arXiv subjects

Roland Masson

Publications and source records attributed to Roland Masson.

15 recordsLinked to original sources

arXiv2026

A higher order polytopal method for contact mechanics with Tresca friction

In this work, we design and analyze a Discrete de Rham (DDR) scheme for a contact mechanics problem involving fractures along which a model of Tresca friction is considered. Our approach is based on a mixed formulation involving a displacement field and a Lagrange multiplier, enforcing the contact conditions, representing tractions at fractures. The approximation space for the displacement is made of vectors values attached to each vertex, edge, face, and element, while the Lagrange multiplier space is approximated by piecewise constant vectors on each fracture face. The displacement degrees of freedom allow reconstruct piecewise quadratic approximations of this field. We prove a discrete Korn inequality that account for the fractures, as well as an inf-sup condition (in a non-standard $H^{-1/2}$-norm) between the discrete Lagrange multiplier space and the discrete displacement space. We provide an in-depth error analysis of the scheme and show that, contrary to usual low-order nodal-based schemes, our method is robust in the quasi-incompressible limit for the primal variable~(displacement). An extensive set of numerical experiments confirms the theoretical analysis and demonstrate the practical accuracy and robustness of the scheme.

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arXiv2024

Numerical analysis of a mixed-dimensional poromechanical model with frictionless contact at matrix-fracture interfaces

We present a complete numerical analysis for a general discretization of a coupled flow-mechanics model in fractured porous media, considering single-phase flows and including frictionless contact at matrix-fracture interfaces, as well as nonlinear poromechanical coupling. Fractures are described as planar surfaces, yielding the so-called mixed- or hybrid-dimensional models. Small displacements and a linear elastic behavior are considered for the matrix. The model accounts for discontinuous fluid pressures at matrix-fracture interfaces in order to cover a wide range of normal fracture conductivities. The numerical analysis is carried out in the Gradient Discretization framework, encompassing a large family of conforming and nonconforming discretizations. The convergence result also yields, as a by-product, the existence of a weak solution to the continuous model. A numerical experiment in 2D is presented to support the obtained result, employing a Hybrid Finite Volume scheme for the flow and second-order finite elements ($\mathbb P_2$) for the mechanical displacement coupled with face-wise constant ($\mathbb P_0$) Lagrange multipliers on fractures, representing normal stresses, to discretize the contact conditions.

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arXiv2024

Analysis of a VEM-fully discrete polytopal scheme with bubble stabilisation for contact mechanics with Tresca friction

This work performs the convergence analysis of the polytopal nodal discretisation of contact-mechanics (with Tresca friction) recently introduced in [18] in the framework of poro-elastic models in fractured porous media. The scheme is based on a mixed formulation, using face-wise constant approximations of the Lagrange multipliers along the fracture network and a fully discrete first order nodal approximation of the displacement field. The displacement field is enriched with additional bubble degrees of freedom along the fractures to ensure the inf-sup stability with the Lagrange multiplier space. It is presented in a fully discrete formulation, which makes its study more straightforward, but also has a Virtual Element interpretation. The analysis establishes an abstract error estimate accounting for the fully discrete framework and the non-conformity of the discretisation. A first order error estimate is deduced for sufficiently smooth solutions both for the gradient of the displacement field and the Lagrange multiplier. A key difficulty of the numerical analysis is the proof of a discrete inf-sup condition, which is based on a non-standard $H^{-1/2}$-norm (to deal with fracture networks) and involves the jump of the displacements, not their traces. The analysis also requires the proof of a discrete Korn inequality for the discrete displacement field which takes into account fracture networks. Numerical experiments based on analytical solutions confirm our theoretical findings

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arXiv2024

A bubble VEM-fully discrete polytopal scheme for mixed-dimensional poromechanics with frictional contact at matrix fracture interfaces

