Synthetic experimentsfor understanding and upscaling flow and transport processes in heterogeneous mediaJean-Raynald de Dreuzy, Tanguy Le Borgne

Classical modeling protocol

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Example of numerical experimenation Investigation of transport processes in heterogeneous mediaMacro-dispersion, Mixing and Reactivity

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

Physical model

lengthncorrelatio

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xxxYxY

YxYxY

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MODELITYHETEROGENE

Y

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varlog:

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Flow model 0 hK

Transport model

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Periodic boundary conditions

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Reflecting boundary conditions

c=0

Adsorbing boundary conditions

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Initial conditionsc(x,t=0)=0

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

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

Macrodispersion

Macrodispersion

Permeability variance 0.25 < y2 < 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 stime for flow = 213stime for transport = 1605s

Example of CPU times :

Performances

Temporal evolution of the dimensionless longitudinal effective dispersivity L(t) for various values of y²

Validation against analytical predictions sY2<1

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

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 heterogeneitysY

2=1

High heterogeneitysY

2=6.25

Mixing

Simulation and analysis of concentration distributions

Probability distribution of concentrations

macrodispersion model

Simulations at different times

Definition and validation of a new effective mixing model

Lamella representation

Villermaux, Cargèse summer school 2010

Quantification of fluid deformation processes

Map of fluid deformation Distribution of elongations

Le Borgne et al., JFM 2015

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

Adapted numerical method for accurate

gradient simulations

Numerical experimentation projects during the summer

school

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

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

▪ …

Time schedule: first week

Time schedule: second week