flow in heterogeneous porous media

Numerical simulation of two-phase inertial flow in heterogeneous porous media

Fig. 9 Evolution of the saturation fields at three given times: a t = 200 s, b t = 400 s, c t = 600 s for two-dimensional flow in a two-layer medium. The Darcy–Forchheimer model is used with imposed pressure at the inlet, Re = 6.3 × 10 −3 Stratified Case: Saturation fields obtained for a low Reynolds number (Re = 6.3 × 10 −3 ) are first presented (Fig. 9 ). Pressures imposed at the inlet and outlet faces are respectively 1 .05×10 5 and 10 5 Pa. In this particular case of low pressure gradient in the medium, contrasts of the capillary pressure and capillary pressure gradient in the two layers lead to a perturbed saturation profile at the interface between these two layers. This effect was reproduced with finer grids and smaller time steps highlighting the signature of a physical mechanism. In fact, this can be explained by a transverse capillary suction from the more permeable η-region featuring low capillary effects towards the ω-region where capillary effects are ten times larger. Since the longitudinal flow rate is small, the saturation profile is strongly affected by this capillary cross flow. To be convinced of that, simulations with smaller capillary pressure contrasts were performed leading to smooth saturation fronts.
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Immiscible two-phase Darcy flow model accounting for vanishing and discontinuous capillary pressures: application to the flow in fractured porous media

Immiscible two-phase Darcy flow model accounting for vanishing and discontinuous capillary pressures: application to the flow in fractured porous media

In several recent works [11], [5], the Vertex Approx- imate Gradient (VAG) discretization, employing phase pressures formulation, was applied to model two-phase Darcy flows in heterogeneous porous media. In the con- text of vertex-centered schemes the phase pressures for- mulation allows to capture the saturation jump condi- tion at the interface between different rocktypes with- out introducing any additional unknowns at these inter- faces. It is, however, limited to strictly increasing capil- lary pressure curves and lacks robustness compared to pressure-saturation formulations. In this article we ex- tend the scheme introduced in [5] to the case of general increasing capillary pressure curves.
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Modeling two-phase flow of immiscible fluids in porous media: Buckley-Leverett theory with explicit coupling terms

Modeling two-phase flow of immiscible fluids in porous media: Buckley-Leverett theory with explicit coupling terms

model can be further extended to account for inertial effects by including additional drag terms [ 35 ], but we limit our analysis to creeping flow in this paper. In fact, in such cases, the velocity-dependent terms are not compatible with the Buckley-Leverett theory. The derivation of these models requires several important assumptions. One of these is that the interface between the immiscible fluids remains locally quasistatic, i.e., that the flow at the pore-scale relaxes quickly compared to characteristic time scales of the macroscale process. Another important assumption is that the capillary and Bond numbers, which respectively compare the viscous and gravity effects to surface tension, are much smaller than unity. Alternative models have been proposed to account for dynamic effects (see, for example, [ 36 , 37 ] for the use of pseudofunctions, [ 11 , 13 , 14 ] for other forms of laws accounting for dynamic effects induced by heterogeneities, multizones, or [ 10 ] for the use of the theory of irreversible thermodynamics). However, it is probable that less restrictive assumptions in the upscaling may still yield equations similar to Eqs. ( 2a ) and ( 2b ) for momentum transport, with the same effective parameters but capturing additional physical effects. Further, the expression in Eqs. ( 2 ) is used, in a variety of different forms, in engineering applications, where it is successful in describing many different systems [ 38 ]. We therefore base our model on the system of equations ( 1 ), ( 2a ), and ( 2b ), with simplifications that are described in the next section.
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A hybrid-dimensional compositional two-phase flow model in fractured porous media with phase transitions and Fickian diffusion

A hybrid-dimensional compositional two-phase flow model in fractured porous media with phase transitions and Fickian diffusion

