Formulation of the Problem of Upscaling of Solute Transport in Highly Heterogeneous Formations

1. 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, Italy 2 Dept. of Civil, Structural and Environmental Engineering, SUNY, Buffalo, USA 3 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

2. 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

3. 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, σY2, IY); • As a consequence, V(x) and C are also random.

4. 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, σY2, 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?

5. Assumptions • 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) • A thin, ergodic plume of planar dimensions >> IY is injected on a plane at x=0 • Transport is quantified by the first <> and second σ2 temporal moments of the breakthrough curve (BTC) at a control plane at x>>IY

6. Upscaling problem • How can we ensure that <>=<> and σ2=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.

7. 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

8. The “Fine-scale” solution Lfs<<IY • 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

9. The “Fine-scale” solution: Previous results • Semi-analytical solution:

10. The Upscaled solution L>Lfs • 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?

11. Upscaling methodology (1) • Flow is upscaled such that <V>=U and Kef=Kef • The variance is calculated by the Cauchy Algorithm • The integral scale of V is calculated by same procedure

12. Upscaling methodology (2) • The upscaled structure is made of cubical blocks of side 2IY of independent lognormal conductivities Ywith mean <Y>=<Y> and variance σY2 • The resulting longitudinal dispersivity L is obtained by same procedure as the fine scale solution. • 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

13. Results

14. 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.

15. 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.