highly heterogeneous porous and fractured media Jean Raynald de Dreuzy Géosciences Rennes

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1 Transient flow and particle transport in highly heterogeneous porous and fractured media Jean Raynald de Dreuzy Géosciences Rennes

2 Outline 1. Well tests in fractal and multifractal synthetic media 2. Lagrangian volume method for the simulation of reactive transport 3. HYDROLAB: collaborative simulation platform for flow and transport in porous and fractured media

3 Objectives Influence of heterogeneity on flow and transport processes. Influence of heterogeneity range and correlations How to define equivalent and simpler models Reduction of complexity and parameterization Differences of medium signatures subjected to flow or transport processes How can we switch between reaction systems or from flow to transport bounded correlations large correlations fractured media

4 Outline 1. Well tests in fractal and multifractal synthetic media 2. Lagrangian volume method for the simulation of reactive transport 3. HYDROLAB: collaborative simulation platform for flow and transport in porous and fractured media

5 Well test interpretation models 10 h 100 h Classical flow equation 3 2, D = 1 or n=1 1<n<2 n=2 Generalized flow equation 3 1 D = r h r r r T t h S D D 1 1 ( ) = r h r r r T t h S w d D D Drawdown solution t t h D r 1 ~ ) ( 1 = D ith n t t h 2 1 ^ ) ( Generalized drawdown solution () t t h D t t h D r r 1 1 ) ( 3 ~ ln ) ( 2 ) ( = = ( ) w c r d n with r t t h = 1 2 ^ ~ ) ( ( ) w d t t r 2 2 ~ ( ) w d r r t ~ t t h D r 1 ~ 1 ) ( 3 = ( ) w t c t c r ~ Barker [1988], Acuna and Yortsos [1996] ( ) w c r r t ~

6 Primary interest: well test t interpretation t ti in fractured media h 0 t) ~ ( t 0.2 Well test in the cristalline aquifer of Ploemeur (Brittany, France) [Le Borgne et al., 2004]

7 Nature of diffusion Δh ( r, t) ~ h ( r) 0 4T t S t c n 1 2 α Model Diffusion Theis t c ~ r 2 Barker (1988) Acuna and t c ~ r 2 t c ~ r dw Yortsos (1990) h o ~ r dw.(1-n/2)

8 Exponent n form field data

$Well tests in fractured media Fractional flow dimension or$ Transition of dimension 10 1 [Barker et al.

$3D 1D 10-1 1D 10-2 10-1 10 0 10 1 10 2 t/tc A fractional$

9 Well tests in fractured media Fractional flow dimension or transition of dimension? Transition of dimension 10 1 [Barker et al., 1988] Consistent Fractional flow? /Δh(tc) Δh(t) D 2D 1.3D 1D D t/tc A fractional flow dimension should be consistent on several orders of magnitude reflect a geometric or hydraulic organization

10 Fractional flow and anomalous diffusion dimensions (n,dw), 33 D 2 1 Ploemeur σ(lnk)=3.6 σ(lnk)=1.8 σ(lnk)= dw de Dreuzy,PhD, 1999

11 Motivations for studying porous scaling structures like fractals and multifractals rather than fractured media Structure with ih multiple li l scale of heterogeneities i Structure defined by simple parameters, basically the multifractal dimensions (moments of the permeability distribution) Perfect scaling media that can be used as reference for more complex natural or fracture synthetic cases

12 Sierpinski like Fractals and Multifractals Fractal D q = ln Pi i q 1 ln q l ( ) ( ) Multifractal

13 Multifractals compared to other scale free structures Fractals Multifractals generalized fractals Fractional Brownian motion (fbm) Succesive random multiplication versus successive random addition Different correlation functions fbm : S(ω)~ω (D+2H)( ) Multifractal : S(ω)~ω D2 Universal Multifractal

14 FBM compared to multifractals FBM H= H=0.55 H= Multifractals p=0.4 p=0.3 p=

15 Illustration of fbms Permeability maps where the lowest values have been removed [Saadatfar, 2002] H=0.3 H=0.8

16 Connectivity of multifractals (D 0,DD 2 ) How to connect multifractals? Regular multifractals with D 0 <2 Selection of the sole connected become disconnected with multifractals increasing scale Thresholding multifractals generated with D 0 =2 maintains D 0 =2. Enforce connectivity by specifying the fundamental pattern

