N m = 72) : Gauche : l'allure des indicateurs de failles associés à toutes les failles candidates après l'étape de prolongation pour une première itération. Droite : les failles candidates , portées par C tracées en cyan, retenues pour l'étape d'optimisation pour une première itération. La faille tracée en vert a la meilleure valeur d'indicateur, Test, vol.3, issue.2, p.98 ,
N m = 72) : Gauche : L'allure des indicateurs de failles associés à toutes les failles candidates après l'étape de prolongation pour une deuxième itération. Droite : L'ensemble de failles candidates, portées par la grille de recherche de fractures C tracées en cyan, retenues pour l'étape d'optimisation pour une deuxième itération. La faille tracée en vert présente la meilleure valeur d'indicateur, Test, vol.3, issue.2, p.100 ,
Barrières obtenues par l'algorithme pour N m = 8, en faisant varier ? ? {2%, 5% 8%}. La barrière à identifier est en bleu au centre du domaine. Colonne de gauche : pour mesures de pressions et de haut en bas : ? = 2%, 5%, 8%. Colonne de droite : pour des mesures de vitesses de Darcy et de haut en bas : ? = 2%, 5%, pp.8-130 ,
Barrières obtenues par l'algorithme, quand la fracture n'est pas portée par C . Cas où les mesures sont de vitesses de Darcy. La barrière à identifier est en bleu au centre du domaine Colonne de gauche : sans bruit (? = 0%) et de haut en bas : N m = 72, Colonne de droite : avec bruit (? = 2%) et de haut en bas : N m = 72, p.135 ,
Résultat de l'algorithme. Le type et le nombre de mesures varient, ainsi que le niveau de bruit. On donne le nombre de barrières que trouve l'algorithme ainsi que les valeurs de ? ? associées. Pour N m = 72, la localisation de la barrière donnée par l'algorithme est exacte, Test, vol.2, issue.8 ?, p.128 ,
Résultat de l'algorithme, quand la barrière n'est pas portée par C . Le type et le nombre de mesures varient, ainsi que le niveau de bruit. On donne le nombre de barrières que trouve l'algorithme ainsi que les valeurs de ? ? associées. La localisation des barrières est indiquée dans les Figures 5, Test, vol.270, issue.12 8, p.133 ,
Résultat de l'algorithme, quand la fracture n'est pas portée par C . Le type et le nombre de mesures varient, ainsi que le niveau de bruit ? = 0%, 2% et 5%. On donne le nombre de barrières que trouve l'algorithme ainsi que les valeurs de ? ? associées. . . . . . . . . . . . . . . 144 ,
Résultat de l'algorithme, quand la fracture n'est pas portée par C . Le type et le nombre de mesures varient, ainsi que le niveau de bruit ? = 0%, 2%. On donne le nombre de barrières que trouve l'algorithme ainsi que les valeurs de ? ? associées. La localisation des fractures est indiquée dans les Figures 5, Test, vol.3, issue.12 8, p.150 ,
A reduced fracture model for two-phase flow with different rock types. working paper or preprint, 2015. ,
URL : https://hal.archives-ouvertes.fr/hal-01119986
Control-volume distributed multi-point flux approximation coupled with a lower-dimensional fracture model, Journal of Computational Physics, vol.284, pp.462-489, 2015. ,
DOI : 10.1016/j.jcp.2014.12.047
Domain decomposition for flow in porous fractured media, in Domain Decomposition Methods in Sciences and Engineering, Domain Decomposition Methods in Sciences and Engineering, pp.365-373, 1999. ,
Modeling fractures as interfaces for flow and transport in porous media, in Fluid flow and transport in porous media : mathematical and numerical treatment, Contemp. Math, issue.295, pp.13-24, 2001. ,
Stable determination of conductivity by boundary measurements, Applicable Analysis, vol.1975, issue.1-3, pp.153-172, 1988. ,
DOI : 10.2307/1971291
Determining Linear Cracks by Boundary Measurements: Lipschitz Stability, SIAM Journal on Mathematical Analysis, vol.27, issue.2, pp.361-375, 1996. ,
DOI : 10.1137/S0036141094265791
Determining 2-dimensional cracks in 3- dimensional bodies : uniqueness and stability, Math. J, vol.46, pp.1-82, 1997. ,
Cracks with impedance ; stable determination from boundary data, Indiana Univ, Math. J, vol.62, pp.947-989, 2013. ,
