Talk at the 2015 Cargese summer school on Flow and Transport in Porous and Fractured media: Development, Protection, Management and Sequestration of Subsurface Fluids

1. Synthetic experiments
for understanding and upscaling flow and transport processes in
heterogeneous media
Jean-Raynald de Dreuzy, Tanguy Le Borgne

2. Classical modeling protocol
Anderson,M.P.,andW.W.Woessner(1990),APPLIEDGROUNDWATER
MODELING:SimulationofFlowandAdvectiveTransport,AcademicPress.

3. Example of numerical
experimenation
Investigation of transport processes in
heterogeneous media
Macro-dispersion, Mixing and Reactivity

4. Upscaling solute transport processes (heterogeneous media)
▪ Purposes
▪ Effect of heterogeneity on inert and reactive solute processes
▪ Enhancement of dispersion and mixing induced by permeability heterogeneity
▪ Determine effective, upscale laws: Multiple scales in the same simulations
▪ Conceptual model (Assumptions)
▪ Stochasticly well-defined heterogeneity fields with evolving levels of complexity
▪ Simplification of boundary and initial conditions to focus on the processes
▪ Stochastic simulations
▪ Mathematical model
▪ Advection-diffusion-dispersion equations

5. Physical model
    
   
   
lengthncorrelatio
Kofiancenormal
xx
xYxY
YxYxY
xKxY
MODELITYHETEROGENE
Y
Y
:
varlog:
'
exp'''
'
ln
2
2
2



















 




6. Flow model
  0 hK

7. Transport model
   
   
v
vv
dvD
cDuc
t
c
ji
TLijTij  



0
Periodic boundary
conditions
extendedinjection
Reflecting
boundary
conditions
c=0
Adsorbing
boundary
conditions
point-source
Initial conditions
c(x,t=0)=0

8. Numerical methods
▪ Multi-scale stochastic simulations
▪ requires parallel computation
▪ Flow equation
▪ finite volume discretization
▪ algebraic multigrid linear solver
▪ Transport equation
▪ Lagrangian method: random walks
▪ Numerical strategy
▪ Macrodispersion: stochastic simulations with limited number of particles
▪ Mixing: few simulations with large number of particles
Beaudoin, A., J. R. de Dreuzy, and J. Erhel (2007), An efficient parallel tracker for advection-diffusion simulations in
heterogeneous porous media, paper presented at Europar, Rennes, France, 28-31 August 2007, Lecture Notes in
Computer Science 4641 705-714 Springer-Verlag, Berlin, Heidelberg

9. Some examples of software for porous and fractured media
▪ Classical hydrogeological models
▪ MODFLOW
▪ FEEFLOW
▪ HYDROGEOSPHERE
▪ Specialized modelling plateforms
▪ Tough, Berkeley, reactive transport
▪ DUMUX, DUNE, Stutgart, Multiphase flow, Multiphysics
▪ GEOSYS, UFZ, THMC
▪ PROOST, Barcelona
▪ H20lab, Rennes, heterogeneity (porous,fracture) and transport
▪ Multiphysics models
▪ COMSOL
▪ ABACUS
▪ Fluid mechanics models
▪ Open foam

10. Macrodispersion

11. Macrodispersion

12. Permeability variance 0.25 < y
2 < 9
Domain size Nx = 16384, Ny  Nz = 128
500 Monte Carlo simulations
10 000 particles
Extensive parameter study :
Cluster = 64 nodes of 2 processors Intel Quad Core
x5472. Each processor is composed of 4 cores
(Harpertown 3GHz) and 4GB of memory per core.
permeability generation = 20 s
time for flow = 213s
time for transport = 1605s
Example of CPU times :
Performances

13. Temporal evolution of the dimensionless longitudinal effective
dispersivity L(t) for various values of y²
Validation against analytical predictions Y
2<1

14. Predictions
2D and 3D longitudinal macro dispersivities
LA as function of y²
3D transverse macro dispersivity TA for
various values of y²
A. Beaudoin and J.R. de Dreuzy, Numerical assessment of 3D macro dispersion in heterogeneous porous media, Water Resources Research, Vol. 43, 2013

15. A. Beaudoin and J.R. de Dreuzy, Numerical assessment of 3D macro dispersion in heterogeneous porous media, Water Resources Research, Vol. 43, 2013
Presentation of results
Low heterogeneity
Y
2=1
High heterogeneity
Y
2=6.25

16. Mixing

17. Simulation and analysis of concentration distributions
Probability distribution of concentrations
macrodispersion model
Simulations at different times

18. Definition and validation of a new effective mixing model
Lamella representation
Villermaux, Cargèse summer school 2010

19. Quantification of fluid deformation processes
Map of fluid deformation Distribution of elongations
Le Borgne et al., JFM 2015

20. Definition and validation of a new effective mixing model
t1
t2
𝒑 𝒄, 𝒕
t2 t3
Fluid deformation Concentrations
𝒑(𝒄|𝝆)
Lamella representation Concentration PDF
Le Borgne et al. PRL 2013
macrodispersion model

21. Adapted numerical method
for accurate
gradient simulations

22. Numerical experimentation
projects during the summer
school

23. Scope: develop simulation projects in interaction with lecturers
(on a voluntary basis)
▪ Projects linked to practical courses
▪ Simulation of saltwater/freshwater interface
▪ Simulation of heat transport and potential fiber optic signal
▪ Direct modelling of geophysical signals (Resistivity, Spontaneaous Potential…)
▪ Projects linked to lectures
▪ Transport in heterogeneous media
▪ Reactive transport, colloid transport
▪ Multiphase or Non-Newtonian flows..
▪ Hydro-mechanics
▪ Projects linked to students PhD topics

24. Tool: COMSOL multiphysics
▪ Advantages
▪ Easy to learn in a week (friendly interface)
▪ Handles a large spectrum of coupled flow and transport processes
▪ Disadvantages
▪ Commercial licence
▪ Limited in terms of simulation size
▪ Free alternatives
▪ OpenFoam
▪ FreeFem
▪ …

25. Time schedule: first week

26. Time schedule: second week