The objective of this article is to address the discretisation of fractured/faulted poromechanical models using 3D polyhedral meshes in order to cope with the geometrical complexity of faulted geological models. A polytopal scheme is proposed for contact-mechanics, based on a mixed formulation combining a fully discrete space and suitable reconstruction operators for the displacement field with a face-wise constant approximation of the Lagrange multiplier accounting for the surface tractions along the fracture/fault network. To ensure the inf--sup stability of the mixed formulation, a bubble-like degree of freedom is included in the discrete space of displacements (and taken into account in the reconstruction operators). It is proved that this fully discrete scheme for the displacement is equivalent to a low-order Virtual Element scheme, with a bubble enrichment of the VEM space. This $\mathbb{P}^1$-bubble VEM--$\mathbb{P}^0$ mixed discretization is combined with an Hybrid Finite Volume scheme for the Darcy flow. All together, the proposed approach is adapted to complex geometry accounting for network of planar faults/fractures including corners, tips and intersections; it leads to efficient semi-smooth Newton solvers for the contact-mechanics and preserve the dissipative properties of the fully coupled model. Our approach is investigated in terms of convergence and robustness on several 2D and 3D test cases using either analytical or numerical reference solutions both for the stand alone static contact mechanical model and the fully coupled poromechanical model.

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arXiv2024

Discretisations of mixed-dimensional Thermo-Hydro-Mechanical models preserving energy estimates

In this study, we explore mixed-dimensional Thermo-Hydro-Mechanical (THM) models in fractured porous media accounting for Coulomb frictional contact at matrix fracture interfaces. The simulation of such models plays an important role in many applications such as hydraulic stimulation in deep geothermal systems and assessing induced seismic risks in CO2 storage. We first extend to the mixed-dimensional framework the thermodynamically consistent THM models derived in [16] based on first and second principles of thermodynamics. Two formulations of the energy equation will be considered based either on energy conservation or on the entropy balance, assuming a vanishing thermo-poro-elastic dissipation. Our focus is on space time discretisations preserving energy estimates for both types of formulations and for a general single phase fluid thermodynamical model. This is achieved by a Finite Volume discretisation of the non-isothermal flow based on coercive fluxes and a tailored discretisation of the non-conservative convective terms. It is combined with a mixed Finite Element formulation of the contact-mechanical model with face-wise constant Lagrange multipliers accounting for the surface tractions, which preserves the dissipative properties of the contact terms. The discretisations of both THM formulations are investigated and compared in terms of convergence, accuracy and robustness on 2D test cases. It includes a Discrete Fracture Matrix model with a convection dominated thermal regime, and either a weakly compressible liquid or a highly compressible gas thermodynamical model.

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arXiv2024

Multi-segmented non-isothermal compositional liquid gas well model for geothermal processes

We consider a non-isothermal compositional gas liquid model for the simulation of well operations in geothermal processes. The model accounts for phase transitions assumed to be at thermodynamical equilibrium and is based on an hydrodynamical Drift Flux Model (DFM) combined with a No Pressure Wave approximation of the momentum equation. The focus of this work is on the design of a robust discretization accounting for slanted and multibranch wells with the ability to simulate both transient behavior such as well opening as well as coupled simulations at the time scale of the reservoir. It is based on a staggered finite volume scheme in space combined with a fully implicit Euler time integration. The construction of consistent and stable numerical fluxes is a key feature for a robust numerical method. It is achieved by combining a monotone flux approximation for the phase superficial velocities with an upwind approximation of the phase molar fractions, density and enthalpy. In order to facilitate the coupling of the well and reservoir models, the Newton linearization accounts for the elimination of the hydrodynamical unknowns leading to Jacobian systems using the same primary unknowns than those of the reservoir model. The efficiency of our approach is investigated on both stand alone well test cases without and with cross flow, and on a fully coupled well-reservoir simulation.