17 D 2 Connected multifractal systems D 0

18 Flow patterns

19 Flow computation h 0 L Q h 1 Equivalent permeability Q = K eq h. S. 0 h L 1

$: Keff(L) Fractal multifractal$ media (D 0 <2) K eff ( L) ~ L^[

20 Effective permeability scaling : Keff(L) Fractal multifractal media (D 0 <2) K eff ( L) ~ L^[ α ( D D )] 0 2 with D 2 D and α = D 2 0 Permeability scaling induced by the gap between D 0 and D 2 y g y gp 0 2 Permeability always decreases with scale

21 Obtention of n and d w exponents n t r c ( ) h Generalized drawdown solution ( r, t ) = h ( ) r. Γ 1, 2 t Scaling analysis Characteristic Flow amplitude dimension Characteristic time Model Flow dimension Diffusion Head at the well steady state amplitude Theis Barker (1988) Acuna and Yortsos (1995) n=2 1<n<3 1<n<3 n=2d/dw t c ~ r 2 t c ~ r 2 t c ~ r dw h 0 ~ ln(t)t h ~ r 0 h 0 ~t -(n/2-1) h ~ r 2-n h 0 ~t -(n/2-1) h ~ r (2-n).dw/2

22 Lognormal K fields 100 1,00-0,6 06 1/d w 0,75 0, ,25 Normal Transport 0, σ -D/d w -0,8-1,0-1,2-1,4-1,6 Normal Transport σ 1. d w =2 normal transport 2. <D>=[2,2.3]

23 Illustration of n and d w on the percolation cluster Euclidean dimension d=2 d=3 Fractal dimension D=1.9 D=2.53 Permeability exponent μ/ν μ/ν= μ/ν= 2.27 d w =μ/ν+d+(2-d) d w =2.86 d w =3.8 Fractional lflow dimension i n=2d/d w n= n=

24 Illustration of n and d w on the percolation cluster x )0.5 ) log ((R 2 /R 2 10 max 0,0 0,5 L=67-0,5-1,0 10 R 2 (t)~t^(2/2.86) -1, log 10 t /d w =1/2.86= /d w D/d D/d w w =-1.9/2.86= L

$n and d w on classical fractal structures n=d s 3D Ifii 3,5 Cayley Trees 2D Infinite$ 30 3,0 Sierpinski Gasket Continuum percola Sierpinski Lattice [Karasaki,2002] Continuum

30 3,0 Sierpinski Gasket Continuum percola Sierpinski Lattice [Karasaki,2002] Continuum

$[Sahimi, [Shi i1996] Generalized Radial Flow [Barker, 1988] 2,0 Continuous Multifractal$

25 n and d w on classical fractal structures n=d s 3D Ifii 3,5 Cayley Trees 2D Infinite cluster Infinite cluster Sierpinski Gasket Homogeneous Fragmentation Frac Backbone Backbone 30 3,0 Sierpinski Gasket Continuum percola Sierpinski Lattice [Karasaki,2002] Continuum percolation 2,5 Correlated percolation [Prakash, 1992] Correlated ltdpercolation lti [Sahimi, [Shi i1996] Generalized Radial Flow [Barker, 1988] 2,0 Continuous Multifractal Self affine [Saadatfar, 2002] Sierpinski lattice 1,5 AO Conjecture 1,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0 d w

26 Comparison with Ploemeur s data 3.5 Cayley Trees d s n=d AO Conjecture Ploemeur d w

27 Relation between the hydraulic exponents (n,d w ) and the geometrical ti dimensions i (D 0,DD 2, ) Simplified structure characterization: i D 0 for the structure D 2 for the permeability bl correlation on the structure Relation between geometrical and hydraulic exponents? How does d w depend on D 0 and D 2? does d w increase with D 0 and D 2? is d w necessarily larger than 2? Are n and d w global field properties or dependent on the well location? Is there any general relation between n and d w? Is n=2d/d w verified whatever the structure?

