the fractures are included individually in the model. However, because they are so thin they are treated as interfaces. Such models are called discrete models (since the fractures are taken into account individually and not averaged) or reduced models (since the fractures are modeled as surfaces of co-dimension 1). This type of model has been much studied in the mathematical and engineering literature for many types of flow problems and for many different types of numerical schemes. Using the techniques of domain decomposition , a first reduced model has been proposed for flow infractured porous mediain the case of very permeable fracture in . Later on, a generalization of this model considering also the case of fracture with low permeability has been proposed in , where the fracture can be seen as geological barrier. This approach has been also used to modeling the two phase flow in  and , where Darcy’s law is still applicable separately for each phase and coupling conditions were added. This model has been extended to the case of flow and transport in porous mediain and . Finally porous media with a networks of fracture was also treated in  and .
computational challenges of traditional techniques, we develop in this work a GSA based on the non-intrusive PCE. In particular, we apply an efficient sparse technique to construct the PCE with a reduced number of model evaluations, based on Kashyap information criterion. In the literature, GSA has been recently applied to SWI but previous studies are limited to homogeneous domain. Two configurations of the fractured Henry Problem, dealing with a single horizontal fracture (SHF) and a network of orthogonal fractures (NOF), are considered as conceptual models. The simulations required to construct the PCE are performed using a finite element model developed in the framework of COMSOL software. Boussinesq approximation is implemented to improve the computational efficiency of the COMSOL model. From technical point of view, this work shows several novelties that are important for the simulation of SWI. It shows the ability of COMSOL to accurately simulate SWI in simple and fractured aquifers. It also proves that the dimension reduction of fracturesin the frame of the DFMM model is a valid approachto simulate SWI in FCAs and confirms the validity of the Boussinesq approximation in such a case. Regarding uncertainty analysis, this study presents an efficient (low cost) methodology to understand uncertainty propagation into SWI models. This methodology is generic and can be efficiently applied to real field investigations. In hydrogeological applications, GSA is often applied to investigate uncertainty propagation associated with hydrogeological parameters. This work shows that GSA is generic and can be a valuable tool for different kinds of uncertainties. The GSA results showed that, for the SHF configuration, the uncertainty associated with the fracture hydraulic conductivity and depth is the first sources of uncertainty on the salinity distribution. The spatial distributions of the SIs are given as maps. This represents an important feature of this study as these maps are not only important for uncertainty analysis but also provide relevant locations for measurement required for aquifer characterization. Fracture hydraulic conductivity and depth are also important parameters for the toe position ( L toe ), thickness of the freshwater discharge zone ( ) Z I , the mass of salt persisting in the aquifer ( M S ) and the flux of saltwater entering the aquifer
The discretization of hybrid-dimensional Darcy flow models has been the object of several works. For an exhaustive review of existing methods, we refer to [65, 68]. For two-phase Darcy flow, a cell-centred Finite Volume scheme using a Two Point Flux Ap- proximation (TPFA) is proposed in , assuming the orthogonality of the mesh and isotropic permeability fields. Cell-centred Finite Volume schemes can be extended to gen- eral meshes and anisotropic permeability fields using MultiPoint Flux Approximations (MPFA) following the ideas introduced in  for discontinuous pressure models. Nev- ertheless, 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 to nodal discretizations on tetrahedral meshes. In , the two-phase flow equations are solved in an IMPES framework, using a Mixed Hybrid Finite Element (MHFE) dis- cretization for the pressure equation and a Discontinuous Galerkin discretization of the saturation equation. The paper also contains a review on the most common numerical approaches, when dealing with discrete fractures. The Hybrid Finite Volume discretiza- tion (HFV, see ) is extended to two phase Darcy flow infracturedmediain . These approaches are adapted to general meshes and anisotropy but require as many degrees of freedom as faces. An early paper to use a Control Volume Finite Element method (CVFE) for the discretization of hybrid-dimensional two-phase flow is . In , a CVFE scheme is proposed that use reconstruction operators for the saturations that de- pend on the rock characteristic capillary pressure curves. In this way, the saturation jumps (due to discontinuous capillary pressure) at the material interfaces are respected. A similar approach can be found in  in phase pressure formulation. However, the rigid choice of the control volumes (that are the dual cells) leads to the need of small matrix cells at the DFN neighbourhood, in order not to enlarge the drain artificially. In , the VAG scheme is used, which is very flexible in the distribution of control volumes and hence circumvents this problem. To the author’s knowledge, there has not yet appeared a comparison of different hybrid-dimensional models with the generic equi-dimensional model for two phase flow. This is one of the achievements of the present chapter.
