HYDROGEOLOGIE
ECOULEMENT EN MILIEU HETEROGENE
J. Erhel – INRIA / RENNES
J-R. de Dreuzy – CAREN / RENNES
P. Davy – CAREN / RENNES
Chaire UNESCO - Calcul numérique intensif
TUNIS - Mars 2004
Well test interpretation in heterogeneous mediaJ-R De Dreuzy(1), P. Davy(1), J. Erhel(2)
(1)UMR 6118 CNRS, Université de Rennes 1, France(2)IRISA/INRIA Rennes
How does heterogeneity influence transient flow?
Approach
- Evaluation of the classical flow equation on a field experiment (Ploemeur).
- Which heterogeneous media follow the same flow equation ?
- Numerical simulation of transient flow in heterogeneous media
What is the relevant diffusion equation (Theis, Barker, …) ?
A field example of heterogeneous medium
Ploemeur (Brittany): Aquifer in a highly fractured zone
on the contact between granite and micaschiste
Granite
Micaschiste
Well tests in Ploemeur
Barker
Theis
Generalized flow models
Model Dimension exponent
Anomalous diffusion exp
Radius of diffusion
Drawdown at the well
Theis
Barker (1988)
Acuna and Yortsos (1995)
D=2
1<D<3
1<D<3
dw=2
dw=2
dw>2
R2~t
R2~t
R2~t2/dw
ho ~ t-1
ho ~ t-D/2
ho ~ t-D/dw
)( 11 r
hKr
rr
T
t
hS dwD
D
Generalized diffusivity equation
ww
w
dd
D
d
ttRtth
R
rthtrs
2
20
0
~ and ~
exp).(),(
Generalized drawdown solution
Drawdown at the well Radius of diffusion
Relevant models and exponents at Ploemeur
normal fault zone
contact zone
Anomalous diffusion exponent
dw= 2.8
Dimension exponent
D=2.2
Dimension exponent
D=1.6
It appears possible to define a mean equivalent flow model
at least for one of the major fault zone
The relevant model implies: - a fractional flow dimension
- an anomalous diffusion
Influence of the heterogeneity on the flow equationValidity of the generalized flow equation?
Sierpinski Gasket
]4,0[)(log K
D=D0=1.58
dw=2.3
D0=2
1<D2<2
D, dw?
D0=2
D, dw ?
Heterogeneous logK fields with
Fractal fields
Multifractal fields
1<D0<2
D, dw?
Fractal correlation pattern : Generation
D2=1.8 D2=1.2D2=1.5
Dimension D2
nested generation
probability field
0,01 0,1 11E-4
1E-3
0,01
0,1
1
r/lmin
C(r)
r/L
0
1
2
pent
e lo
cale
C(r)~rD2
Correlation function
Transient flow model
Darcy Law
Mass conservation law
Porous media
h/t + .u = f
u = -k h
Boundary conditions
Numerical simulation
space and time discretisations : stiff system of ODEs
scale effects : large grid sizestochastic modelling : many simulations
Need for high performanceschemes and software
Finite Volume Method
mass is conserved locally
it can be simply extended to unstructured 2D and 3D grids
the linear system to solve is positive definite
the scheme is monotone
number of degrees of freedom = number of nodes
velocity is not accuratefull tensors of permeability are not easily handledlarge sparse ill-conditioned linear system at each time stepthe ODE system is stiff
BUT
Mixed Finite Element Method
mass is conserved locally
it can be simply extended to unstructured 2D and 3D grids
the linear system to solve is positive definite
pressure and velocity are approximated simultaneously
full tensors of permeability are easily handled
the scheme is non monotonenumber of degrees of freedom = number of faces + number of nodeslarge sparse ill-conditioned linear system at each time stepthe system is stiff
BUT
Mass conservation law :
S dP/dt + D P - R T = F
Darcy law :
- RT P + M T = V
M large sparse ill-conditioned matrixR large sparse rectangular matrixS and D diagonal matrices
Mixed Hybrid Finite Element Method
Simplified scheme using mass lumping
Elimination of T : S dP/dt + (D - R M-1 RT) P = F + R M-1 V
Exact solution : P = exp(-t (D - R M-1 RT) ) P0 + P1
Sufficient conditions for positivity :(R M-1 RT)
KK ’ 0, MEE ’ 0 and RKE 0
Mass lumping : diagonal elementary matrices
the scheme is monotone