Crack detection by the topological gradient method, Control and Cybernetics, vol.34, p.81, 2005. ,
??cart ?? la r??ciprocit?? et identification de fissures en thermo??lasticit?? isotrope transitoire, Comptes Rendus M??canique, vol.334, issue.4, pp.225-229, 2006. ,
DOI : 10.1016/j.crme.2006.02.004
Asymptotic and numerical modelling of flows in fractured porous media, ESAIM: Mathematical Modelling and Numerical Analysis, vol.43, issue.2, pp.239-275, 2009. ,
DOI : 10.1051/m2an/2008052
URL : https://hal.archives-ouvertes.fr/hal-00127023
The double porosity model for single phase flow in naturally fractured reservoirs, in Numerical Similation in Oil Recovery, IMA volumes in Mathematics and its Applications, pp.23-45, 1988. ,
Analysis of the Simulation of Single Phase Flow Through a Naturally Fractured Reservoir, SIAM Journal on Numerical Analysis, vol.26, issue.1, pp.12-29, 1989. ,
DOI : 10.1137/0726002
Derivation of the Double Porosity Model of Single Phase Flow via Homogenization Theory, SIAM Journal on Mathematical Analysis, vol.21, issue.4, pp.823-836, 1990. ,
DOI : 10.1137/0521046
Double conductivity media: a comparison between phenomenological and homogenization approaches, International Journal of Heat and Mass Transfer, vol.36, issue.10, pp.2613-2621, 1993. ,
DOI : 10.1016/S0017-9310(05)80198-X
Modelling fluid flow in fractured-porous rock masses by finite-element techniques, International Journal for Numerical Methods in Fluids, vol.17, issue.4, pp.337-348, 1984. ,
DOI : 10.1002/fld.1650040404
Uniqueness for the determination of unknown boundary and impedance with the homogeneous Robin condition, Inverse Problems, vol.25, issue.1, p.15004, 2009. ,
DOI : 10.1088/0266-5611/25/1/015004
A parabolic inverse problem with mixed boundary data. Stability estimates for the unknown boundary and impedance, Transactions of the American Mathematical Society, vol.366, issue.8, pp.3965-3995, 2014. ,
DOI : 10.1090/S0002-9947-2014-05807-8
URL : http://arxiv.org/abs/1107.5151
Corrosion detection in a 2D domain with a polygonal boundary, J. Inverse Ill-Posed Probl, pp.281-305, 2010. ,
Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks [strata], Journal of Applied Mathematics and Mechanics, vol.24, issue.5, pp.1286-1303, 1960. ,
DOI : 10.1016/0021-8928(60)90107-6
Identification of 2D cracks by elastic boundary measurements, Inverse Problems, vol.15, issue.1, p.67, 1999. ,
DOI : 10.1088/0266-5611/15/1/011
URL : https://hal.archives-ouvertes.fr/inria-00073252
Level set methods for geometric inverse problems in linear elasticity, Inverse Problems, vol.20, issue.3, pp.673-696, 2004. ,
DOI : 10.1088/0266-5611/20/3/003
Cavity identification in linear elasticity and thermoelasticity, Mathematical Methods in the Applied Sciences, vol.38, issue.6, pp.625-647, 2007. ,
DOI : 10.1002/mma.772
First-order indicators for the estimation of discrete fractures in porous media, submitted , see, 2016. ,
First-order indicators for the estimation of discrete fractures in porous media, Inverse Problems in Science and Engineering, vol.365, issue.1, 2016. ,
DOI : 10.1016/0309-1708(91)90020-O
URL : https://hal.archives-ouvertes.fr/hal-01279503
Refinement and coarsening indicators for adaptive parametrization: application to the estimation of hydraulic transmissivities, Inverse Problems, vol.18, issue.3, pp.775-794, 2002. ,
DOI : 10.1088/0266-5611/18/3/317
The multidimensional refinement indicators algorithm for optimal parameterization, J. Inverse Ill-Posed Probl, vol.16, pp.107-126, 2008. ,
URL : https://hal.archives-ouvertes.fr/inria-00079668
Characterizing flow and transport in fractured geological media : A review Advances in water resources, pp.861-884, 2002. ,
Mixed finite element methods and applications, no. 44 in Springer series in computational mathematics, 2013. ,
Convergence of the Homogenization Process for a Double-Porosity Model of Immiscible Two-Phase Flow, SIAM Journal on Mathematical Analysis, vol.27, issue.6, pp.1520-1543, 1996. ,
