Adaptation of a mortar method to model flow fractured media G - PowerPoint PPT Presentation

Adaptation of a mortar method to model flow in large-scale Adaptation of a mortar method to model flow fractured media G´ eraldine in large-scale fractured media Pichot, Jocelyne Erhel , Jean-Raynald De Dreuzy Outline eraldine Pichot 1 , Jocelyne Erhel 2 , G´ Meshing process and local Jean-Raynald De Dreuzy 1 corrections Mortar MHFEM Implementation 1 CNRS, UMR6118 G´ eosciences Rennes, France features and simulations 2 Inria Rennes, France Conclusions Scaling up and Modeling for Transport and Flow in Porous Media October 15, 2008

Motivation The simulation of the flow in Discrete Fracture Networks (DFNs). Adaptation of a mortar method to model flow in large-scale Fracture network characteristics : fractured media Many fractures intersecting each other ( ≈ 10 4 fractures, ≈ 10 5 G´ eraldine Pichot, Jocelyne intersections), Erhel , Jean-Raynald De Dreuzy Fractures with broad ranges of length, shape, orientation and position ⇒ A stochastic discrete approach to model fractures Outline A set of 2D domains (fractures) intersecting each other. Meshing process and local corrections Mortar MHFEM Implementation features and simulations Conclusions 30 fractures/125 intersections 8064 fractures/12943 intersections

Some assumptions : Adaptation of a mortar method The rock matrix is impervious : flow is only simulated in the fractures, to model flow in large-scale fractured media Study of steady state flow, G´ eraldine There is no longitudinal flux in the intersections of fractures. Pichot, Jocelyne Erhel , Jean-Raynald De Dreuzy Numerical method : a Mixed Hybrid Finite Element Method Makes it easy to deal with complex geometry (triangular elements) ; Outline Meshing process A linear system with only trace of pressure unknowns, the flux at the and local corrections edges and the mean pressure are then easily derived locally on each Mortar MHFEM triangle. Implementation features and simulations Two main difficulties : Conclusions Classical mesh generation can be insufficient due to the amount of 1 intersections between fractures (FE with bad aspect ratio), e.g. success in only 222 networks for 1620 generated networks ⇒ Local corrections are required, J. Erhel et al., submitted 2008 Matching grids at the intersection can be very costly (e.g. consider a 2 small fracture with a fine mesh intersecting a large one).

Some assumptions : Adaptation of a mortar method The rock matrix is impervious : flow is only simulated in the fractures, to model flow in large-scale fractured media Study of steady state flow, G´ eraldine There is no longitudinal flux in the intersections of fractures. Pichot, Jocelyne Erhel , Jean-Raynald De Dreuzy Numerical method : a Mixed Hybrid Finite Element Method Makes it easy to deal with complex geometry (triangular elements) ; Outline Meshing process A linear system with only trace of pressure unknowns, the flux at the and local corrections edges and the mean pressure are then easily derived locally on each Mortar MHFEM triangle. Implementation features and simulations Two main difficulties : Conclusions Classical mesh generation can be insufficient due to the amount of 1 intersections between fractures (FE with bad aspect ratio), e.g. success in only 222 networks for 1620 generated networks ⇒ Local corrections are required, J. Erhel et al., submitted 2008 Matching grids at the intersection can be very costly (e.g. consider a 2 small fracture with a fine mesh intersecting a large one).

Our objective : Adaptation of a mortar method to model flow in large-scale Allowing independent mesh generation within the fractures fractured media ⇒ Non matching grid at the intersections between fractures G´ eraldine Pichot, Jocelyne Erhel , Jean-Raynald De The challenge : Dreuzy Implementation of a Mortar method for each intersection Outline between fractures to ensure continuity of the flux and trace of Meshing process and local pressure at the intersections, corrections Mortar MHFEM The number of fractures and intersections can be large so that Implementation we have to deal with numerous cases of non matching grids, features and simulations Handling the large variety of configurations leading to numerical Conclusions difficulties. Mortar method : Bernardi, Maday et Patera, 1992 ; Arbogast, Cowsar, Wheeler et Yotov, 2000

Adaptation of a mortar method to model flow in large-scale fractured media G´ eraldine Pichot, Jocelyne Erhel , Meshing process and local corrections 1 Jean-Raynald De Dreuzy Outline Meshing process and local Mortar MHFEM 2 corrections Mortar MHFEM Implementation features and simulations Implementation features and simulations 3 Conclusions

Creating 3D Discrete Fracture Networks (DFNs) and meshing Adaptation of a A software included in the scientific platform Hydrolab (written in C + +) mortar method to model flow in ( Erhel et al., 2007 ) : large-scale fractured media Allows the generation of random DFNs in a 3D domain, G´ eraldine Pichot, Jocelyne Includes a mesh generator for general DFN structures (using a Erhel , Jean-Raynald De procedure extracted from the software FreeFem++) Dreuzy Includes projection of the intersections on a regular grid (H. Outline Mustapha, PhD, 2005) and local corrections to remove configurations Meshing process and local that would lead to triangles with bad aspect ratio, ( J. Erhel, J.R. De corrections Dreuzy and B. Poirriez, submitted 2008 ) Mortar MHFEM Implementation features and simulations Conclusions