the matrix M is diagonal, easy to invert
the system of ODE is of size N
Additive Runge-Kutta scheme
S dP/dt + (D - R M-1 RT) P = F + R M-1 V
D 0 and R M-1 RT 0
Stiff part in D : implicit for D and explicit for R M-1 RT
No sparse linear system to solve
High performance compact scheme
Example : ARK of order 1 (Euler)
(S + dt D) P n+1 - R M-1 RT P n = dt (F n+1 + R M-1 V n+1)
Numerical experiments
Currently, finite volume scheme
for transient computations, use of LSODES package • BDF scheme and direct sparse linear solver• high memory requirements
for steady flow computations, use of UMFPACK solver
Steady flow in porous media : numerical results
Lognormal distributionwell test simulation
Steady flow in porous media : numerical results
Fractal with D = 1.5well test simulation
100 101 102 103
10-2
10-1
100
K
L
Equivalent permeability
Steady flow in porous media : physical interpretation
1,0 1,5 2,00,0
0,5
1,0
K(L)~L^[-(D-2)]=2-D
expo
nent
D: fractal dimension
Validation of the transient flow simulator
Percolation network Anomalous medium
0 25 50 750,00
0,25
0,50
-1,0
-0,5
0,0
1/dw=1/2.86=0.35
1/dw
L
-D/dw
-D/dw=-1.9/2.86=-0.66
K(r)~rx
-2 -1 0 1 20,0
0,5
1,0
1,5
2,0
dw(théorique)=1/(2-x)
1/d w
x
0 2 4 6-2
-1
0
0.61/dw=0.35 0.5
log
(R2 (t
))
log(t)
Transient flow simulation and determination of the exponents
Pattern generation
D=1.5Fit on h0(t)
Flow simulationFit on R2(t)
0 2 4 6-4
-3
-2
-1
0
-0.7-D/dw=-1 -0.7
log
(h0(t
))
log(t)
Distribution of exponents for multifractals D0=2 and D2=1.5
0,00 0,25 0,50 0,75 1,00 1,250,0
0,1
0,2
0,3
Mean value d
w=2
Normal diffusion
Anomalously fast diffusion
dw<2
Anomalously slow diffusion
dw>2
1/dw Exponent of R(t)
R2(t)~t^(2/dw)
-2,00 -1,75 -1,50 -1,25 -1,00 -0,75 -0,500,0
0,1
0,2 Mean value -D/d
w=-1
-D/dw Exponent of h
0(t)
h0(t)~t^(-D/dw)
Mean exponents : dw=2, D=D0=2
Exponent mean and stds for multifractals
1,00 1,25 1,50 1,75 2,00-1,6
-1,4
-1,2
-1,0
-0,8
-0,6
Normal Transport
-D/d
w
D2
Conclusions <D>=D0 (support dimension)
<dw>=2 (normal transport)
Large variability around the mean D=[1.5,2.5] and dw=[1.5,3]
1,00 1,25 1,50 1,75 2,000,00
0,25
0,50
0,75
1,00
Normal Transport1/d w
D2
Why is the mean transport normal in multi-fractal media?
Porous medium Flow
1,0 1,5 2,00,0
0,5
1,0 K(L)~L^(-)~(2-D
2)
D2
=2-D2
dw= 2-D2+D?
dw=2
With D?=D2
Einstein Relation in 2D : dw=D?+
Comparison between fractal and multi-fractal media
Multifractal Fractal
Support dimension D0=2 D0=[1,2]
Correlation dimension D2=[1,2] D2=D0
Permeability exponent =2-D2 ?
Diffusion exponent dw=2 ([1.5,3]) ?
Hydraulic Dimension D=D0 ([1.5,2.5]) ?
Characteristic exponents for fractal media
1,50 1,75 2,00
-1,00
-0,75
-0,50
-D0/d
w
Percolation Network
Sierpinski gasket-D
/dw
D0
1,50 1,75 2,000,250
0,375
0,500
Percolation Network
Normal transport
Sierpinski gasket1/d
w
D0
Comparison between fractal and multi-fractal media
Multifractal Fractal
Support dimension D0=2 D0=[1,2]
Correlation dimension D2=[1,2] D2=D0
Permeability exponent =2-D2 =dw-D0
Diffusion exponent dw=2 ([1.5,3]) dw=2.3 0.2
Hydraulic Dimension D=D0 ([1.5,2.5]) D=D0 0.1
Heterogeneous logK fields
0 1 2 3 4
-1,6
-1,4
-1,2
-1,0
-0,8
-0,6
Normal Transport
-D/d
w
0 1 2 3 4
0,00
0,25
0,50
0,75
1,00
Normal Transport1/d
w
1. Large exponent variability
2. dw=2 normal transport
3. <D>=[2,2.3]
Conclusions
The relation of Einstein is verified
The average transport is normal <dw>~2
The average hydraulic dimension is the fractal dimensionand more precisely the support dimension D0.
Individual media have a large variabilitydw=[1.5,3]D=[1.5,2.5]
Average anomalous diffusion is to be searched in medium having a highly heterogeneous structure like percolation network at threshold (dw=2.86)
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