. Then, these models write the half two-phase Darcy flux continuity equations on both sides of the mf interface using a TPFA of the half fluxes on the fracture side. It leads to the so-called mf nonlinear hybrid-dimensional models [34, 35, 25, 26, 36, 37] according to the terminology used in [26]. The main objective of this paper is to extend these types of mf nonlinear hybrid- dimensional models to compositional two-phase Darcy flow accounting for phase transitions and Fickian diffusion. The transmission conditions are designed to be consistent with the physical processes at mf interfaces. They account in particular for the saturation jump in- duced by the different rock types, for the Fickian diffusion in the fracture width, as well as for the thermodynamical equilibrium. They are based on flux continuity equations for each component using a TPFA of the fluxes in the fracture width. The saturation jumps at mf interfaces are captured for general capillary pressure curves thanks to a parametrization of the matrix and fracture capillary pressure graphs as introduced in [26, 23]. The thermo- dynamical equilibrium is formulated at mf interfaces using complementary constraints and taking into account the saturation jumps.
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A robust VAG scheme for a two-phase flow problem in heterogeneous porous media

A robust VAG scheme for a two-phase flow problem in heterogeneous porous media

assigned to each cell k ∈ M. We denote by χ s = {rt k , k ∈ M s } the set of rock types surrounding the node s ∈ V, and we set χ k = {rt k } for all k ∈ M. The choice of the primary variables follows the variable switching strategy introduced in [3]. We use the pressure of the non-wetting phase as the first primary variable for all d.o.f.; then for the cells and the nodal d.o.f. associated with a single rock type the second primary unknown is the saturation, while for the nodes s located at rock type interfaces we invoke the variable switching based on a parametrization of e P c,rt , rt ∈ χ s .
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The gradient flow structure for incompressible immiscible two-phase flows in porous media

The gradient flow structure for incompressible immiscible two-phase flows in porous media

In this note, we will focus on the model governing the motion of an incompressible immiscible two-phase flow in a possibly heterogeneous porous medium, that will appear in the sequel as (3) and (11)–(13). This model is relevant for instance for describing the flow of oil and water, whence the subscripts o and w appearing in the sequel of this note, within a rock that is possibly made of several rock-types. Our goal is to show that, at least formally, this model can be reinterpreted as the gradient flow of some singular energy. This will motivate the use of structure- preserving numerical methods inspired from [9] to this model in the future.
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Coupling of a two phase gas liquid 3D Darcy flow in fractured porous media with a 1D free gas flow

Coupling of a two phase gas liquid 3D Darcy flow in fractured porous media with a 1D free gas flow

1 Introduction Flow and transport processes in domains composed of a porous medium and an adjacent free- flow region appear in a wide range of industrial and environmental applications. This is in particular the case for radioactive waste deep geological repositories where such models must be used to predict the mass and energy exchanges occuring at the interface between the repos- itory and the ventilation excavated galleries. Typically, in this example, the porous medium initially saturated with the liquid phase is dried by suction in the neighbourhood of the inter- face. To model such physical processes, one needs to account in the porous medium for the flow of the liquid and gas phases including the vaporization of the water component in the gas phase and the dissolution of the gaseous component in the liquid phase. In the gallery, a single phase gas free flow can be considered assuming that the liquid phase is instantanneously vaporized at the interface. This single phase gas free flow has to be compositional to account for the change
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One-phase and Two-phase Flow In Highly Permeable Porous Media

One-phase and Two-phase Flow In Highly Permeable Porous Media

(12) where hϕi stands for the spatial averaging of a variable ϕ while ϕ stands for the time average, may lead to different macro-scale models. The fundamental reason is due to the fact that each up- scaling or averaging step introduces approximations which do not in general commute. Scheme I, favored by [23, 24] among others, involves a first spatial averaging. Assuming that the closed macro-scale equations have the form of a generalized Forchheimer equation, it is subsequently time averaged. Since it was found that the assumption in step 1 is difficult to justify theoretically, most researchers follow scheme II. In this case, it is assumed that the Navier-Stokes equations may be time aver- aged and the resulting equations are subsequently spatially aver- aged [22, 25–27]). The result is in general some sort of effective RANS macro-scale model. It must be emphasized that, in the two cases, it is difficult to justify the possibility of a decoupled closure for both averages. A complete validation through exper- iments or numerical modeling is still an open problem. The need for such models has not also been fully assessed. Interestingly, it has been observed using direct numerical simulations that turbu- lence generated in a fluid layer or at the interface between fluid and porous layers may penetrate a distance of several unit cells into the porous domain and that in this area a macro-scale turbu- lence model is needed [28].
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Upscaling multi-component two-phase flow in porous media with partitioning coefficient