♦ Transport problem
Two distinct transport modelling strategies were implemented for simulating transport inafractured rock with asmeared fracture approach: a Lagrangian method and an Eulerian one.
The Lagrangian method is not well suited to the actual flow field for the smeared fracture approach. Indeed, for Y-type meshes, the variability of Darcy velocity over the whole grid is very large, leading to extremely high and low transit times depending on the location of the particle entering the mesh. This difficulty was circumvented by constraining transport paths to controlled flow lines. This is achieved for a release position at the centre of the element edge leading for X and Y type elements to exit positions similarly located at the centre of the edges. Associated travel times are easily computed and yield values of the equivalent porosity consistent with volume ratios. This method was implemented in Cast3M for pure advection allowing for the computation of streamlines. It is computer efficient and was tested with success as regard reference simulations. It requires a theoretical derivation of equivalent porosity (based on summation of transit times associated with N X X-type meshes and N Y Y-type meshes along a single fracture) as well as slight modifications of the particle tracking procedure available in the Cast3M code. However, this approach cannot be extended to other processes such as a diffusion because particles would move to other flow lines than the one selected. Thus, the Lagrangian approach is limited to purely advective transport in the fracture network. The Eulerian approach has more potential in this situtation and better fits into the expected transport regimes for long-term transfersina low permeability zone hosting a waste storage. These regimes include purely diffusive process for the matrix blocks
a parametrization of the capillary pressure graphs. The thermodynamical equilibrium is formulated using complementary constraints and taking into account the saturation jumps at mf interfaces. The hybrid-dimensional model is discretized using a TPFA combined with a redistribution of the porous volume both at mf interfaces and at edges shared by more than three fracture faces. The hybrid-dimensional model is shown to provide basically the same accuracy as the reference equi-dimensional model and to be more accurate and physi- cally consistent than the usual approach based on harmonic averaging of the transmissivities at mf interfaces combined with a two-point upwinding of the mobilities jumping over the mf interfaces. This hybrid-dimensional model is used to investigate the desaturation of afractured Callovo-Oxfordian argilite at the interface with a ventilation tunnel. The Fickian diffusion in the gas phase and in the fracture width is shown to remove the barrier effect induced by the gas filled fractures. In perspective, the discretization will be extended to account for general meshes and anisotropy of the permeability tensor using a face based approachin the spirit of .
fracturesin the disposal areas due to, for example, tunnel excavation, can drastically accelerate the migration process of radionuclides.