28 Continuous multifractal fields D 0 =2, D 2 < log (L/2)] [<R 2 (t)>/(l/2)] log 10 [h w (t)] <d w > and <n> <d > d w <n> log 10 (t) D 2 R 2 and h w display consistent it t scale dependence d for each simulation <d w > and <n> remain close to those of the 2D homogeneous medium d w and n are highly variable in [1.25,3]

29 d w related to the permeability scaling ( ) h T dw 2 S h D 1 = r K r t r D 1 r r S 1 ( ) D 1 Relation between K(r) and d w? What is the relevant K(r)? K ~ 3 () 0 0 r r θ Testing <K 0 (r)>, the annular average log 10 <K> log 10 r

30 d w related to the permeability scaling K 0 ( ) θ0 r ~ r d w D 2 =1.9 D 2 =1.75 D 2 = D 2 =1.25 D 2 =1.125 d w =2+θ 0 +2-D θ 0 +2-D 2 d w is given by purely geometrical exponents D 2 induces an increase of d w by (2 D 2 ) when compared to the annular case Through θ 0, d w depends on a local property : the permeability at the well

31 Relation between n and d w n D 2 =1.25 D 2 = D 2 =1.75 D 2 = n=2d 0 /d w d w

$Fractal multifractal fields D 2 < D 0 <2 3.0 1.2 1.1 d w /(2+(2-D 0 )(2.$ 0 15 1.

32 Fractal multifractal fields D 2 < D 0 < d w /(2+(2-D 0 )(2.75D D 2-2)) d w D 0 = D 0 = d w increases for more heterogeneous systems (D 2 decreasing) D 2 d w ( 2 D ) [ 2.5 D ] 0. 1 = D ± n = d D d w 2 0 ± ( )

33 <d w >(D 0, D 2 ) d w Maximum of d w against D 0 for a fixed D 2 <1.6 w g 0 2 D 0max = D 2

34 Relation between permeability scaling and d w 4 increasing permeability decreasing permeability 3 d w =2+θ 0 d w θ 0 K 0 ( r) ~ r θ

35 Conclusions Assumptions Multifractal description : superposition of fractals having different dimensions Multifractal media parameterized by their fractal and correlation dimensions D 0 and D 2 Results Bulk permeability decrease for multifractal media decrease speed of the gap between D 0 and D 2 Transient state response Anomalous model characterized by d w and n <d w > in the interval [2;3] d w in the interval [1.5;4] n = 2D 0 /d w 0 w d w related to the permeability scaling conditioned by the permeability at the well

36 Outline 1. Wll Well tests t in fractal and multifractal t l synthetic ti media 2. Lagrangian volume method for the simulation of reactive transport General advantages and drawbacks of particle methods A LagrangianVolume g (LV) method for simulating reactive transport The LV method for Freundlich adsorption in 1D Analysis of Local Equilibrium i Assumption (LEA) Sorption simulation under kinetic and non linear conditions Simulation of mixing and dispersion 3. HYDROLAB: collaborative simulation platform for flow and transport in porous and fractured media

37 General advantages and drawbacks of particle methods Advantages No numerical diffusion for simulating the advective term Existing methods for treating particle diffusion and dispersion Time effective for obtaining coarse solutions Relative ease of implementation Drawbacks Improvement of precision Coupling between transport and reaction Definition of concentration from particles

38 Lagrangian Volumes model in 1D Segmentation of flow in elementary volumes (water slices) LV are tracked in the domain like in classical particles tracking schemes LV contain solutes as concentrations LV interact with the solid and between themselves Eit Existence of two scales: LV scale and mesh scale Increase of precision without additional intra mesh discretization of S C Injected concentrations (pulse) Elementary Lagrangian volumes C 0 u, flow velocity time C = 0 C = C 0 C = 0 Meshes

concentration C according to ϕ ϕ : amount of solute sorbing per unit

39 Lagrangian Volumes with Adsorption Computation of the chemical flux ϕ at the LV scale along the pore surface Modification of the solute concentration C according to ϕ ϕ : amount of solute sorbing per unit surface area per unit time (ϕ <0: desorption) dc dt = γ ϕ γ : surface to volume ratio

