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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1 Introduction
**Two**-**phase** **Darcy** **flows** are widely used in many subsurface applications such as oil and gas recovery, basin modeling, geological storage, geothermal energy or hydrogeology. These models lead to Partial Differential Equations (PDEs) accounting for strongly coupled nonlinear processes typically involving viscous, buoyancy and capillary forces [6, 18, 39]. The high heterogeneity of natural porous media entails a large range of velocities and time scales in the transport of the **phase** saturations. Capillary driven flow dynamics can also occur at interfaces between different rock types where highly nonlinear transmission conditions take place. The abrupt change of the pore sizes at such interfaces induces spatially discontinuous capillary pressure curves at the **Darcy** scale triggering the discontinuity of the **phase** saturation and pressure solutions [44, 21, 16, 17, 11]. These discontinuities play a major role and should be accurately captured in many important processes such as capillary driven imbibition in oil recovery or capillary barrier effects in oil migration and gas storage. This is particularly enhanced in the case of Discrete Fracture Matrix (DFM) models which exhibit highly contrasted permeabilities and capillary pressure curves between the fracture network and the surrounding matrix domains [9, 35, 42, 38, 34, 12, 13, 2, 31, 14, 1, 3, 45, 4].

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Abstract. This paper concerns the discretization of multiphase **Darcy** **flows**, in the case of heteroge- neous anisotropic porous media and general 3D meshes used in practice to represent reservoir and basin geometries. An unconditionally coercive and symmetric vertex centred approach is introduced in this paper. This scheme extends the Vertex Approximate Gradient scheme (VAG), already introduced for single **phase** diffusive problems in [9], to multiphase **Darcy** **flows**. The convergence of the VAG scheme is proved for a simplified **two**-**phase** **Darcy** flow model, coupling an elliptic equation for the pressure and a linear hyperbolic equation for the saturation. The ability for the VAG scheme to efficiently deal with highly heterogeneous media and complex meshes is exhibited on immiscible and miscible **two** **phase** **Darcy** flow models.

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January 20, 2021
Abstract
This work deals with sequential implicit schemes for incompressible and immis- cible **two**-**phase** **Darcy** **flows** which are commonly used and well understood in the case of spatially homogeneous capillary pressure functions. To our knowledge, the stability of this type of splitting schemes solving sequentially a pressure equation followed by the saturation equation has not been investigated so far in the case of discontinuous capillary pressure curves at different rock type interfaces. It will be shown here to raise severe stability issues for which stabilization strategies are investigated in this work. To fix ideas, the spatial discretization is based on the Vertex Approximate Gradient (VAG) scheme accounting for unstructured polyhe- dral meshes combined with an Hybrid Upwinding (HU) of the transport term and an upwind positive approximation of the capillary and gravity fluxes. The sequen- tial implicit schemes are built from the total velocity formulation of the **two**-**phase** flow model and only differ in the way the conservative VAG total velocity fluxes are approximated. The stability, accuracy and computational cost of the sequen- tial implicit schemes studied in this work are tested on oil migration test cases in 1D, 2D and 3D basins with a large range of capillary pressure parameters for the drain and barrier rock types. It will be shown that usual splitting strategies fail to capture the right solutions for highly contrasted rock types and that it can be fixed by maintaining locally the pressure saturation coupling at different rock type interfaces in the definition of the conservative total velocity fluxes.

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Tel.: +33 4 92 07 62 32 roland.masson@unice.fr
Abstract
This article analyses the convergence of the Vertex Approximate Gradi- ent (VAG) scheme recently introduced in Eymard et al. 2012 for the discretization of multiphase **Darcy** **flows** on general polyhedral meshes. The convergence of the scheme to a weak solution is shown in the par- ticular case of an incompressible immiscible **two**-**phase** **Darcy** flow model with capillary diffusion using a global pressure formulation. A remarkable property in practice is that the convergence is proven whatever the dis- tribution of the volumes at the cell centers and at the vertices used in the control volume discretization of the saturation equation. The numerical experiments carried out for various families of 2D and 3D meshes confirm this result on a one-dimensional Buckley Leverett solution.