A common feature of fracturesin porous media is the variety of length scales. While the presence of smaller fractures may be accounted for by using homogenization or other upscaling techniques, fractures with larger extension have to be modelled explicitly, and there are several possible ways to incorporate their presence. Our focus is here on the approach developed in [29 29 ], where a reduced model for the flow in the fracture is obtained by an averaging process, and the fracture is treated as an interface inside the bulk region. The fracture is assumed to be filled of debris, so that the flow therein can still be modelled by Darcy’s law. The problem is closed by interface conditions that relate the average and jump of the bulk pressure to the normal flux and pressure in the fracture. In [15 15 ] we have designed and analysed a Hybrid High-Order (HHO) method to discretize this model, and proved stability and order O(h k +1 ) convergence of the discretization error measured in an energy-like norm,
simulations based on a mechanistic modelto predict the formation and propagation of fractures and validated their results with experimental observations. Different numerical methods have been used to generate DFN’s inspired from geomechanics and based on linear elastic fracture mechanics. The process of fracture generation is performed in four main steps in an iterative manner [ Paluszny et al. (2009) ]: (i) generation of initial mechani- cal state of the rock to mimic the process by which natural fractures initiate from micro cracks, (ii) calculation of the perturbed stress field in the rock under imposed boundary conditions, (iii) derivation of the stress intensity factor at the tip of each fracture and (iv) propagation of fractures which satisfy a growth criterion (e.g., the criterion of sub-critical law [ Atkinson (1984) ]). Many studies used the mechanical approachto generate physically coherent fracture networks and reviews on the subject can be found in the literature (see, for instance, an extended review by [ Jing (2003) ]). Additionally, some pseudo-mechanical procedures can be used. The pseudo-mechanical methods for fracture network generation are constrained stochastic methods in which mechanical conditions for fracture formation and propagation are integrated in the stochastic generator. For example, [ Bonneau et al. (2013) ] developed a pseudo-genetic method to generate stochastic fracture models that are consistent with patterns observed on outcrops and fracture growth principles (Fig. ( 1.5 )). More recently, [ Bruel (2018) ] has developed a geo-mechanically based approach aiming at reproducing the fracture network developing in an extensional context by capturing some pattern inherited from mechanical processes.
Abstract Fully implicit time-space discretizations ap- plied to the two-phase Darcy flow problem lead to the systems of nonlinear equations, which are traditionally solved by some variant of Newton’s method. The effi- ciency of the resulting algorithms heavily depends on the choice of the primary unknowns since Newton’s method is not invariant with respect toa nonlinear change of variable. In this regard the role of capillary pressure/saturation relation is paramount because the choice of primary unknowns is restricted by its shape. We propose an elegant mathematical framework for two- phase flow in heterogeneous porous media resulting ina family of formulations, which apply to general mono- tone capillary pressure/saturation relations and handle the saturation jumps at rocktype interfaces. The pre- sented approach is applied to the hybrid dimensional model of two phase water-gas Darcy flow infractured porous media for which the fractures are modeled as in-
case : mainly from bottom to top of the domain. The initial transport conditions are : unity concentration in one mesh of the domain. Several release positions were studied corresponding to the different zones mentioned on figure 2. We focus here on the Zone 0 case, located upstream at the intersection of two main fractures (numbered 3 and 4). Globally, the plume first migrates within both conductors at different velocities, separates at following intersections and reaches outlets at the top right of the domain following different paths. This leads to dispersion process of the plume through the network. The quantity measured are breakthrough curves at the limits of the domain, concentration fields at different times, temporal evolution of the masses in the fractures and fracture sections. A picture of the concentration field at time 7.8 10 4
of the pollutant profiles.
