Flow simulation in 3D Discrete Fracture Networks (DFN)

Transcription

1 Reconstructed Fracture Network Flow simulation (Connected fractures) Flow simulation in 3D Discrete Fracture Networks (DFN) Jean-Raynald de Dreuzy, Géraldine Pichot, Patrick Laug, Jocelyne Erhel Géosciences Rennes, INRIA Stripa Fracture Structures Scales, Organization and Diversity Granite (Sweden) 1m Sandstones (Norway) 0 m Coal Cleats (Australia) 5 cm Odling, N. E. (1997), Scaling and connectivity of joint systems in sandstones from western Norway, Journal of Structural Geology, 19(), Busse, J., J.-R. de Dreuzy, S. G. Torres, D. Bringemeier, and A. Scheuermann (2016), Image processing based characterisation of coal cleat networks, International Journal of Coal Geology. Shale (US) 1m Sandstones (Norway) 1 km Energy Minerals Division; Gas shale tricky to understand Brian Cardott (EMD Gas Shale Committee member).

2 Fracture reconstruction DFN models Widely-distributed fracture lengths Stress-limited connectivity a=3.5, p=1.2 p c, L/l min =, Poissonian a=3.5, p=1.2 p c, L/l min =, UFM Single-Phase flow Assumptions Steady-state Only in fractures, impervious rock matrix Flow equation in each fracture γ ν (resp. µ) outward normal unit vectors T(x): transmissivity field [m 2.s -1 ] f(x): sources/sinks Continuity conditions at intersections Σ a=3.5, p=1.2 p c, L/l min =, Poissonian On Σ k intersection between fractures γ of F k

3 Computational domain Fracture Statistics Multiple scales of heterogeneity a=3.5 Power-law length distribution n(l)~l-a (2 a 4) Stress-induced correlations, p=1.2 =, p c, L/l min onian Poiss a=3.5 connectivity =, p c, L/l min UFM Stress-limited connectivity Widely-distributed fracture lengths connected connectivity scale Lc Fracture density, p=1.2 Heterogeneous 3 and increasing with scale to 5 fractures Fracture apertures disconnected Self-affine truncated Gaussian distribution Fracture Transmissivity T~a3 For networks with a broad fracture length distribution Fracture orientations Fisher or uniform distributions Secondary correlations Aperture-Length Aperture-Orientation Length-Density pt (T ) = ( ) (T / β )1/ 3 Γ c exp 1/ 3 2 / 3 2Γ 2 2πΓ 2 3β T a Aperture Transmissivity c=1.5 c=1.5 0 PDF Orthogonal scale L -2-2 de Dreuzy, J. R., Y. Meheust, and G. Pichot (2012), Influence of fracture scale heterogeneity on the flow properties of three-dimensional discrete fracture networks (DFN), Journal of Geophysical Research-Solid Earth, T l /T lm 2 Numerical Challenges of DFNs Multi-Scale, Robust, Efficient Multiple scales of heterogeneity MP_FRAC Few large fractures Bulk contribution of small fractures 1D channels, 2D fractures, 3D space 3 L=50 lmin Large domains to 5 fractures High heterogeneity Widely distributed transmissivities close to some critical state Topology Weak constrains for high complexity Stochastic modelling Intricate local configurations Local clustering of fractures Fracture intersecting by their tips Broad power-law length distribu4on n(l)~l-a with lmin<l<l Large number of fractures: ~2 4

4 Numerical Challenges of DFNs Multi-Scale, Robust, Efficient Multiple scales of heterogeneity Few large fractures Bulk contribution of small fractures 1D channels, 2D fractures, 3D space Large domains 3 to 5 fractures High heterogeneity Widely distributed transmissivities Topology close to some critical state Weak constrains for high complexity Stochastic modelling Intricate local configurations Local clustering of fractures Fracture intersecting by their tips Stochastically-issued intricate configurations Stress-limited connectivity Fracture-Network Decomposition Method Sparse flow structure Stripa, 1980s 2D Discretization Fracture boundaries and intersections Removes the locally intricate configurations Remains local to the fracture Mesh generation 2D in the fracture plane Standard mesh generation techniques Discretization scheme in fractures Mixed-Hybrid Finite Element Method Continuity at fracture intersections Mortar conditions Linear system solver Domain decomposition

5 Fracture-Network Decomposition Method Sparse flow structure Stripa, 1980s 2D Discretization Fracture boundaries and intersections Removes the locally intricate configurations Remains local to the fracture Mesh generation 2D in the fracture plane Standard mesh generation techniques Discretization scheme in fractures Mixed-Hybrid Finite Element Method Continuity at fracture intersections Mortar conditions Linear system solver Domain decomposition 2D Discretization and Mesh generation Non-conforming Mesh Stochastically-issued intricate configurations Does the mesh quality matter? Hoteit, H., J. Erhel, R. Mosé, B. Philippe, and P. Ackerer (2002), Numerical Reliability for Mixed Methods Applied to Flow Problems in Porous Media, Computational Geosciences, 6(2),

6 Mesh quality after discretization Quality mesh criterion Q K [0;1] for each triangle K: S K : surface of K h s : mean edge length Non-conforming Mesh Optimal triangle quality: Q K =1 Mesh quality with discretization Number of triangles: Minimum of Q K : 0.51 Mesh quality without discretization Number of triangles: Minimum of Q K : Borouchaki, H., P. Laug, and P. L. George (2000), Parametric surface meshing using a combined advancing-front generalized Delaunay approach, International Journal for Numerical Methods in Engineering, 49(1-2), Fracture-Network Decomposition Method Sparse flow structure Stripa, 1980s 2D Discretization Fracture boundaries and intersections Removes the locally intricate configurations Remains local to the fracture Mesh generation 2D in the fracture plane Standard mesh generation techniques Discretization scheme in fractures Mixed-Hybrid Finite Element Method Continuity at fracture intersections Mortar conditions Linear system solver Domain decomposition

7 Mixed-Hybrid Mortar method Non-conforming mesh For each of the intersections: arbitrary choice of master (m) and slave (s) sides Mixed-Hybrid Mortar method Continuity conditions at fracture intersections Notations Continuity conditions C: L 2 -projection from master to slave side Network scale system A: (Number of master or slave edges) -1

8 Mixed-Hybrid Mortar method Convergence Test 50 fractures, 315 intersections edges Head field Fracture-Network Decomposition Method Sparse flow structure 2D Discretization Fracture boundaries and intersections Removes the locally intricate configurations Remains local to the fracture Mesh generation 2D in the fracture plane Standard mesh generation techniques Discretization scheme in fractures Mixed-Hybrid Finite Element Method Continuity at fracture intersections Mortar conditions Linear system solver Domain decomposition Advantages High quality mesh Decouples fracture and network scales Avoids global operations before solving Enables parallelization Prepares for mesh refinement

9 Conclusions and perspectives Optimization A posteriori estimators for mesh refinement According to flow sparsity Method combination for DFN flows Classic: mesh generation, pde discretization Specific: discretization, Mortar Multi-scale DFNs Parallel computation and scalability Upscaling rules Process coupling Transport (fracture/matrix) Mechanics (fracture/matrix) Conformal 3D Mesh Generation, P. Laug, G. Pichot