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ically arise.
The main advantage of this formulation is to use the natural set of unknowns for the hydrodynamical and thermodynamical laws and to extend to a large class of compositional **Darcy** flow models ranging from immiscibility to full miscibility (see [11]). On the other hand, its main drawbacks are an additional complexity to deal with sets of unknowns and equations depending on the set Q, and the use of a fixed point algorithm to compute the set of present phases Q at each point of the space time domain. The efficiency of this formulation has mainly been shown for reservoir simulation test cases with complex thermodynamics, **two** and tri **phase** **Darcy** **flows**, but with usually small capillary effects and the use of a reference pressure in the thermodynamical state laws rather than the **phase** pressures. In the next section it will be assessed and compared with the **two** other formulations on test cases with both strong or weak capillary effects.

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such that Γ i = S σ ∈F Γi σ . We will denote by F Γ the set of fracture faces
S
i∈I F Γ i .
This geometrical discretization of Ω and Γ is denoted in the following by D. The VAG discretization has been introduced in [3] for diffusive problems on het- erogeneous anisotropic media. Its extension to the hybrid dimensional **Darcy** model is based on the following vector space of degrees of freedom:

We consider in this section the extension of the previous combined VAG-HFV discretization to the case of a non-isothermal compositional **two**-**phase** **Darcy** flow. This extension is based on the formulation of the model introduced in [5]. Its main advantages compared with the related Coats’ variable switch formulation [9] is to be based on a fixed set of unknowns using extended **phase** molar fractions and to express the thermodynamical equilibrium as comple- mentary constraints for both phases α ∈ P. Previous works have considered the VAG discretization of isothermal and of non-isothermal compositional **two**-**phase** **Darcy** **flows** in respectively [18] and [26]. The HFV discretization of isothermal **two** component **two**-**phase** **Darcy** **flows** is also derived in [3] and the related Mixed-Hybrid Finite Element discretization of three-**phase** **Darcy** **flows** in [2]. Our extension to the combined VAG-HFV discretization follows the methodology presented in the previous section for immiscible isothermal **two**-**phase** **flows** which takes advantage of the cell based definition of the fluxes shared by the VAG, the HFV and by the modified scheme at interface cells.

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1 Introduction
This work has **two** aims: Providing a reduced model for **two** **phase** **flows** in porous media with complex Discrete Fracture Networks (DFN) and validating the reduced model by comparing numerically derived solutions of different test cases with the solutions of the full (non reduced) model. More precisely, we are concerned with the modelling and the discretization of **two** **phase** **Darcy** **flows** in fractured porous media, for which the fractures are represented as interfaces of codimension one. In this framework, the d −1 dimensional flow in the fractures is coupled with the d dimensional flow in the matrix leading to the so called, hybrid dimensional **Darcy** flow model. These models are derived from the so called equi-dimensional model, where fractures are represented as geological layers of equal dimension as the matrix, by averaging fracture quantities over the fracture width. We consider the case for which the pressure can be discontinuous at the matrix-fracture (mf ) interfaces in order to account for fractures acting either as drains or as barriers as described in [14], [17], [4], contrary to the continuous pressure model described in [2] developed for highly conductive fractures. A hybrid dimensional discontinuous pressure model for **two** **phase** flow in global pressure formulation has been derived in [18]. The model presented in this work, in pressure-pressure formulation, provides features like an upwind coupling condition for mf mass exchange fluxes and the incorporation of gravitational force in these fluxes, which is a novelty. Subsequently, in this work, we use numerically derived solutions of different test cases, to compare our model with the equi-dimensional model and with the hybrid dimensional model for complex DFN, presented in [7], which assumes pressure continuity accross the fractures.

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The outline of the paper is the following. Section 2 introduces a general formulation for compositional mul- tiphase **Darcy** flow models, accounting for an arbitrary number of phases and **phase** appearance and disappear- ance. Section 3 details the vertex-centred discretiza- tion of compositional multiphase flow models. The VAG scheme is first recalled for diffusive equations in sub- section 3.1. Then, the VAG scheme fluxes are derived and used in subsection 3.2 to discretize the composi- tional multiphase **Darcy** flow model, and the pore vol- ume assignment procedure is detailed. Subsection 3.3 briefly discusses the algorithms to solve the nonlinear and the linear systems arising from the VAG discretiza- tion of the compositional models. The last section 4 ex- hibits the efficiency of the VAG discretization which is compared to the solutions obtained with the MPFA O scheme. The first test cases deal with **two** **phase** flow examples including highly heterogeneous cases and dis- continuous capillary pressures. Then, the last test case considers the nearwell injection of miscible CO2 in a saline aquifer, taking into account the vaporization of H2O in the gas **phase** as well as the precipitation of salt.