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arXiv2022

Energy-stable discretization of two-phase flows in deformable porous media with frictional contact at matrix-fracture interfaces

We address the discretization of two-phase Darcy flows in a fractured and deformable porous medium, including frictional contact between the matrix-fracture interfaces. Fractures are described as a network of planar surfaces leading to the so-called mixed- or hybrid-dimensional models. Small displacements and a linear elastic behavior are considered for the matrix. Phase pressures are supposed to be discontinuous at matrix-fracture interfaces, as they provide a better accuracy than continuous pressure models even for high fracture permeabilities. The general gradient discretization framework is employed for the numerical analysis, allowing for a generic stability analysis and including several conforming and nonconforming discretizations. We establish energy estimates for the discretization, and prove existence of a solution. To simulate the coupled model, we employ a Two-Point Flux Approximation (TPFA) finite volume scheme for the flow and second-order ($\mathbb P_2$) finite elements for the mechanical displacement coupled with face-wise constant ($\mathbb P_0$) Lagrange multipliers on fractures, representing normal and tangential stresses, to discretize the frictional contact conditions. This choice allows to circumvent possible singularities at tips, corners, and intersections between fractures, and provides a local expression of the contact conditions. We present numerical simulations of two benchmark examples and one realistic test case based on a drying model in a radioactive waste geological storage structure.

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arXiv2021

A numerical algorithm based on probing to find optimized transmission conditions

Optimized Schwarz Methods (OSMs) are based on optimized transmission conditions along the interfaces between the subdomains. Optimized transmission conditions are derived at the theoretical level, using techniques developed in the last decades. The hypothesis behind these analyses are quite strong, so that the applicability of OSMs is still limited. In this manuscript, we present a numerical algorithm to obtain optimized transmission conditions for any given problem at hand. This algorithm requires few subdomain solves to be performed in an offline phase. This additional cost is usually negligible due to the resulting faster convergence, even in a single-query context.

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arXiv2021

Gradient discretization of two-phase poro-mechanical models with discontinuous pressures at matrix fracture interfaces

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 [10], 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 [16, 2], 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 [22], 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 [10] 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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arXiv2021

Gradient discretization of two-phase flows coupled with mechanical deformation in fractured porous media

We consider a two-phase Darcy flow in a fractured porous medium consisting in a matrix flow coupled with a tangential flow in the fractures, described as a network of planar surfaces. This flow model is also coupled with the mechanical deformation of the matrix assuming that the fractures are open and filled by the fluids, as well as small deformations and a linear elastic constitutive law. The model is discretized using the gradient discretization method [26], 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. Here, we describe the model together with its numerical discretization, and we prove a convergence result assuming the non-degeneracy of the phase mobilities and that the discrete solutions remain physical in the sense that, roughly speaking, the porosity does not vanish and the fractures remain open. This is, to our knowledge, the first convergence result for this type of models taking into account two-phase flows in fractured porous media and the non-linear poromechanical coupling. Previous related works consider a linear approximation obtained for a single phase flow by freezing the fracture conductivity [36, 37]. Numerical tests employing the Two-Point Flux Approximation (TPFA) finite volume scheme for the flows and $\mathbb P_2$ finite elements for the mechanical deformation are also provided to illustrate the behavior of the solution to the model.

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arXiv2021

Two-phase geothermal model with fracture network and multi-branch wells

This paper focuses on the numerical simulation of geothermal systems in complex geological settings. The physical model is based on two-phase Darcy flows coupling the mass conservation of the water component with the energy conservation and the liquid vapor thermodynamical equilibrium. The discretization exploits the flexibility of unstructured meshes to model complex geology including conductive faults as well as complex wells. The polytopal and essentially nodal Vertex Approximate Gradient scheme is used for the approximation of the Darcy and Fourier fluxes combined with a Control Volume approach for the transport of mass and energy. Particular attention is paid to the faults which are modelled as two-dimensional interfaces defined as collection of faces of the mesh and to the flow inside deviated or multi-branch wells defined as collection of edges of the mesh with rooted tree data structure. By using an explicit pressure drop calculation, the well model reduces to a single equation based on complementarity constraints with only one well implicit unknown. The coupled systems are solved fully implicitely at each time step using efficient nonlinear and linear solvers on parallel distributed architectures. The convergence of the discrete model is investigated numerically on a simple test case with a Cartesian geometry and a single vertical producer well. Then, the ability of our approach to deal efficiently with realistic test cases is assessed on a high energy faulted geothermal reservoir operated using a doublet of two deviated wells.

