Formulation of the Problem of Upscaling of Solute Transport in

Highly Heterogeneous Formations

A. FIORI1 , I. JANKOVIC2, G. DAGAN3

1 Dept. of Civil Engineering, Università di Roma Tre, Rome, Italy2 Dept. of Civil, Structural and Environmental Engineering, SUNY, Buffalo, USA3 School of Mechanical Engineering, Tel Aviv University, Ramat Aviv, Israel

Scaling Up and Modeling for Transport and Flow in Porous Media Dubrovnik, Croatia, 13-16 October 2008

Problem statement

Transport of a conservative solute in porous media is governed by

C(x,t) local concentration V(x) Eulerian steady velocity field D local dispersion coefficient

C(x,0)=C0(x) Initial condition

CDCt

C

V

The role of heterogeneity

Quantification of transport is usually carried out by the spatial or temporal moments of C, mainly the first two (sufficient for a Gaussian plume).

Flow and transport in natural aquifers are largely determined by the spatial distribution of the hydraulic conductivity K; It is convenient to describe Y=lnK as a space random function, with assigned statistical properties (KG, σY

2, IY);

As a consequence, V(x) and C are also random.

Formulation of the upscaling problem

Transport is solved generally numerically by discretization of space by elements of scale L.

“Fine scale” solution: Lfs<<IY (e.g. 1/10 or less); this is viewed as "exact". It requires considerable computational effort for 3D problems.

Upscaled solution: L> Lfs. For a selected upscaled medium permeability field Y (KG, σY

2, IY); the solutions are V(x), C(x,t).

The upscaling problem: what is the relationship between Y and Y to render V, C "good" approximations of V, C?

Assumptions

1. Flow is uniform in the mean, i.e. <V>=U(U,0,0)=const and the domain is large at IY scale (the procedure is supposedly applicable to complex flows that are slowly varying in space)

2. A thin, ergodic plume of planar dimensions >> IY is injected on a plane at x=0

3. Transport is quantified by the first <> and second σ2

temporal moments of the breakthrough curve (BTC) at a control plane at x>>IY

Upscaling problem

How can we ensure that <>=<> and σ2=σ

2 (i.e.L=L)?

Previous studies (Dagan, 1994; Rubin, 1999) have solved advective transport for weakly heterogeneous media. It was found that L<L.

In order to compensate for loss of "spreading", a fictitious upscaled induced dispersivity equal to L - L was added to L.

The medium structure Medium is modelled by cubes

of side 2R=2IY, of independent random conductivities K, drawn from a lognormal pdf. The Y covariance is linear.

An applied constant mean head gradient –J results in mean uniform velocity U.

A thin plume is injected over a large area A. Spreading is characterized by the first two moments of the BTC, i.e.: Mean arrival time <>x/U Longitudinal dispersivity

xUL 2/22

The “Fine-scale” solution

The “fine-scale” solution was obtained by us in previous works (e.g. Jankovic, Fiori, Dagan, AWR, in press)

The method is based on the solution for an isolated spherical element and the adoption of the self-consistent argument

Lfs<<IY

The “Fine-scale” solution: Previous results

Semi-analytical solution: dKKfPeK

UKRL ,

8

30

22

zyzyRR dbdbbbPeK ,,,22

0

5

10

15

20

25

30

35

40

45

50

0 1 2 3 4 5 6 7 8 9 10

Pe=inf

Pe=10000

Pe=1000

2Y

Y

L

I

D

UIPe Y

The Upscaled solution

Upscaled cubical elements of size L are used for numerical solution.

How do we model the structure so that flow and transport solutions lead to same U and L?

L>Lfs

Upscaling methodology (1)

1. Flow is upscaled such that <V>=U and Kef=Kef

2. The variance is calculated by the Cauchy Algorithm

3. The integral scale of V is calculated by same procedure

L L L

dL 0 0 03

1xxVV

0

2

2

2

/2

1/2dKKf

KK

KK

U Kef

efV

L L L

VV

V drLrLrLL 0 0 0 32132

28

rr

3/1

2

2

V

VVV II

Upscaling methodology (2)

4. The upscaled structure is made of cubical blocks of side 2IY of independent lognormal conductivities Y with mean <Y>=<Y> and variance σY

2

5. The resulting longitudinal dispersivity L is obtained by same procedure as the fine scale solution.

6. The procedure allows the calculation of the longitudinal dispersivity that needs to be supplemented to the upscaled medium in order to recover the fine-scale L

Results

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5

L/R

First Order

sy2=0.2

sy2=0.5

sy2=1

sy2=2

sy2=4

sy2=6

sy2=8

Pe=1000L

LL

Conclusions Upscaling causes smoothing of conductivity spatial

variations at scales smaller than that of discretization blocks. This results in a reduction of rate of spreading of solutes.

In order to correct for this loss, a fictitious upscaling macrodispersivity is introduced.

It is determined quantitatively for mean uniform flow, simplified formation structure and approximate solutions of flow and transport obtained in the past.

It is found that the value of the induced longitudinal macrodispersivity is enhanced by high degree of heterogeneity.

The breakthrough curve may be skewed for high heterogeneity and characterization by the second moment is not sufficient.

References

Dagan, G., Upscaling of dispersion coefficients in transport through heterogeneous formations, Computational Methods in Water Resources X, Kluwer Academic Publishers, Vol. 1, pp. 431-440, 1994

Rubin, Y., et al. The concept of block-effective macrodispersivity and a unified approach for grid-scale- and plume-scale-dependent transport, Journ. Fluid Mechanics, 395, pp 161-180, 1999

Fiori, A., I. Jankovic, G. Dagan, and V. Cvetkovic. Ergodic transport through aquifers of non-Gaussian logconductivity distribution and occurrence of anomalous behavior, Water Resour. Res., 43, W09407, doi:10.1029/2007WR005976, 2007.

Jankovic, I., A. Fiori, G. Dagan, The impact of local diffusion on longitudinal macrodispersivity and its major effect upon anomalous transport in highly heterogeneous aquifers, Advances in Water Resources, 2008.