DOI : 10.1137/S0036141094276457
Gradient discretization of hybrid dimensional Darcy flows in fractured porous media, Numer. Math, 2015. ,
URL : https://hal.archives-ouvertes.fr/hal-01097704
Mixed and Hybrid Finite Element Methods, 1991. ,
DOI : 10.1007/978-1-4612-3172-1
A Uniqueness Result Concerning the Identification of a Collection of Cracks from Finitely Many Electrostatic Boundary Measurements, SIAM Journal on Mathematical Analysis, vol.23, issue.4, 1992. ,
DOI : 10.1137/0523051
Simultaneous reconstruction of shape and generalized impedance functions in electrostatic imaging, Inverse Problems, vol.30, issue.10, p.105009, 2014. ,
DOI : 10.1088/0266-5611/30/10/105009
Integral equations for inverse problems in corrosion detection from partial Cauchy data, Inverse Probl. Imaging, vol.1, pp.229-245, 2007. ,
Integral equations for shape and impedance reconstruction in corrosion detection Simultaneous reconstruction of shape and impedance in corrosion detection, Inverse Problems Methods Appl. Anal, vol.26, pp.17-357, 2010. ,
Discretized Tikhonov regularization for Robin boundaries localization, Applied Mathematics and Computation, vol.226, pp.374-385, 2014. ,
DOI : 10.1016/j.amc.2013.10.036
A stable recovery method for the Robin inverse problem, Mathematics and Computers in Simulation, vol.66, issue.4-5, pp.66-367, 2004. ,
DOI : 10.1016/j.matcom.2004.02.016
Numerical reconstruction of a piecewise constant Robin parameter in the two- or three-dimensional case, Inverse Problems, vol.28, issue.6, p.65016, 2012. ,
DOI : 10.1088/0266-5611/28/6/065016
Logarithmic stability estimates for a Robin coefficient in two-dimensional Laplace inverse problems, Inverse Problems, vol.20, issue.1, pp.47-59, 2004. ,
DOI : 10.1088/0266-5611/20/1/003
Identification of Robin coefficients by the means of boundary measurements, Inverse Problems, vol.15, issue.6, pp.1425-1438, 1999. ,
DOI : 10.1088/0266-5611/15/6/303
On the theory and practice of non-linear least-squares, Advances in Water Resources, vol.14, issue.2, pp.55-63, 1991. ,
DOI : 10.1016/0309-1708(91)90051-O
Nonlinear least squares for inverse problems, Scientific Computation Theoretical foundations and step-by-step guide for ap- plications, 2009. ,
MULTISCALE PARAMETRIZATION FOR THE ESTIMATION OF A DIFFUSION COEFFICIENT IN ELLIPTIC AND PARABOLIC PROBLEMS, 5th IFAC Symposium on Control of Distributed Parameter Systems, pp.315-324, 1989. ,
DOI : 10.1016/B978-0-08-037036-1.50038-0
A unified physical presentation of mixed, mixed-hybrid finite elements and standard finite difference approximations for the determination of velocities in waterflow problems, Advances in Water Ressources, pp.329-348, 1991. ,
DOI : 10.1016/0309-1708(91)90020-O
A mixed finite element method for Darcy flow in fractured porous media with non-matching grids, ESAIM : M2AN, pp.465-489, 2012. ,
Anomalous diffusion exponents in continuous two-dimensional multifractal media, Physical Review E, vol.70, issue.1, p.70, 2004. ,
DOI : 10.1103/PhysRevE.70.016306
Efficient algorithms for the determination of the connected fracture network and the solution to the steady-state flow equation in fracture networks, Computers & Geosciences, vol.29, issue.1, pp.107-111, 2003. ,
DOI : 10.1016/S0098-3004(02)00081-X
40 years of inverse problems in hydrogeology, Comptes Rendus de l'Academie des Sciences Series IIA Earth and Planetary Science, pp.73-87, 1999. ,
A UNIFIED APPROACH TO MIMETIC FINITE DIFFERENCE, HYBRID FINITE VOLUME AND MIXED FINITE VOLUME METHODS, Mathematical Models and Methods in Applied Sciences, vol.20, issue.02, pp.265-295, 2010. ,
DOI : 10.1142/S0218202510004222
URL : https://hal.archives-ouvertes.fr/hal-00346077
GRADIENT SCHEMES: A GENERIC FRAMEWORK FOR THE DISCRETISATION OF LINEAR, NONLINEAR AND NONLOCAL ELLIPTIC AND PARABOLIC EQUATIONS, Mathematical Models and Methods in Applied Sciences, vol.23, issue.13, pp.2395-2432, 2013. ,
DOI : 10.1142/S0218202513500358
URL : https://hal.archives-ouvertes.fr/hal-00751551
Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: a scheme using stabilization and hybrid interfaces, IMA Journal of Numerical Analysis, vol.30, issue.4, pp.1009-1043, 2010. ,
DOI : 10.1093/imanum/drn084
A new fault model in geological basin modelling, application to finite volume scheme and domain decomposition methods, in Finite Volumes for Complex Applications III, Hermés Penton Sci, pp.543-550, 2002. ,
Model reduction and discretization using hybrid finite volumes for flow in porous media containing faults, Computational Geosciences, vol.16, issue.7, 2016. ,
DOI : 10.1007/s10596-016-9558-3
URL : https://hal.archives-ouvertes.fr/hal-01395454
Determining cracks by boundary measurements, Indiana Uni. Math. Jou, p.38, 1989. ,
Modeling fractures as interfaces with nonmatching grids, Computational Geosciences, vol.16, issue.2, pp.1043-1060, 2012. ,
DOI : 10.1007/s10596-012-9302-6
URL : https://hal.archives-ouvertes.fr/inria-00561601
Forchheimer pour un écoulement dans un milieu poreux fracturé, pp.129-143, 2006. ,
Modeling fractures as interfaces: a model for Forchheimer fractures, Computational Geosciences, vol.25, issue.7, pp.91-104, 2008. ,
DOI : 10.1007/s10596-007-9062-x
URL : https://hal.archives-ouvertes.fr/inria-00207993
Numerical modelling of multiphase subsurface flow in the presence of fractures, Commun. Appl. Ind. Math, vol.3, pp.1-23, 2011. ,
Lectures on Cauchy's problem in linear partial differential equations, Courier Corporation, 2014. ,
An Adaptive Subdivision Algorithm for the Identification of the Diffusion Coefficient in Two-dimensional Elliptic Problems, Journal of Mathematical Modelling and Algorithms, vol.17, issue.3, pp.529-545, 2007. ,
DOI : 10.1007/s10852-006-9046-1
A new refinement indicator for adaptive parameterization: Application to the estimation of the diffusion coefficient in an elliptic problem, Journal of Computational and Applied Mathematics, vol.224, issue.1, pp.307-319, 2009. ,
DOI : 10.1016/j.cam.2008.05.006
An application of the geostatistical approach to the inverse problem in two-dimensional groundwater modeling, Water Resources Research, vol.20, pp.1003-1020, 1984. ,
Multicomponent fluid flow by discontinuous Galerkin and mixed methods in unfractured and fractured media, Water Resources Research, vol.3, issue.2, p.41, 2005. ,
DOI : 10.1029/2005WR004339
An efficient numerical model for incompressible twophase flow in fractured media, Adv Water Resourc, pp.31-891, 2008. ,
Phénomènes de transmission à travers des couches minces de conductivité élevée, J. Math. Anal. Appl, pp.47-284, 1974. ,
Generalized cell-centered finite volume methods for flow in porous media with faults, in Finite volumes for complex applications, pp.343-350, 2002. ,
A discrete fracture model for two-phase flow with matrix-fracture interaction, Procedia Computer Science, vol.4, pp.967-973, 2011. ,
DOI : 10.1016/j.procs.2011.04.102
Introduction to Shape Optimization, 1992. ,
An efficient discrete fracture model applicable for general purpose reservoir simulators, SPE J, pp.227-263, 2004. ,
DOI : 10.2118/88812-pa
Numerical Simulation of Water Injection in Fractured Media Using the Discrete-Fracture Model and the Galerkin Method, SPE Reservoir Evaluation & Engineering, vol.6, issue.02, pp.117-126, 2003. ,
DOI : 10.2118/83633-PA
Problèmes inverses, To be published, 2016. ,
Finite element discrete-fracture model for multiphase flow in porous media, AIChE Journal, vol.463, pp.1120-1130, 2006. ,
The detection of holes by elasto-static measurements ,
Mathematical analysis of a discrete fracture model coupling Darcy flow in the matrix with Darcy???Forchheimer flow in the fracture, ESAIM: Mathematical Modelling and Numerical Analysis, vol.48, issue.5, pp.2-48, 2014. ,
DOI : 10.1051/m2an/2014003
URL : https://hal.archives-ouvertes.fr/hal-00945028
Flow modelling of coupled fracture-matrix porous media systems with a two mesh concept, Master's thesis, 2013. ,
Logarithmic convergence rates for the identification of a nonlinear Robin coefficient, Journal of Mathematical Analysis and Applications, vol.359, issue.2, pp.451-463, 2009. ,