Upscaling multi-component two-phase flow in porous media with partitioning coefficient

 the closure problems allow to map the concentration devia- tions onto macro-scale concentrations and gradients (similar relations exist for the velocity deviations). The closure problems are time-dependent because of the evolution of the interface (term involving w bg and A bg ðx, tÞ), and, also, because of the accumulation terms. Macro-scale equations and pore-scale closure problems are fully coupled, which in fact leads to memory (history) effects. Is it possible to simplify these closure problems in order to decouple macro and pore-scale problems? Considering that the diffusion term near the interface is dominant versus the flux proportional to n bg:ðv b ÿw bg Þ and that the mass fraction field relaxes faster than the evolution of the interface, a first possibility could be to discard all terms involving n bg:ðv b ÿw bgÞ and the accumulation term in the closure problems. The coupling between the macro and pore-scale equations remains through the evolution of the interface A bg ðx, tÞ. Therefore, even after having removed the n bg:ðv b ÿw bg Þ and the accumula- tion terms, the effective properties are associated to a specific
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Vertex Approximate Gradient Scheme for Hybrid Dimensional Two-Phase Darcy Flows in Fractured Porous Media

Vertex Approximate Gradient Scheme for Hybrid Dimensional Two-Phase Darcy Flows in Fractured Porous Media

Abstract This paper presents the Vertex Approximate Gradient (VAG) discretiza- tion of a two-phase Darcy flow in discrete fracture networks (DFN) taking into account the mass exchange between the matrix and the fracture. We consider the asymptotic model for which the fractures are represented as interfaces of codi- mension one immersed in the matrix domain with continuous pressures at the ma- trix fracture interface. Compared with Control Volume Finite Element (CVFE) ap- proaches, the VAG scheme has the advantage to avoid the mixing of the fracture and matrix rocktypes at the interfaces between the matrix and the fractures, while keeping the low cost of a nodal discretization on unstructured meshes. The conver- gence of the scheme is proved under the assumption that the relative permeabilities are bounded from below by a strictly positive constant but cover the case of discon- tinuous capillary pressures. The efficiency of our approach compared with CVFE discretizations is shown on a 3D fracture network with very low matrix permeabil- ity.
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Modeling two-phase flow of immiscible fluids in porous media: Buckley-Leverett theory with explicit coupling terms

Modeling two-phase flow of immiscible fluids in porous media: Buckley-Leverett theory with explicit coupling terms

model can be further extended to account for inertial effects by including additional drag terms [ 35 ], but we limit our analysis to creeping flow in this paper. In fact, in such cases, the velocity-dependent terms are not compatible with the Buckley-Leverett theory. The derivation of these models requires several important assumptions. One of these is that the interface between the immiscible fluids remains locally quasistatic, i.e., that the flow at the pore-scale relaxes quickly compared to characteristic time scales of the macroscale process. Another important assumption is that the capillary and Bond numbers, which respectively compare the viscous and gravity effects to surface tension, are much smaller than unity. Alternative models have been proposed to account for dynamic effects (see, for example, [ 36 , 37 ] for the use of pseudofunctions, [ 11 , 13 , 14 ] for other forms of laws accounting for dynamic effects induced by heterogeneities, multizones, or [ 10 ] for the use of the theory of irreversible thermodynamics). However, it is probable that less restrictive assumptions in the upscaling may still yield equations similar to Eqs. ( 2a ) and ( 2b ) for momentum transport, with the same effective parameters but capturing additional physical effects. Further, the expression in Eqs. ( 2 ) is used, in a variety of different forms, in engineering applications, where it is successful in describing many different systems [ 38 ]. We therefore base our model on the system of equations ( 1 ), ( 2a ), and ( 2b ), with simplifications that are described in the next section.
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Vertex Approximate Gradient Discretization preserving positivity for two-phase Darcy flows in heterogeneous porous media

