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

pdf

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

pdf

-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)