40 Kinetically controlled controlled Freundlich sorption S = t α ( n K C S ) a C concentration ti in solution (mass/m 3 ) S concentration sorbed (mass/m 2 ) a kinetic constant of reaction (s 1 ) K a reaction capacity n Freundlich (n=1 or n < 1) 1,0 α = 0,1 20 2,0 n = 1 α = 0,5 α = 1 1,5 n = 0.8 n = 0.5 n = 0.2 C/C 0 0,5 S (solid) 1,0 0,5 00 0, Temps 0,0 0,0 0,5 1,0 1,5 2,0 C (fluid)

41 Transport and Sorption coupling C t S t S + γ t = u C x + D 2 = α ( n K C S ) a 2 x C C aqueous concentration (in mass/m 3 ) S sorbed concentration adsorbée (in mass/m 2 ) u flow velocity (m/s), γ surface/volume ratio (m 1 ), a reaction rate (s 1 ), K a distribution ib ti coefficient i n Freundlich coefficient (n < 1) Advective dispersive term simulated like in classical particle tracking Water mass conservation remains valid globally but not necessarily locally Solute mass is always conserved both globally and locally

Within the mesh Decoupling of transport and reactivity at LV scale Batch reaction at LV scale

differential equation on [t i,t i+1 [ dc C ( W ) n ( K C ( t) S ( t) ) ( W )( t) = α γ a ( W ) (

t n 1 numerical integration on [t i,t i+1 [ C [ γ K C ( t) + C ( t) ( C ( t ) +γ S ( t ))] ti +Δt

integrations between t 0 and t n 1 i dt Mesh scale homogenization of S when the LV leaves the

42 Within the mesh Decoupling of transport and reactivity at LV scale Batch reaction at LV scale Contact time between LV and corresponding solid Δt=Δx/u Δx/u Δx: characteristic length of the LV differential equation on [t i,t i+1 [ dc C ( W ) n ( K C ( t) S ( t) ) ( W )( t) = α γ a ( W ) ( W ) d () t + γ S( )() t = C ( t ) + γ S( )( t ) W ( W ) i W i t t t i 0 Δx t i+1 S W ( W ) C( W ) t n 1 numerical integration on [t i,t i+1 [ C [ γ K C ( t) + C ( t) ( C ( t ) +γ S ( t ))] ti +Δt n ( W )( t i +Δt) = C ( W )( t i ) α ( M ) a ( W ) ( W ) ( W ) i ( M ) ( W ) t ti n successive integrations between t 0 and t n 1 i dt Mesh scale homogenization of S when the LV leaves the mesh Mesh scale homogenization of S at t n 1 {Jump+React} i=0,,n 1 + homogenization of S S( M ) C W ( W )

43 Validation cases 3 parameters: Freundlich exponent (n), kinetic constant (α), heterogeneity (σ)

44 Kinetically controlled controlled linear sorption n = 1, γka=4, α=1 days 1, analytical solutions for spatial moments [e.g. Dagan et Cvetkovic, 1993] Water velo ocity Plume velocity / 1,0 0,8 0,6 0,4 0,2 analytic numeric 1E-3 0,01 0, Time (days) Lo ongitudina al variance (m 2 ) 3000 analytic 2500 numeric Time (days)

45 Asymptotic concentration profiles C/C 0 scales asymptotically as t 1/(1 n) Jaekel et al. [1996] n=0.9, α=1 day 1, u=1 m/day, γk a C 0 n 1 =4, LV/Mesh volume= C/C 0 X = 20 m 10-6 X = 30 m X = 40 m X = 50 m ~t Time (days)

S 0 where C 0 Right retardation factor LV induce a spreading D/U=0,1 m Other cause of

46 Analysis of Local Equilibrium Assumption τ reaction <τ advection First steps of LV algorithm n=1, Ka=1 Analytical solution Injection form constant Velocity reduced by 1+Ka S 0 where C 0 Right retardation factor LV induce a spreading D/U=0,1 m Other cause of numerical diffusion homogenization of S at the mesh scale Induces a slight reduction of diffusion

47 Analysis of Local Equilibrium Assumption Solutions to lower the numerical diffusion Mesh refinement τ reaction >τ advection Use of an analytical solution of the reactive transport equation at the mesh scale What does the LEA assumption imply? S 0 where C 0 although solutes are delayed instantaneous homogenization of solute concentration within the fluid phase