IV. COMPARISON WITH EXPERIMENTAL RESULTS
In this section, we test the proposed random-walk model on some experimental measurements of dense contaminant transport obtained at the Physical-Chemistry Department 共DPC兲, CEA/Saclay. The experimental device, named BEETI, consists of a dichromatic x-ray source 共20–40 keV, 50–75 keV兲, applied toa vertical column of height H = 80 cm and diameter D = 5 cm 共the aspect ratio is therefore H /D=16Ⰷ1兲. The x-ray transmitted countings allow quan- titatively assessing the contaminant concentration inside the column 共as a function of time兲 at various sections ᐉ: we denote this quantity by c ᐉ 共t兲. The different positions are ex- plored by means of a remotely controlled rack rail that dis- places the x-ray emitter and the coupled NaI detector. At the exit of the column, c ᐉ=H 共t兲 coincides with the breakthrough curve, which is the most frequently measured variable in contaminant migration experiments 关 1 兴. In the specific con-
Although this research focuses on a 2D model, by unifying information obtained by analyzing images of longitudinal and transversal cross-sections of glass canisters, it is realistic to construct a realistic 3D model. This is especially appealing, since almost all tools necessary to accomplish this exercise are already prepared. First, the knowledge of the fracture pattern of a transverse plane of a fracture package can be obtained by applying the proposed workflow. Second, the data required for geostatistical modelling are identified. Moreover, the maps of the solidification front arrival times necessary to support the image analysis results can be procured easily and the way in which they should be analyzed is already known. Besides, the algorithm used to generate the anisotropic Voronoї tessellation (realized in RGeostats package) should potentially be able to do it in 3D after some proper adaptations are made. These additional implementations concern the creation of the connected 3D fracture planes. However, certain caution must be applied because the fracture network is likely to vary along the z-axis owing to the presence of the different thermo-mechanical environments such as a zone of re-liquefaction or a stress-free surface. Nevertheless, it could be interesting to create realizations of 3D equivalent tessellation based on the data obtained by analyzing two transversal cross sections and one longitudinal cross section. The transversal cross sections should belong to two different castings, in order to capture the difference of the fracture network morphology related to the two-stage manufacturing procedure.
Apromising way tomodel fracture mechanics with the use of an original Discrete Element Method (DEM) is proposed. After proving the ability of the method to capture kinetic damage induced by cracking phenomena in brittle materials such as silica , taking advantage of the method for composite materials applications is the main purpose of this work. This paper highlights recent developments to prove capabilities of the DEM and to give some answers to challenges : i) use the present DEM tomodel damage mechanisms (matrix cracking, debonding, fiber break and delamination) ina composite material ii) deal with impact applications using the DEM. All developments are made in the home made software GRANOO (GRANular Objet Oriented) . The capability of the DEM tomodel matrix cracking, debonding and fiber break is first demonstrated on a so-called representative elementary volume (REV) made of a fiber flooded ina matrix. Modelize the REV with DEM and retrieve suitable homogenized properties is the first challenge reached. Secondly, the ability of the method to capture matrix cracking, debonding and fiber break is qualitatively demonstrated through basic static simulations performed on the REV. The ongoing developments to improve are presented. Then, the Double Cantilever Beam (DCB) test using Discrete Element (DE) is investigated. Contact cohesive laws are identified from experiments and implemented in GRANOO. Simulations of DCB test using DEM are then performed. Results are discussed and ways of improvements are proposed. Finally, the ability of the DEM to simulate impact damage on textile is pointed out. Numerical investigations are based on Ha-Minh & co. Works in [3, 4] taken for reference. The weaving is exactly reproduced with DE. The contact between yarns is naturally taken into account in the DEM. The promising results are commented and the on going developments are exposed.
In this paper, we develop a simple method for ap- plying model checking without incurring the cost of modelling concurrency by interleaving. Our method yields results identical to those of methods based on interleaving semantics, it just avoids most of the as- sociated combinatorial explosion. It is quite orthog- onal tomodel checking based on partial-order logics [32, 27, 31]. Indeed, these logics are designed to be semantically more powerful. We are “only” more effi- cient. The idea that the cost of modelling concurrency by interleaving can be avoided in finite-state verifica- tion already appears in [34, 39, 40, 22]. We build upon this earlier work, specifically that of , and bring to it the full capabilities of model checking.
b(m, u) 6= 0 and for each 0 6= m ∈ M there is a u ∈ U with b(m, u) 6= 0. One
concludes that (M, U ; b) is a strict dual pair. See Marshall et al. (2011), Page 99 or Muller (2013), Page 50, for related discussions.
We are now ready to present our main result. Equivalence between statements (a) and (b) in Theorem 2 below is a full discretization of Theorem 2.4.1. provided by Muller (2013), Page 51. Indeed our distribution are discrete in the sense that we only count individuals (value in Z) in each possible discrete and finite outcome, whereas Muller (2013) considers probability measures (value in R). It follows that the sequence of transfers we consider is defined as a discrete cone, not a convex cone. Consequently, we do not have to refer toa weak closure argument for the cone we consider, as done by Muller (2013). The core argument in the proof is nevertheless the same, that is an application of the bipolar theorem but with an originality, the introduction of the notion of Hilbert basis. Notice that a closely related result – equivalence between statements (a) and (b) – can be found in Proposition 3.4, Page 237 in Marshall (1991), in the continuous case.
