HYDROGEOLOGIE ECOULEMENT EN MILIEU FRACTURE 2D
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Transcript of HYDROGEOLOGIE ECOULEMENT EN MILIEU FRACTURE 2D
HYDROGEOLOGIEECOULEMENT EN MILIEU FRACTURE 2D
J. Erhel – INRIA / RENNESJ-R. de Dreuzy – CAREN / RENNES
P. Davy – CAREN / RENNESChaire UNESCO - Calcul numérique intensif
TUNIS - Mars 2004
Modelling Flow and Transport in Subsurface Complex Fracture Networks
Jocelyne Erhel, Jean-Raynald de Dreuzy, Philippe DavyIRISA / INRIA Rennes
Géosciences Rennes / CAREN
2D Model 3D Model
Channeling in natural fractured media
80 % of flow
100 % of flowOlsson [1992]
Flow arrival in a mine gallery at Stripa (Sweden)
50 m
10 m
Fluid flows only in a very limited number of fractures
Fracture networks geometry
2D Outcrop
50 mHornelen Basin
Synthetic image
Gylling et al. [2000]
100 mÄspö
Influence of geometry on hydraulics
length distribution has a great impactpower law n(l) = l-a
3 types of networks based on the moments of length distribution
mean std variation3 < a < 4
mean std variation2 < a < 3
mean variationa > 4
Flow in 2D fracture networks
1. Darcy law and mass conservation law : Div(K grad h) = 0 2. stochastic modelling 3. High numerical requirements : large sparse matrix
Papers: Dreuzy and al, WRR, [2000a;2000b;2001]
h = 1
h = 0dh/dn = 0
dh/dn = 0
Flow
Linear solver for permanent flow computation
Paper: Dreuzy and Erhel, CG [20002]
563
CPU requirements a=2.5 - 10 000 points in infinite cluster - 4 000 points in backbone
Linear solver infinitecluster Backbone
CG - No preconditioning 370
PCG with Jacobi preconditioning 48
PCG with ILU preconditioning 175 19
Sparse LU from Petsc 2 1
Sparse Multifrontal LU from UMFPACK 0.07
Permanent flow computation - PCG solver
-3 -2 -1 0 1 2 30
100
200
n(log10vp)
log10vp100 101 102 103 104
10-5
10-4
10-3
10-2
10-1
100Number of elements
gc+ilu
gc
Rel
ativ
e er
ror
Iteration number
Solid : constant fracture permeabilitydashed : lognormal fracture permeability distribution
About 400 nodesCG : ~6000 iterations for 10-5
PGC with ILU : ~50 iterations for 10-5
Eigenvalue distributionConvergence history
Complexity analysis of linear solver
Solver Complexity nz(A)=kn=nz(L)
Direct 2nz(L)2/n+5nz(L) 2k(k+2.5)n
CG 2(nz(A)+5n)nit(CG)
2(k+5)n nit(CG)
PCG 2(2nz(A)+5n)nit(PCG)
2(2k+5)n nit(PCG)
Example : k = 5 , niter(CG)=1500, niter(PCG) = 50Direct solver : 1 , CG : 400 , PCG : 20
SOME RESULTS FOR PERMANENT FLOW
Computation of equivalent permeability or network permeability
Percolation parameter : p (related to the fracture mass density) percolation threshold : pcpower law exponent : aconstant aperture in fractures
Domain of validity of classical approaches
• a > 3 : percolation theory• p > pc and a < 3 : homogeneous media• p < pc and a < 3 : unique fracture system• a < 2 and p ~ pc : network of infinite fractures• 2 < a < 3 and p ~ pc : multi-path multi-segment network
Transient flow computation
dh / dt = Div (K grad h)Boundary conditions : h = 0
Initial condition : h = 0 excepted h = -1 in centre
BDF scheme and sparse linear solverLSODE package
Complex model with simple equations
How to get a simplified model with adapted equations ?
Transient flows
Dim
ensi
on c
him
ique
Dimension hydraulique
1 2 3
2
3
4
2D 3DPercolation
Homogeneous modelsMultiscale models Ploemeur
Simplified modeladapted equations
hydraulic dimension chemical dimension
3D models
First approach :3D finite element or finite volume method
very high numerical requirements
Objective : 100 000 fractures with 1 000 elements in each
3D models
Second approach : network of links - not accurate enough
Current work : multilevel method based on a subdomain approach