DOI : 10.1016/j.jmaa.2009.06.004
Statistical characteristics of flow as indicators of channeling in heterogeneous porous and fractured media, Advances in Water Resources, vol.33, issue.3, pp.257-269, 2010. ,
DOI : 10.1016/j.advwatres.2009.12.002
URL : https://hal.archives-ouvertes.fr/insu-00577996
A multiscale Darcy???Brinkman model for fluid flow in fractured porous media, Numerische Mathematik, vol.30, issue.3, pp.717-752, 2011. ,
DOI : 10.1007/s00211-010-0343-2
A Multiresolution Method for Distributed Parameter Estimation, SIAM Journal on Scientific Computing, vol.14, issue.2, pp.389-405, 1993. ,
DOI : 10.1137/0914024
Ecoulements monophasiques en milieu poreux, Rev, pp.1471-1509, 1967. ,
Modeling Fractures and Barriers as Interfaces for Flow in Porous Media, SIAM Journal on Scientific Computing, vol.26, issue.5, pp.1667-1691, 2005. ,
DOI : 10.1137/S1064827503429363
URL : https://hal.archives-ouvertes.fr/inria-00071735
Numerical simulation of fracture flow with a mixed-hybrid FEM stochastic discrete fracture network model, Computational Geosciences, vol.31, issue.3, pp.217-234, 2004. ,
DOI : 10.1007/s10596-005-0152-3
A level set method for inverse problems, Inverse Problems, vol.17, issue.5, pp.1327-1356, 2001. ,
DOI : 10.1088/0266-5611/17/5/307
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.645.100
The Multiscale Analysis of Saturated Flow in Porous Media with an Adjacent Thin Channel, 2011. ,
The narrow fracture approximation by channeled flow, Journal of Mathematical Analysis and Applications, vol.365, issue.1, pp.365-320, 2010. ,
DOI : 10.1016/j.jmaa.2009.10.042
Identifiability problems of defects with the Robin condition, Inverse Problems, vol.25, issue.5, pp.55007-55019, 2009. ,
DOI : 10.1088/0266-5611/25/5/055007
A mixed hybrid Mortar method for solving flow in discrete fracture networks, Applicable Analysis, vol.19, issue.10, pp.1629-1643, 2010. ,
DOI : 10.1137/S1064827503429363
URL : https://hal.archives-ouvertes.fr/insu-00681662
Performance of fractured oil reservoirs, Bull. Amer. Assoc. Petrol. Geologists, vol.37, pp.232-244, 1953. ,
A mixed finite element method for 2-nd order elliptic problems, Mathematical aspects of finite element methods, pp.292-315, 1977. ,
A mixed-dimensional finite volume method for multiphase flow in fractured porous media, Adv. Water Resources, pp.29-1020, 2006. ,
Mixed and hybrid methods, in Handbook of Numerical Analysis, pp.523-639, 1991. ,
Recovering an obstacle and a nonlinear conductivity from Cauchy data, Inverse Problems, vol.24, issue.5, pp.55015-55027, 2008. ,
DOI : 10.1088/0266-5611/24/5/055015
Flow and Transport in Porous Media and in Fractured Rock : From Classical Methods to Modern Approaches, Second, Revised and Enlarged Edition, 2011. ,
DOI : 10.1002/9783527636693
Problèmes de perturbations liés aux phénomènes de conduction à travers des couches minces de grande résistivité, J. Math. Pures Appl, issue.9, pp.53-251, 1974. ,
An efficient multi-point flux approximation method for Discrete Fracture???Matrix simulations, Journal of Computational Physics, vol.231, issue.9, pp.231-3784, 2012. ,
DOI : 10.1016/j.jcp.2012.01.023
Stability for the Determination of Unknown Boundary and Impedance with a Robin Boundary Condition, SIAM Journal on Mathematical Analysis, vol.42, issue.6, pp.2922-2943, 2010. ,
DOI : 10.1137/100788343
Inverse problems in groundwater modeling, 1994. ,
DOI : 10.1007/978-94-017-1970-4
A model for conductive faults with non-matching grids, Computational Geosciences, vol.81, issue.6, pp.1-20, 2011. ,
DOI : 10.1007/s10596-011-9267-x
Modelling of fluide flow in fractured porous media by the ingular integral equations method / Modélisation des écoulements dans des milieux poreux fissurés par la méthode des équations intégrales singulières, theses, 2012. ,
The Behavior of Naturally Fractured Reservoirs, Society of Petroleum Engineers Journal, vol.3, issue.03, pp.245-255, 1963. ,
DOI : 10.2118/426-PA