Numerical and stochastic models of flow Pichot, B. Poirriez in 3D - PowerPoint PPT Presentation

Numerical and stochastic models of flow in 3D Discrete Fracture Networks J-R. de Dreuzy , J. Erhel , G. Numerical and stochastic models of flow Pichot, B. Poirriez in 3D Discrete Fracture Networks Outline Introduction Problem J-R. de Dreuzy 1 , J. Erhel 2 , G. Pichot 2 , B. Poirriez 3 description Derivation of the linear 1 CNRS, G´ eosciences Rennes, 2 INRIA, Rennes, 3 INSA, IRISA, Rennes system Linear solvers Domain Special Semester on Multiscale Simulation and Analysis in Energy and Decomposition method the Environment Conclusion Linz, October 2011

Context of the SAGE team Numerical and common team to INRIA and IRISA in Rennes, Brittany stochastic models of flow http ://www.irisa.fr/sage in 3D Discrete Fracture MICAS project (funded by ANR) Networks collaboration with Geosciences Rennes, University of Lyon, University of J-R. de Dreuzy , J. Erhel , G. Poitiers, University of Rennes. Pichot, B. Poirriez study of flow in fractured media and fractured porous media Outline study of flow and transport in heterogeneous porous media Introduction uncertainty quantification : random transmissivty fields and random networks Problem of fractures description software platform H2OLab, with libraries MPFRAC and PARADIS Derivation of the linear Some other related projects system GEOFRAC, with Estime team at Inria Rocquencourt, on fractured porous Linear solvers media Domain Decomposition GRT3D, global reactive transport model, with Andra and Momas group method (nuclear waste disposal) Conclusion H2OGuilde, a user interface for H2OLab Hydromed, with Morocco and Tunisia, on uncertainty quantification and inverse problems Arphymat, with Archeosciences Rennes and IPR, on prehistoric fires

Outline Numerical and stochastic models of flow in 3D Discrete Problem description 1 Fracture Networks Geometry J-R. de Dreuzy Flow equations , J. Erhel , G. Pichot, B. Poirriez Derivation of the linear system 2 Outline Mixed Hybrid Finite Element Method Introduction Mortar method Problem description Derivation of Linear solvers 3 the linear system Linear solvers Domain Decomposition method 4 Domain Domain decomposition of a DFN Decomposition method PCG applied to the Schur complement Conclusion Results with Schur Conclusion 5

Stochastic Generation of DFN Numerical and stochastic models of flow in 3D Discrete Fracture Following a Discrete Fracture Network approach, Networks fractures are planes with the following random properties : J-R. de Dreuzy , J. Erhel , G. Pichot, B. Poirriez Outline Parameter Random distribution Introduction length power law Problem shape disks / ellipses description Geometry position uniform Flow equations orientation uniform Derivation of Conductivity homogeneous the linear system / correlated log-normal Linear solvers Domain Decomposition method Example of DFN with 217 fractures Conclusion

Stochastic Generation of DFN Numerical and stochastic The broad natural fracture length distribution is modeled by a power law models of flow distribution (Bour et al, 2002) : in 3D Discrete Fracture Networks l − a 1 p ( l ) dl = dl , J-R. de Dreuzy l − a +1 a − 1 , J. Erhel , G. min Pichot, B. where p ( l ) dl is the probability of observing a fracture with a length in the interval Poirriez [ l , l + dl ], l min is the smallest fracture length, and a is a characteristic exponent. Outline Introduction Problem description Geometry Flow equations Derivation of the linear system Linear solvers Domain Decomposition method Conclusion

Flow model Current assumptions : Numerical and stochastic The rock matrix is impervious : flow is only simulated in the fractures, models of flow in 3D Discrete Study of steady state one phase flow, Fracture Networks There is no longitudinal flux in the intersections of fractures. J-R. de Dreuzy , J. Erhel , G. Flow equations within each fracture Ω f : Pichot, B. Poirriez ∇ · u ( x ) = f ( x ) , for x ∈ Ω f , Outline u ( x ) = −T ( x ) ∇ p ( x ) , for x ∈ Ω f , Introduction p ( x ) = p D ( x ) , on Γ D ∩ Γ f , Problem description u ( x ) . ν = q N ( x ) , on Γ N ∩ Γ f , Geometry Flow equations u ( x ) . µ = 0 , on Γ f \{ (Γ f ∩ Γ D ) ∪ (Γ f ∩ Γ N ) } , Derivation of the linear system ν (resp. µ ) outward normal unit vectors Linear solvers T ( x ) a given transmissivity field (unit [ m 2 . s − 1 ]), f ( x ) ∈ L 2 (Ω f ) sources/sinks. Domain Decomposition method Continuity conditions in each intersection : Conclusion p k , f = p k , on Σ k , ∀ f ∈ F k , � u k , f . n k , f = 0 , on Σ k , f ∈ F k with F k the set of fractures with Σ k (the k-th intersections) on the boundary,

