1. 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

2. Well test interpretation in heterogeneous media J-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, …) ?

3. A field example of heterogeneous medium Ploemeur (Brittany): Aquifer in a highly fractured zone on the contact between granite and micaschiste Granite Micaschiste

4. Well tests in Ploemeur Barker Theis

5. Generalized flow models ModelDimension exponent Anomalous diffusion exp Radius of diffusion Drawdown at the well Theis Barker (1988) Acuna and Yortsos (1995) D=2 1<D<3 d w =2 d w >2 R 2~t R 2 ~t 2/dw h o ~ t -1 h o ~ t -D/2 h o ~ t -D/dw  Generalized diffusivity equation  Generalized drawdown solution Drawdown at the well Radius of diffusion

6. Relevant models and exponents at Ploemeur normal fault zone contact zone Anomalous diffusion exponent d w = 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

7. Influence of the heterogeneity on the flow equation Validity of the generalized flow equation? Sierpinski Gasket D=D 0 =1.58 d w =2.3 D 0 =2 1<D 2 <2 D, d w ? D 0 =2 D, d w ? Heterogeneous logK fields with Fractal fields Multifractal fields 1<D 0 <2 D, d w ?

8. Fractal correlation pattern : Generation D 2 =1.8D 2 =1.2D 2 =1.5 Dimension D 2 nested generation probability field C(r)~r D2 Correlation function

9. Transient flow model Darcy Law Mass conservation law Porous media  h/  t + .u = f u = -k  h Boundary conditions

10. Numerical simulation  space and time discretisations : stiff system of ODEs  scale effects : large grid size  stochastic modelling : many simulations Need for high performance schemes and software

11. 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 accurate  full tensors of permeability are not easily handled  large sparse ill-conditioned linear system at each time step  the ODE system is stiff BUT

12. 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 monotone  number of degrees of freedom = number of faces + number of nodes  large sparse ill-conditioned linear system at each time step  the system is stiff BUT

13. Mass conservation law : S dP/dt + D P - R T = F Darcy law : - R T P + M T = V M large sparse ill-conditioned matrix R large sparse rectangular matrix S and D diagonal matrices Mixed Hybrid Finite Element Method

14. Simplified scheme using mass lumping Elimination of T : S dP/dt + (D - R M -1 R T ) P = F + R M -1 V Exact solution : P = exp(-t (D - R M -1 R T ) ) P 0 + P 1 Sufficient conditions for positivity : (R M -1 R T) KK ’  0, M EE ’  0 and R KE  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

15. Additive Runge-Kutta scheme S dP/dt + (D - R M -1 R T ) P = F + R M -1 V D  0 and R M -1 R T  0 Stiff part in D : implicit for D and explicit for R M -1 R T 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 R T P n = dt (F n+1 + R M -1 V n+1 )

16. 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

17. Steady flow in porous media : numerical results Lognormal distribution well test simulation

18. Steady flow in porous media : numerical results Fractal with D = 1.5 well test simulation

19. Equivalent permeability Steady flow in porous media : physical interpretation

20. Validation of the transient flow simulator Percolation network Anomalous medium K(r)~r x

21. Transient flow simulation and determination of the exponents Pattern generation D=1.5 Fit on h 0 (t) Flow simulation Fit on R 2 (t)

22. Distribution of exponents for multifractals D 0 =2 and D 2 =1.5 R 2 (t)~t^(2/d w ) h 0 (t)~t^(-D/d w ) Mean exponents : dw=2, D=D 0 =2

23. Exponent mean and stds for multifractals Conclusions =D 0 (support dimension) =2 (normal transport) Large variability around the mean D=[1.5,2.5] and dw=[1.5,3]

24. Why is the mean transport normal in multi-fractal media? Porous medium Flow  =2-D 2 d w = 2-D 2 +D ? d w =2 With D?=D 2 Einstein Relation in 2D : d w =D ? + 

25. Comparison between fractal and multi-fractal media MultifractalFractal Support dimensionD 0 =2D 0 =[1,2] Correlation dimensionD 2 =[1,2]D 2 =D 0 Permeability exponent  =2-D 2 ? Diffusion exponentd w =2 ([1.5,3]) ? Hydraulic DimensionD=D 0 ([1.5,2.5]) ?

26. Characteristic exponents for fractal media

27. Comparison between fractal and multi-fractal media MultifractalFractal Support dimensionD 0 =2D 0 =[1,2] Correlation dimensionD 2 =[1,2]D 2 =D 0 Permeability exponent  =2-D 2  =d w -D 0 Diffusion exponentd w =2 ([1.5,3]) d w =2.3  0.2 Hydraulic DimensionD=D 0 ([1.5,2.5]) D=D 0  0.1

28. Heterogeneous logK fields 1. Large exponent variability 2. d w =2 normal transport 3. =[2,2.3]

29. Conclusions  The relation of Einstein is verified  The average transport is normal ~2  The average hydraulic dimension is the fractal dimension and more precisely the support dimension D 0.  Individual media have a large variability d w =[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 (d w =2.86)