The Virtual Element Method for Discrete Fracture Network flow - PowerPoint PPT Presentation

Intro Model VEM Conforming Mortar Transport References The Virtual Element Method for Discrete Fracture Network flow simulations Stefano Berrone Dipartimento di Scienze Matematiche “ Giuseppe Luigi Lagrange ” Politecnico di Torino stefano.berrone@polito.it joint work with Matias Benedetto, Andrea Borio, Sandra Pieraccini, Stefano Scial` o Georgia Tech Atlanta, October 26th, 2015 Stefano Berrone Polytopal Element Methods in Mathematics and Engineering 1

Intro Model VEM Conforming Mortar Transport References Discrete fracture network and flow model: 3D network of intersecting fractures in the rock matrix Fractures are represented as planar polygons The flow only /mainly occurs in the fractures, i.e. Rock matrix is considered impervious Flow modeled by Darcy law in the fractures Flux balance and hydraulic head continuity imposed across Figure : Example of DFN fracture intersections ( traces ) No changes on the given DFN geometry stocastically generated (position, orientation, size, shape) → uncertainty quantification Many approaches in literature require some geometry modification Complex domain: difficulties in good quality mesh generation Stefano Berrone Polytopal Element Methods in Mathematics and Engineering 2

Intro Model VEM Conforming Mortar Transport References Fracture model Insulated fracture formulation find H ∈ H 1 D ( F ) such that: (K ∇ H, ∇ v ) = ( q, v ) 2 (Γ N ) , ∀ v ∈ V =H 1 + � G N , v | Γ N � 0 , D ( F ) H − 1 1 2 (Γ N ) , H H is the hydraulic head on the fracture F ; K is the fracture transmissivity tensor: a symmetric and uniformly positive definite tensor containing the hydrological properties; n t K ∇ H = G N is the outward co-normal derivative of the ∂H ν = ˆ ∂ ˆ hydraulic head and ˆ n the unit outward vector normal to the boundary Γ N . Stefano Berrone Polytopal Element Methods in Mathematics and Engineering 3

Intro Model VEM Conforming Mortar Transport References Fracture model Coupled fracture formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S . . . . . . . . . . . . . . . . . . . . . . � � � � . . . . . . . . . . . . . . . . . ∂H 1 . . . . . . . . . U S . . . . . . . . . . . . 1 = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F 1 . . . . . . ν 1 . . . . . . . . . . . . . . . . . . . . . . . ∂ ˆ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S S . . . . . . . . . . . . . . . . . . . . . . . . . S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ω . . . . . . . . . . . . . . . . . . . . . . . � � � � . . . . . . . . . . . . . . ∂H 2 U S 2 = . . . . . . . . . . . . . . . . . . . . . . . . . . ν 2 . . . . . ∂ ˆ . . . . . . . . . . . . . . . . . . . . . . . S . . . . . S . . . . . . . . . . . . . . . . . . . . . . . . . . . . S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . For each trace S ∈ S on the fracture F i i = 1 , . . . , N , let use denote by � � ∂H i � � S ⊆ H − 1 U S U S 2 ( S ) i := i ∈ U ν i ∂ ˆ S S the flux entering in the fracture through the trace S , and U i ∈ U S i the tuple of fluxes U S i ∀ S ∈ S i Stefano Berrone Polytopal Element Methods in Mathematics and Engineering 4

Intro Model VEM Conforming Mortar Transport References Fracture model Full fracture formulation Solving ∀ i ∈ I the problem: find H i ∈ H 1 D ( F i ) and U i ∈ S i such that: (K i ∇ H i , ∇ v ) = ( q i , v ) + � U i , v | S i � U S i , U S i ′ 2 (Γ iN ) , ∀ v ∈ V i =H 1 + � G iN , v | Γ iN � 0 , D ( F i ) H − 1 1 2 (Γ iN ) , H with additional conditions H i | S − H j | S = 0 , for i, j ∈ I S , ∀ S ∈ S , U S i + U S = 0 , for i, j ∈ I S , ∀ S ∈ S , j provides the hydraulic head H ∈ V = H 1 D (Ω) . H i is the hydraulic head on the fracture F i , H is the hydraulic head on Ω ; K i is the fracture transmissivity tensor: a symmetric and uniformly positive definite tensor containing the hydrological properties; ∂H i n t ν i = ˆ i K i ∇ H i = G iN is the outward co-normal derivative of the ∂ ˆ hydraulic head and ˆ n i the unit outward vector normal to the boundary Γ iN . Stefano Berrone Polytopal Element Methods in Mathematics and Engineering 5

Intro Model VEM Conforming Mortar Transport References Fracture model Totally/Partially conforming meshes Figure : Totally conforming mesh Figure : Partially conforming mesh Stefano Berrone Polytopal Element Methods in Mathematics and Engineering 6

Intro Model VEM Conforming Mortar Transport References Fracture model non-conforming meshes: 120 fractures, 256 traces DFN Figure : Non-conforming mesh on a 120 fracture DFN A reformulation as a PDE constrained optimization problem allows non conforming meshes. Stefano Berrone Polytopal Element Methods in Mathematics and Engineering 7