Vertex Approximate Gradient Discretization preserving positivity for two-phase Darcy flows in heterogeneous porous media

Abstract In this article, a new nodal discretization is proposed for two-phase Darcy flows in heterogeneous porous media. The scheme combines the Vertex Approximate Gradient (VAG) scheme for the approximation of the gradient fluxes with an Hybrid Upwind (HU) approximation of the mobility terms in the saturation equation. The discretization in space incorporates naturally nodal interface degrees of freedom (d.o.f.) allowing to capture the transmission conditions at the interface between different rock types for general capillary pressure curves. It is shown to guarantee the physical bounds for the saturation unknowns as well as a nonnegative lower bound on the capillary energy flux term. Numerical experiments on several test cases exhibit that the scheme is more robust compared with previous approaches allowing the simulation of 3D large Discrete Fracture Matrix (DFM) models.
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On the effects of discontinuous capillarities for immiscible two-phase flows in porous media made of several rock-types

On the effects of discontinuous capillarities for immiscible two-phase flows in porous media made of several rock-types

i (s) = λ i (s)π ′ i (s), the ∂ x ϕ i (u) belongs to L ∞ (Ω i × R + ), then ϕ i (u) admits a strong trace on {x = 0}. Thanks to assumptions (H2)-(H3), ϕ −1 i is continuous, then u admits also strong traces. As already stressed, u represents the saturation in oil of the fluid, then it has naturally to stay bounded between 0 and 1, as required in the first point of Defi- nition 2.1 . The denomination bounded-flux solution clearly comes from the second point. The connection of the capillary pressures 5 is required by the third point. The equations 1 , the connection of the fluxes 4 and the respect of the initial value are required in a weak sense by the formulation 8 .
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Vertex Approximate Gradient Scheme for Hybrid Dimensional Two-Phase Darcy Flows in Fractured Porous Media

Vertex Approximate Gradient Scheme for Hybrid Dimensional Two-Phase Darcy Flows in Fractured Porous Media

The discretization of the hybrid dimensional Darcy flow model with continuous pressures has been the object of several works. In [24] a cell-centred Finite Volume scheme using a Two Point Flux Ap- proximation (TPFA) is proposed assuming the orthogonality of the mesh and isotropic permeability fields. Cell-centred Finite Volume schemes can be extended to general meshes and anisotropic perme- ability fields using MultiPoint Flux Approximations (MPFA) following the ideas introduced in [31] for discontinuous pressure models. Nevertheless, MPFA schemes can lack robustness on distorted meshes and large anisotropies due to the non symmetry of the discretization. They are also very expensive compared with nodal discretizations on tetrahedral meshes. In [1], a Mixed Finite Element (MFE) method is proposed for single Darcy flows. It is extended to two-phase flows in [21] in an IMPES framework using a Mixed Hybrid Finite Element (MHFE) discretization for the pressure equation and a Discontinuous Galerkin discretization of the saturation equation. These approaches are adapted to general meshes and anisotropy but require as many degrees of freedom as faces. Control Volume Finite Element Methods (CVFE) [29], [28] have the advantage to use only nodal unknowns leading to much fewer degrees of freedom than MPFA and MHFE schemes on tetrahedral meshes. On the other hand, at the matrix fracture interfaces, the control volumes have the drawback to be shared between the matrix and the fractures. It results that a strong refinement of the mesh is needed at these interfaces in the case of large contrasts between the permeabilities of the matrix and of the fractures.
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Finite volume approximation for an immiscible two-phase flow in porous media with discontinuous capillary pressure

Finite volume approximation for an immiscible two-phase flow in porous media with discontinuous capillary pressure

We consider an immiscible incompressible two-phase flow in a porous medium composed of two different rocks so that the capillary pressure field is discontinuous at the interface between the rocks. This leads us to apply a concept of multi-valued phase pressures and a notion of weak solution for the flow which have been introduced in [Canc`es & Pierre, SIAM J. Math. Anal, 44(2):966–992, 2012]. We discretize the prob- lem by means of a numerical algorithm which reduces to a standard finite volume scheme in each rock and prove the convergence of the approximate solution to a weak solution of the two-phase flow problem. The numerical experiments show in particular that this scheme permits to reproduce the oil trapping phenomenon. Keywords : Finite volume schemes, degenerate parabolic, two-phase flow in porous media, discontinu- ous capillarity
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Two-phase non-Darcy flow in heterogeneous porous media: A numerical investigation