Conforming mesh at the intersections Mixed-Hybrid Finite Element Method (MHFEM) for DFNs Numerical and stochastic Ref. J. Erhel et al., SISC, Vol. 31, No. 4, pp. 2688-2705, 2009 models of flow in 3D Discrete Makes it easy to deal with complex geometry ; Fracture Networks Conforming mesh at the fracture intersections ; J-R. de Dreuzy , J. Erhel , G. A linear system with only trace of pressure unknowns : Pichot, B. Poirriez AΛ = b , Outline with A a symmetric positive definite matrix, the flux at the edges and the Introduction mean pressure are then easily derived locally on each triangle. Problem Specific mesh generation : description Derivation of A first discretization of boundaries and 1 the linear intersections is done in 3D by using elementary system cubes. Mixed Hybrid Finite Element Method The discretization of the boundaries and 2 Mortar method intersections within the fracture f is obtained by Linear solvers a projection of the previous voxel discretization Domain within the fracture plane. Decomposition method Some local corrections are made to ensure 3 Conclusion topological properties. Once the borders and intersections are 4 discretized, a 2D mesh of each fracture is generated, using triangular elements.

Channelling in large fractures Numerical and stochastic models of flow in 3D Discrete In DFN, flow is highly channelled = an opportunity to reduce the number of Fracture Networks unknowns and the computational cost, by using a non conforming mesh at J-R. de Dreuzy intersections. , J. Erhel , G. Pichot, B. Poirriez Outline Introduction Problem description Derivation of the linear system Mixed Hybrid Finite Element Method Mortar method Linear solvers Domain Decomposition method How to apply a mortar method [Arbogast et al., SINUM 37(4) (2000)] to DFN ? Conclusion

Mixed-Hybrid Mortar Finite Element Method applied to DFN Numerical and Mixed-Hybrid Mortar Finite Element Method (MHMFEM) for DFNs stochastic models of flow Ref. G. Pichot et al., Applicable Analysis, Volume 89, 1629-1643, 2010 in 3D Discrete Ref. G. Pichot et al., SISC, In revision Fracture Networks A method to mesh the fractures independently and to refine the chosen fractures J-R. de Dreuzy , J. Erhel , G. using a posteriori estimators. Pichot, B. Poirriez Same advantages as MHFEM Outline A simple mesh generation Introduction A reduced number of unknowns while keeping a solution of good quality Problem a complex numerical method with Mortar conditions description Derivation of A new specific mesh generation : the linear For each fracture f, choose a mesh step, then system Mixed Hybrid A first discretization of boundaries and 1 Finite Element Method intersections is done in 2D, by using elementary Mortar method squares, it leads to a stair-case like Linear solvers discretizations. Domain Decomposition Some local corrections are made to ensure some 2 method topological properties. Conclusion Once the borders and intersections are 3 discretized, a 2D mesh of each fracture is generated, using triangular elements.

Meshing procedure : Example Numerical and stochastic models of flow in 3D Discrete Fracture Networks J-R. de Dreuzy , J. Erhel , G. Pichot, B. Poirriez Outline Introduction Problem description Derivation of the linear system Mixed Hybrid Finite Element Method Mortar method Linear solvers Domain Decomposition method Conclusion The discretization of intersections is non matching ⇒ Mortar conditions are required to ensure the continuity of heads and fluxes.

Mortar principle Numerical and stochastic models of flow Mortar method principle : in 3D Discrete Fracture Networks It consists in choosing arbitrarily for each intersection a master fracture ( m ) and a J-R. de Dreuzy slave fracture ( s ). , J. Erhel , G. Pichot, B. Poirriez Outline Introduction Problem description Derivation of the linear system Mixed Hybrid Finite Element Method Mortar method Linear solvers Domain Decomposition Classical Mortar approach : each edge is either master or slave method Specific Mortar method : some edges have several master or slave properties Conclusion