The discretization of the hybrid dimensional Darcy flow model with continuous pressures has been the object of several works. In  a cell-centered Finite Volume scheme using a Two Point Flux Approximation (TPFA) is proposed assuming the orthogonality of the mesh and isotropic permeability fields. Cell-centered Finite Volume schemes can be extended to general meshes and anisotropic permeability fields using MultiPoint Flux Approximations (MPFA) following the ideas introduced in , , and  for discontinuous pressure models. In , a Mixed Finite Element (MFE) method is proposed, and Control Volume Finite Element Methods (CVFE) using nodal unknowns have been introduced for such models in  and . A MFE discretization adapted to non-matching fracture and matrix grids is also studied in .
6 Conclusions and Comparison with Other Work
The closest work to the one presented here is certainly that of Valmari [Val90]. His paper also addresses the problem of adapting tomodel checking a method that avoids considering all interleavings of independent events while generating the state space of a concurrent program. It is likewise based on linear-time temporal logic, but uses a different strategy from the one we presented here. In our approach, the fact that the order of actions that appear in the formula cannot be ignored while constructing the trace automaton is handled by treating the property as any other component of the concurrent program. In [Val90], the problem is solved by a less discriminating approach. Precisely, the use of the “next” temporal operator is disallowed and all transitions that can affect the truth value of any state predicate appearing in the formula are considered as dependent. Prohibiting “next” is indeed important in this approach since in the presence of this operator all transition could potentially affect the truth value of the formula and hence would have to be considered as dependent and this would annihilate any benefit coming from the use of a partial-order approach. In our paper, we do handle the full temporal logic, and, actually, we can also handle extended temporal logics like that of [Wol83]. However, it should be noted that our interpretation of “next” is different from the one that causes problems in the method used by Valmari: we interpret “next” as meaning “next action monitored by the formula” rather than “next state of the program”.
[Glc] 3 are inducers of thaxtomin
biosynthesis, deleting an enzyme responsible for their depletion was expected to increase thaxtomin production compared to the wild-type strain. Indeed, S. scabies ∆bglC produced more thaxtomin A, but this mutant was also shown to be a constitutive producer [12.]. This unexpected phenotype is valuable because this strain is even more efficient than the ∆cebR mutant for thaxtomin production. Understanding the mechanisms responsible for this thaxtomin A overproduction in the ∆bglC mutant could help us improving production strains.
Several non-Fickian transport models have been obtained using both the MA and VATs, but these only apply to specific cases and lack generality. For instance, mobile-immobile models have been developed using HT in the case where the unit- cell contains two different domains that differ by the value of the diffusion coefficients [30, 29]. The approach relies on the constraint that these diffusion coefficients are several orders of magnitude different, which is formalized using a scaling of the ratio between both diffusion coefficients in ε. With the VAT, formulations with two equa- tions (one for each domain, and exchange terms between the two) have been obtained by treating the case of two phases connected via transmission (boundary) conditions [1, 32, 14, 13, 35, 38, 44, 20]. Such models seem to be more general as nonequilib- rium effects can be postulated as a working hypothesis without the need to identify a priori the origin of the gradients. These could originate from a particular scaling of dimensionless numbers, but also from non-linearities, boundary, initial, or reactive conditions. Another important aspect of these two-equation models obtained using VAT is that they (1) feature counterintuitive (nondiagonal) coupling terms for the advective and diffusive operators, which are not present in heuristic MRMT models, and (2) they can be easily extended to many situations such as mobile-mobile or reactive conditions, which is not necessarily the case of MRMT because of the lack of link with the microscale physics.