Two-phase non-Darcy flow in heterogeneous porous media: A numerical investigation

For 1D flow in a homogeneous porous medium, a validation is performed by comparing numerical results of the saturation front kinetics with a semi-analytical solution inspired from the “Buckley-Leverett” model extended to take into account inertia. The influence of inertial effects on the saturation profiles and therefore on the breakthrough curves for homogeneous media is analysed for different Reynolds numbers, thus emphasizing the necessity of taking into account this additional energy loss when necessary. For 1D heterogeneous configurations, a thorough analysis of the saturation fronts as well as the saturation jumps at the interface between two media of contrasted properties highlights the influence of inertial effects for different Reynolds and capillary numbers.
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Hybrid-dimensional modelling of two-phase flow through fractured porous media with enhanced matrix fracture transmission conditions

Hybrid-dimensional modelling of two-phase flow through fractured porous media with enhanced matrix fracture transmission conditions

For two-phase Darcy flow, the TPFA discretization is used in [25] with elimination of the mf inter- face pressures when computing the Darcy flux transmissivities. In [24], the two-phase flow equations are solved in an IMPES framework, using a Mixed Hybrid Finite Element (MHFE) discretization for the pressure equation and a Discontinuous Galerkin discretization of the saturation equation. Either a zero flux or the pressure continuity are assumed at mf interfaces. The paper also contains a review on the most common numerical approaches, when dealing with discrete fractures. The Hybrid Finite Volume discretization (HFV, see [15, 11]) is extended in [22] to two-phase Darcy flow in fractured media with continuous pressure at mf interfaces. These approaches are adapted to general meshes and anisotropy but require as many degrees of freedom as faces. An early paper to use a Control Volume Finite Element method (CVFE) for the discretization of hybrid-dimensional two-phase flow is [8] assuming pressure continuity at mf interfaces. Still for continuous pressure models, a CVFE scheme is proposed in [31] that uses reconstruction operators for the saturations that depend on the rock characteristic capillary pressure curves. In this way, the saturation jumps (due to discontinuous capillary pressure) at the material interfaces are respected. A similar approach can be found in [30] combined with a switch of variable technique to account for highly contrasted mf capillary pressures. However, the rigid choice of the control volumes (that are the dual cells) leads to the need of small matrix cells at the DFN neighbourhood, in order not to enlarge the drains artificially. In [9, 10], the VAG scheme is used to discretize the continuous pressure model, which is very flexible in the distribution of control volumes and hence circumvents this problem.
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Pressure drop and average void fraction measurements for two-phase flow through highly permeable porous media

Pressure drop and average void fraction measurements for two-phase flow through highly permeable porous media

through the spillway ( Fig. 5 b). The swollen water level then decreases back to the upper bound of the debris bed, and a part of the water initially present in the test section is recovered in a con- tainer connected to the spillway ( Fig. 5 c). The volume of recovered water corresponds to the volume of air in the test section. It is deduced from its mass m. During this phase of the test, the fre- quency f provided by the capacitance probe is measured by a numerical oscilloscope. Its value is fluctuating together with the two-phase air–water flow. The value is averaged over a time long enough such that the mean value remains constant. A part of the air volume stays under the grid which is supporting the debris bed. To determine this volume, the air flow rate is suddenly stopped ( Fig. 5 d). Then, there is a stratification with two layers – air and water. The grid is fine enough such that air cannot exit by the top through the bed. Finally, the void fraction in the bed can be deduced:
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Derivation of a macroscopic model for two-phase non-Darcy flow in homogeneous porous media using volume averaging

Derivation of a macroscopic model for two-phase non-Darcy flow in homogeneous porous media using volume averaging

Science Arts & Métiers (SAM) is an open access repository that collects the work of Arts et Métiers Institute of Technology researchers and makes it freely available over the web where possible. This is an author-deposited version published in: https://sam.ensam.eu

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