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arXiv2020

Verification benchmarks for single-phase flow in three-dimensional fractured porous media

Flow in fractured porous media occurs in the earth's subsurface, in biological tissues, and in man-made materials. Fractures have a dominating influence on flow processes, and the last decade has seen an extensive development of models and numerical methods that explicitly account for their presence. To support these developments, we present a portfolio of four benchmark cases for single-phase flow in three-dimensional fractured porous media. The cases are specifically designed to test the methods' capabilities in handling various complexities common to the geometrical structures of fracture networks. Based on an open call for participation, results obtained with 17 numerical methods were collected. This paper presents the underlying mathematical model, an overview of the features of the participating numerical methods, and their performance in solving the benchmark cases.

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arXiv2016

Parallel numerical modeling of hybrid-dimensional compositional non-isothermal Darcy flows in fractured porous media

This paper introduces a new discrete fracture model accounting for non-isothermal compositional multiphase Darcy flows and complex networks of fractures with intersecting, immersed and non immersed fractures. The so called hybrid-dimensional model using a 2D model in the fractures coupled with a 3D model in the matrix is first derived rigorously starting from the equi-dimensional matrix fracture model. Then, it is dis-cretized using a fully implicit time integration combined with the Vertex Approximate Gradient (VAG) finite volume scheme which is adapted to polyhedral meshes and anisotropic heterogeneous media. The fully coupled systems are assembled and solved in parallel using the Single Program Multiple Data (SPMD) paradigm with one layer of ghost cells. This strategy allows for a local assembly of the discrete systems. An efficient preconditioner is implemented to solve the linear systems at each time step and each Newton type iteration of the simulation. The numerical efficiency of our approach is assessed on different meshes, fracture networks, and physical settings in terms of parallel scalability, nonlinear convergence and linear convergence.

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arXiv2016

Numerical analysis of a two-phase flow discrete fracture model

We present a new model for two phase Darcy flows in fractured media, in which fractures are modelled as submanifolds of codimension one with respect to the surrounding domain (matrix). Fractures can act as drains or as barriers, since pressure discontinuities at the matrix-fracture interfaces are permitted. Additionally, a layer of damaged rock at the matrix-fracture interfaces is accounted for. The numerical analysis is carried out in the general framework of the Gradient Discretisation Method. Compactness techniques are used to establish convergence results for a wide range of possible numerical schemes; the existence of a solution for the two phase flow model is obtained as a byproduct of the convergence analysis. A series of numerical experiments conclude the paper, with a study of the influence of the damaged layer on the numerical solution.

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arXiv2016

Parallel Vertex Approximate Gradient discretization of hybrid dimensional Darcy flow and transport in discrete fracture networks

This paper proposes a parallel numerical algorithm to simulate the flow and the transport in a discrete fracture network taking into account the mass exchanges with the surrounding matrix. The discretization of the Darcy fluxes is based on the Vertex Approximate Gradient finite volume scheme adapted to polyhedral meshes and to heterogeneous anisotropic media, and the transport equation is discretized by a first order upwind scheme combined with an Euler explicit integration in time. The parallelization is based on the SPMD (Single Program, Multiple Data) paradigm and relies on a distribution of the mesh on the processes with one layer of ghost cells in order to allow for a local assembly of the discrete systems. The linear system for the Darcy flow is solved using different linear solvers and preconditioners implemented in the PETSc and Trilinos libraries. The convergence of the scheme is validated on two original analytical solutions with one and four intersecting fractures. Then, the parallel efficiency of the algorithm is assessed on up to 512 processes with different types of meshes, different matrix fracture permeability ratios, and different levels of complexity of the fracture network.

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