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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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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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. 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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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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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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(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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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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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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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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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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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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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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