Smeared Fractures: a promising approach to model transfers in fractured media Andre Fourno, Grenier Christophe, Fred Delay, Emmanuel Mouche, Hakim Benabderrahmane.. To cite this version:[r]

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 [15], **a** first reduced **model** has been proposed for flow **in** **fractured** porous **media** **in** the case of very permeable fracture **in** [1]. Later on, **a** generalization of this **model** considering also the case of fracture with low permeability has been proposed **in** [21], where the fracture can be seen as geological barrier. This **approach** has been also used **to** modeling the two phase flow **in** [13] and [17], 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 **media** **in**[26] and [13]. Finally porous **media** with **a** networks of fracture was also treated **in** [11] and [25].

En savoir plus
117
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 **fractures** **in** the frame of the DFMM **model** is **a** valid **approach** **to** 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

En savoir plus
217 En savoir plus

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** [54], 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** [69] 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** [50], 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 [37]) is extended **to** two phase Darcy flow **in** **fractured** **media** **in** [48]. 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 [11]. **In** [62], **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** [59] **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** [16], 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.

En savoir plus
136 En savoir plus

♦ Transport problem
Two distinct transport modelling strategies were implemented for simulating transport **in** **a** **fractured** rock with **a** **smeared** 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 **transfers** **in** **a** low permeability zone hosting **a** waste storage. These regimes include purely diffusive process for the matrix blocks

En savoir plus
simulations based on **a** mechanistic **model** **to** 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 **approach** **to** 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.

En savoir plus
145 En savoir plus

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 **to** **a** 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 **in** **a** 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 **in** **fractured** porous **media** for which the **fractures** are modeled as **in**-

En savoir plus
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

En savoir plus
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 **to** **a** 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-

En savoir plus
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.

En savoir plus
219 En savoir plus

ABSTRACT
**A** **promising** way **to** **model** 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 [1], 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 **to** **model** damage mechanisms (matrix cracking, debonding, fiber break and delamination) **in** **a** composite material ii) deal with impact applications using the DEM. All developments are made **in** the home made software GRANOO (GRANular Objet Oriented) [2]. The capability of the DEM **to** **model** matrix cracking, debonding and fiber break is first demonstrated on **a** so-called representative elementary volume (REV) made of **a** fiber flooded **in** **a** 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.

En savoir plus
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 **to** **a** 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.

En savoir plus
The discretization of the hybrid dimensional Darcy flow **model** with continuous pressures has been the object of several works. **In** [19] **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** [30], [28], and [2] for discontinuous pressure models. **In** [3], **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** [25] and [24]. **A** MFE discretization adapted **to** non-matching fracture and matrix grids is also studied **in** [10].

En savoir plus
Modeling streaming potential in porous and fractured media, description and benefits of the effective excess charge density approach.. Arkoprovo Biswas, Shashi Prakash Sharma..[r]

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 **to** **model** 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”.

En savoir plus
[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.

En savoir plus
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.

En savoir plus