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Chapter 3
**Two** **phase** flow analysis
3.1 Introduction
Flow and transport in fractured porous media are of paramount importance for many applications such as petroleum exploration and production, geological storage of carbon dioxide, hydrogeology, or geothermal energy. **Two** classes of models, dual continuum and discrete fracture models, are typically employed and possibly coupled to simulate flow and transport in fractured porous media. Dual continuum models assume that the fracture network is well connected and can be homogenised as a continuum coupled to the matrix continuum using transfer functions. On the other hand, discrete fracture models (DFM), on which this chapter focuses, represent explicitly the fractures as co-dimension one surfaces immersed in the surrounding matrix domain. The use of lower dimensional rather than equi-dimensional entities to represent the fractures has been introduced in [5, 11, 45, 54, 56] to facilitate the grid generation and to reduce the number of degrees of freedom of the discretised model. The reduction of dimension in the fracture network is obtained from the equi-dimensional model by integration and averaging along the width of each fracture. The resulting so called hybrid-dimensional model couple the 3D model in the matrix with a 2D model in the fracture network taking into account the jump of the normal fluxes as well as additional transmission conditions at the matrix-fracture interfaces. These transmission conditions depend on the mathematical nature of the equi- dimensional model and on additional physical assumptions. They are typically derived for a single **phase** **Darcy** flow for which they specify either the continuity of the pressure in the case of fractures acting as drains [5, 15] or Robin type conditions in order to take into account the discontinuity of the pressure for fractures acting either as drains or barriers [6, 45, 56].

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Some numerical studies concerning the ﬂow modeling at the pore scale, i.e. the scale of the passage between tubes, have been performed in an effort to understand more accurately the origin and transitions of the different regimes. The **two**-**phase** ﬂow modeling at low Reynolds numbers, in which interfaces play an important role and need to be explicitly tracked, is a complex case due to numerical issues. Direct numerical methods for tracking interfaces, such as the “Volume-of-Fluid” method (Hirt and Nichols, 1981), have been successfully used to simulate **two**- **phase** ﬂow in tubes (Gupta et al., 2009). However, even with a signiﬁcant reduction of the computation time due to the axial symmetry of the problem, simulations still requires signiﬁcant computation time. More recently, the Volume-Of-Fluid method has been validated in a three-dimensional case, the trickling ﬂow on a stack of a few particles (Augier et al., 2010). However, this was possible for a small number of “grains” only and required long computation times (around one week on 2 processors for 1 s of physical time). The numerical simulation of **two**-**phase** ﬂows at the scale of a tube bundle requires too much computation time under these conditions, which explains the lack of numerical simulations.

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The paper is structured as follows: we first derive a primitive system using classical mechanics arguments: we first use a Least Action Principle to derive the conservative part of the system. Then we add dissipative structures to the system by designing an entropy inequality. This entropy evolution equation will provide the system with source terms that account for the fact that mass transfer and pres- sure are unbalanced between phases. **Two** different systems will then be derived by considering the equilibria which correpond to instantaneous relaxations due to the source terms. After a study of these systems, we will present the **two**-step convection-relaxation numerical strategy we use to approximate the equilibrium systems solution. Finally we present numerical tests involving the simulation of dynamical **phase**-change phenomena.

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Florence Drui 1, 2 , Alexandru Fikl 2 , Pierre Kestener 2 , Samuel Kokh 2, 3 , Adam Larat 1, 4 , Vincent Le Chenadec 1 and Marc Massot 1,4
Abstract. Many physical problems involve spatial and temporal inhomogeneities that require a very fine discretization in order to be accurately simulated. Using an adaptive mesh, a high level of resolution is used in the appropriate areas while keeping a coarse mesh elsewhere. This idea allows to save time and computations, but represents a challenge for distributed-memory environments. The MARS project (for Multiphase Adaptative Refinement Solver) intends to assess the parallel library p4est for adaptive mesh, in a case of a finite volume scheme applied to **two**-**phase** **flows**. Besides testing the library’s performances, particularly for load balancing, its user-friendliness in use and implementation are also exhibited here. First promising 3D simulations are even presented.

