This paper summarises the results of a benchmark study that compares a number of mathematical and numerical models applied to specific problems in the context of carbon dioxide (CO 2 ) storage in geologic formations. The processes modelled comprise ad-H. Class (B) · A. Ebigbo · R. Helmig · M. Darcis · B. Flemisch vective multi-phase flow, compositional effects due to dissolution of CO 2 into the ambient brine and nonisothermal effects due to temperature gradients and the Joule-Thompson effect. The problems deal with leakage through a leaky well, methane recovery enhanced P. Audigane BRGM, French Geological Survey, 410 Comput Geosci (2009) 13:409-434 by CO 2 injection and a reservoir-scale injection scenario into a heterogeneous formation. We give a description of the benchmark problems then briefly introduce the participating codes and finally present and discuss the results of the benchmark study.

This work deals with two-phase Discrete Fracture Matrix models coupling the two-phase Darcy flow in the matrix domain to the two-phase Darcy flow in the network of fractures represented as co-dimension one surfaces. Two classes of such hybrid-dimensional models are investigated either based on nonlinear or linear transmission conditions at the matrix-fracture interfaces. The linear transmission conditions include the cell-centred upwind approximation of the phase mobilities classically used in the porous media flow community as well as a basic extension of the continuous phase pressure model accounting for fractures acting as drains. The nonlinear transmission conditions at the matrix-fracture interfaces are based on the normal flux continuity equation for each phase using additional interface phase pressure unknowns. They are compared both in terms of accuracy and numerical efficiency to a reference equi-dimensional model for which the fractures are represented as full-dimensional subdomains. The discretization focuses on Finite Volume cellcentred Two-Point Flux Approximation which is combined with a local nonlinear solver allowing to eliminate efficiently the additional matrix-fracture interfacial unknowns together with the nonlinear transmission conditions. 2D numerical experiments illustrate the better accuracy provided by the nonlinear transmission conditions compared to their linear approximations with a moderate computational overhead obtained thanks to the local nonlinear elimination at the matrix-fracture interfaces. The numerical section is complemented by a comparison of the reduced models on a 3D test case using the Vertex Approximate Gradient scheme.

This paper proposes an efficient splitting algorithm to solve coupled liquid gas Darcy and free gas flows at the interface between a porous medium and a free-flow domain. This model is compared to the reduced model introduced in [6] using a 1D approximation of the gas free flow. For that purpose, the gas molar fraction diffusive flux at the interface in the free-flow domain is approximated by a two point flux approximation based on a low-frequency diagonal approximation of a Steklov-Poincaré type operator. The splitting algorithm and the reduced model are applied in particular to the modelling of the mass exchanges at the interface between the storage and the ventilation galleries in radioactive waste deposits.

We consider a two-phase Darcy flow in a fractured and deformable porous medium for which the fractures are described as a network of planar surfaces leading to so-called hybrid-dimensional models. The fractures are assumed open and filled by the fluids and small deformations with a linear elastic constitutive law are considered in the matrix. As opposed to \cite{bonaldi:hal-02549111}, the phase pressures are not assumed continuous at matrix fracture interfaces, which raises new challenges in the convergence analysis related to the additional interfacial equations and unknowns for the flow. As shown in \cite{BHMS2018,gem.aghili}, unlike single phase flow, discontinuous pressure models for two-phase flows provide a better accuracy than continuous pressure models even for highly permeable fractures. This is due to the fact that fractures fully filled by one phase can act as barriers for the other phase, resulting in a pressure discontinuity at the matrix fracture interface.    The model is discretized using the gradient discretization method \cite{gdm}, which covers a large class of conforming and non conforming schemes. This framework allows for a generic convergence analysis of the coupled model using a combination of discrete functional tools. In this work, the gradient discretization of \cite{bonaldi:hal-02549111} is extended to the discontinuous pressure model and the convergence to a weak solution is proved.  Numerical solutions provided by the continuous and discontinuous pressure models are compared on gas injection and suction test cases using a Two-Point Flux Approximation (TPFA) finite volume scheme for the flows and $\P_2$ finite elements for the mechanics.

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