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