48 2D Heterogeneous media α=1, n=0.9, D=0, γka=1, σ 2 =1.5 particles concentration

49 Mixing Mixing is required for simulating reactions in solution Implementation Mixing is performed between LagrangianVolumes leaving a mesh and their predecessor on the duration needed for crossing the mesh C C1 C = D Δx Δ 0 d A u mesh

50 Validation on the 1D advection diffusion diffusion case D/(ud mesh )=10 2 D/(ud mesh )=10 3 Δx/d mesh C t (x) 0,04 0,02 numeric analytical variance Longitudinal 0,14 0,07 0,2 0,1 0,05 0,01 0, Dt 0, x 0, time diffusion time in the LV < advection in the mesh 2 D d mesh < Δ x u 2

51 Lagrangian volume software organization

52 CONCLUSION: Advantages and drawbacks of the Lagrangian Volume Vl method Drawbacks Numerical dispersion in equilibrium conditions Advantages Decoupling at the LV scale allowing the direct implementation of various chemical sets and reactions Simulation of large domains Differentiation of dispersion and mixing at the local scale bounded correlations large correlations fractured media

53 HYDROLAB Collaborative numerical environment for simulating underground physical phenomena in complex media Collaboration between Géosciences Rennes (Hydrogeology) INRIA (Computer sciences) University it of Le Havre (Civil il engineering) i University of Lyon (Applied mathematics) People currently involved and developing A. Beaudoin, E. Bresciani, E. Canot, J. R. de Dreuzy, J. Erhel, R. Le Goc, F. Guyomarch, B. Poirriez, S. Pomarède, D. Roubinet, D. Tromeur Dervout Funded by the ANR CIS MICAS, INRIA and CNRS

54 HYDROLAB: objectives Modularity implemented by components Maximize code factorization, minimize validation Easy replacement and addition of components Well identified and simple relations between them Flexibility Segmentation Avoid redundancies Minimize border effects Collaborative development Optimization i Test and comparison of numerical methods Parallel computation (distributed memory)

55 HYDROLAB: organization

56 HYDROLAB: practical aspects Object oriented parallel l software (C++,MPI) Careful share between static and dynamic polymorphism static for optimization (template) dynamic for segmentation (virtual function) Developed under Windows (visual) and Linux (eclipse) Code versioning (svn) Merge, validation Stable tags Internal communication Task manager, bug tracker, frequent meetings Documentations Non regression tests

57 HYDROLAB: 3 levels of documentation User guides How to use the platform (installation, execution, exploitation of results) Scientific foundations of the implemented methods Technical documentation for developers Description of the main common modules (UML modeling, a posteriori documentation) Details of the specific implementations (including automatic documentation extraction) Description of procedures (version control, testing) Scientific documentation (articles, presentations)

58 HYDROLAB: non regression tests Objectives Frequent testing Broadest coverage Automatic Characteristics tests Weekly Daily Automatic with cmake and ctest?

59 HYDROLAB interface XML format Friendly user interface

60 HYDROLAB & HYDROWEB

61 DATA BASE Data base

62 Database specifications Ojectives Long term storage of the results Efficient i and convenient recovering Stored data Results Execution parameters Metadata Hydroweb : database interface Data uploading Results selection by pertinent t criteria i Display of the results Download of the selected results

63 HYDROLAB: Advantages and drawbacks Advantages Re use of code Integration of new functionalities Tool for proof of concepts Optimization Drawbacks Investment time 1 full time engineer permanent supervision Formation Advantage or drawback? Programming guidelines

$HYDROLAB: perspectives p Applications Coupling between porous and fractured media 3D flow$ $fractured media Reactive transport 3D asymptotic macrodispersion computations with Schwarz$

64 HYDROLAB: perspectives p Applications Coupling between porous and fractured media 3D flow computation in fractured media Simulation of well testing in complex media Inverse problem in fractured media Reactive transport 3D asymptotic macrodispersion computations with Schwarz decomposition methods Future of HYDROLAB & HYDROWEB Distribution policy (licensing) How large should it grow?

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