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of the nucleate boiling mechanism.
1.6.4 Kim and Mudawar’s correlation
As written by Kim and Mudawar [ 50 ], after reviewing the literature, it is imperative to evaluate the large body of models for flow boiling in small channels and create a universal model. It is important to have, as they propose on the second part of their study a universal tool that can be used to model and predict heat transfer applicable to a broad range of operating conditions. The development of a predictive tool is presented for saturated flow boiling in mini and microchannels. With the widely discussed **two** mechanisms identified that can dominate the largest fraction of the channel, bubbly and slug flow are mainly present in the channel whereas a significant part of the channel length is dominated by annular flow when convective boiling heat transfer is dominant.

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is set to the carrier **phase** velocity at the location of the particle. This implies, that the Quasi Brownian Motion is initially zero. Fig. 2.9 shows the spectral development of the integral properties of the Eulerian-Lagrangian simulation. The carrier **phase** energy decays due to molecular viscosity (continuous line with circles). The carrier-**phase** dispersed-**phase** correlation follows closely the carrier **phase** energy (dot-dashed line with triangles). Due to particle inertia the dispersed **phase** energy (dashed line with squares) decays more slowly than the carrier **phase** energy or the carrier-**phase** dispersed-**phase** correlation. The uncor- related particle kinetic energy (QBE) (line with diamonds) is initially zero. This is due to the initialization, where the particle velocity is initially equal to the gaseous velocity and therefore entirely correlated. The time evolution behavior of the Quasi Brownian energy can be interpreted as follows: Due to its inertia the particle velocity becomes slightly un- correlated from the gaseous velocity. After about one carrier **phase** turn over time (≈ 4 non-dimensional time units) some particles have been ejected from the vortices. If such particles from different vortices meet, their velocity is at best partially correlated. This leads to the production of Quasi Brownian Energy in such regions. At large scales dispersed **phase** velocity diminishes due to drag force, since the carrier **phase** velocity is effectively decreased by molecular viscosity. Since particle inertia is proportional to its velocity, the production of Quasi Brownian Motion decreases with the decreasing correlated particle velocity. Quasi Brownian Motion decreases as well due to drag with the carrier **phase** and consequently so does Quasi Brownian Energy.

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University Institute of France
Abstract
The present paper aims at building a fast and accurate **phase** transition solver dedicated to unsteady multiphase flow computations. In a previous contribution (Chiapolino et al. 2017), such a solver was successfully developed to compute thermodynamic equilibrium between a liquid **phase** and its corresponding vapor **phase**. The present work extends the solver’s range of application by considering a multicomponent gas **phase** instead of pure vapor, a necessary improvement in most practical applications. The solver proves easy to implement compared to common iterative procedures, and allows systematic CPU savings over 50%, at no cost in terms of accuracy. It is validated against solutions based on an accurate but expensive iterative solver. Its capability to deal with cavitating, evaporating and condensing **two**-**phase** **flows** is highlighted on severe test problems both 1D and 2D.

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A Level-Set type method has been developed to track rigid bodies on unstructured meshes. Thanks to the Overbee limiter of Chiapolino et al. (2017) the method doesn’t need reinitialization, nor interface reconstruction. A solid fluid coupling method has been built and compared to other approaches, based on stiff relaxation and conventional Ghost-Cell extrapolation. It is simple to implement and improves convergence. It has been extended to 2D and validated against 2D computations of supersonic **two**-**phase** flow around blunt body at rest. The overall method has been extended to **two**-way coupling and illustrations have been shown.

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3. Eymard, R., Guichard, C., Herbin, R.: Small-stencil 3d schemes for diffusive **flows** in porous media. ESAIM: M2AN 46, 265–290 (2010)
4. Eymard, R., Herbin, R., Gallouet, T.: Discretisation of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: a scheme using stabilisation and hybrid interfaces. IMA J Numer Anal 30, 1009–1043 (2010). DOI doi: 10.1093/imanum/drn084 5. Reichenberger, V., Jakobs, H., Bastian, P., Helmig, R.: A mixed-dimensional finite volume