Flow equations in each fracture Ω f Adaptation of a mortar method ∇ · u = f ( x ) for x ∈ Ω f to model flow in large-scale fractured media u = −K ( x ) ∇ p ( x ) for x ∈ Ω f G´ eraldine Pichot, Jocelyne Erhel , p ( x ) = p D ( x ) on Γ D Jean-Raynald De Dreuzy u ( x ) .ν = q N ( x ) Outline on Γ N Meshing process and local corrections u ( x ) .µ = 0 on Γ f , Mortar MHFEM ν (resp. µ ) outer normal unit vector to the cube edge (resp.fracture side) ; Implementation K ( x ) is a given 2D permeability field ; f ( x ) ∈ L 2 (Ω f ) represents the features and simulations sources/sinks ; Conclusions + Continuity conditions at each intersection : � p k , h = p k , on Σ k , ∀ f ∈ F k u k , f . n k , f = 0 on Σ k , and f ∈ F k where p k , h is the trace of pressure and n k , f is the normal unit vector on the boundary Σ k of the fracture Ω f , Σ k is the k-st intersection, F k is the set of fractures with Σ k as intersection.

On a simple example with two fractures Adaptation of a mortar method to model flow in large-scale fractured media G´ eraldine Pichot, Jocelyne Erhel , Jean-Raynald De Dreuzy Outline Meshing process and local corrections Mortar MHFEM Implementation Geometry : features and simulations A cubic domain Ω=[0 , L ]x[0 , L ]x[0 , L ], Conclusions Two fractures Ω 1 and Ω 2 , with Γ = Ω 1 ∩ Ω 2 . Ω 1 and Ω 2 independently meshed (mesh step in Ω 1 : 0.08 ; in Ω 2 : 0.2) Choice of a master intersection side (e.g. domain 1) and a slave intersection side (e.g. domain 2) Remark : things will be more complicated with many fractures intersecting each other ...

Weak formulation of the problem Adaptation of a P d ( K ) space of polynoms of total degree d defined on K , K ∈ T h : mortar method to model flow in large-scale RT 0 ( K ) = { s ∈ ( P 1 ( K )) 2 , s = ( a + b x 1 , c + b x 2 ) , a , b , c ∈ R } fractured media G´ eraldine Pichot, Jocelyne RT 0 ( T h ) = { φ ∈ L 2 (Ω) , φ | K ∈ RT 0 ( K ) , ∀ K ∈ T h } Erhel , Jean-Raynald De Dreuzy We also need a space M 0 ( T h ) defined as : Outline Meshing process M 0 ( T h ) = { ϕ ∈ L 2 (Ω) , ϕ | K ∈ P 0 ( K ) , K ∈ T h } and local corrections E h , in set of edges of the two meshes not belonging to Γ, Mortar MHFEM Implementation E G h , m (resp. E G h , s ) : edges belonging to Γ on the master (resp. slave) features and simulations side Conclusions E h = E h , in ∪ E G h , m ∪ E G h , s . We define the multiplier spaces N 0 ( E h ) = { λ ∈ L 2 ( E h ) , λ | E ∈ P 0 ( E ) , ∀ E ∈ E h } N 0 g , D ( E h ) = { λ ∈ N 0 ( E h ) , λ = g on Γ D }

Weak MH Mortar formulation Find ( u h , p h , tp h ) ∈ RT 0 ( T h ) x M 0 ( T h ) x N 0 p D , D ( E h ) such that : Adaptation of a mortar method to model flow in large-scale � K − 1 u h . χ h d x + fractured media G´ eraldine Ω � Pichot, Jocelyne � K p h ∇ · χ h d x , ∀ χ h ∈ RT 0 ( T h ) , � � tp h χ h .ν K dl = Erhel , K ∈T h Jean-Raynald De ∂ K Dreuzy K ∈T h � Ω f ϕ h d x , ∀ ϕ h ∈ M 0 ( T h ) , � ∇ · u h .ϕ h d x = Outline Ω Meshing process � � � ∂ Ω q N λ h dl , ∀ λ h ∈ N 0 and local = 0 , D ( E h , in ) u h .ν K λ h dl corrections ∂ K K ∈T h Mortar MHFEM Implementation � � features and � � E ′ u h .ν E ′ η h , ∀ η h ∈ M m u h .ν E η h = − h , simulations E E ′ ∈E G E ∈E G Conclusions h , m h , s � � � � E ′ tp h β h .ν E ′ dl , ∀ β h ∈ M s tp h β h .ν E dl = h . E E ′ ∈E G E ∈E G h , m h , s h = N 0 ( E G with M m h , m ) and M s h the space spanned by the local RT basis functions on the slave element sides.