caa a22 4500 46316377X CHVBK 20180406164755.0 cr unu---uuuuu 170326e20070601xx s 000 0 eng 10.1007/s11242-006-9040-0 doi (NATIONALLICENCE)springer-10.1007/s11242-006-9040-0 Upscaling, relaxation and reversibility of dispersive flow in stratified porous media [Elektronische Daten] [C. Berentsen, C. van Kruijsdijk, M. Verlaan] Dispersive tracer released in a unidirectional velocity field belonging to a stratified porous of finite height describes a transition, called relaxation, from a convective dominated behaviour for short times to Fickian behaviour for asymptotic long times. The temporal relaxation state of the tracer is controlled by the transverse mixing term. In most practical applications, the orders of the time and length scales of the relaxation mechanism are such that in an upscaled model of a stratified medium the dispersive flux is in a pre-asymptotic state. Explicit modelling of the relaxation of the dispersive flux in the pre-asymptotic region is required to improve the accuracy. This paper derives a pre-asymptotic one-dimensional upscaled model for the transverse averaged tracer concentration. The model generalises Taylor dispersion (Proc. R. Soc. London 219, 186-203 (1953)) and extends the method of Camacho (Phys. Rev. E 47(2), 1049-1053 (1993a); Phys. Rev. E 48 (1993b)) to dispersion tensors that may vary as function of the transverse direction. In the averaging step, the governing two-dimensional equation is first spectrally decomposed in terms of the eigenfunctions of the transverse mixing term. Next, the resulting modal relaxation equations are combined into an effective relaxation equation for the extended dispersive Taylor flux. Contrary to the one-dimensional Fickian approach, the upscaled model approximates the multi-scale relaxation behaviour as a single scale relaxation process and accounts for the partial reversibility of convective dispersion upon reversal of the flow direction. The upscaled model is evaluated against the original two-dimensional model by means of moment analysis. The longitudinal tracer variance predicted by our model is quantitatively correct in the short and long time limits and is qualitatively correct for intermediate times. Springer Science+Business Media B.V., 2007 Relaxation nationallicence Reversibility nationallicence Taylor dispersion nationallicence Non-Fickian dispersion nationallicence Upscaling nationallicence Tracer flow nationallicence Stratified flow nationallicence $${\bf B}/{\tilde{\bf B}}/{\bf B}_N$$ : [m/s] Modal velocity interaction matrix w.r.t. {eigenfunction base of −∂ y (D T (y)∂ y )/cosine Fourier base/eigenfunction base of $${\tilde {\bf D}}_{T,N}$$ } nationallicence c : [kg/m3] 2D concentration nationallicence c 0 : [kg/m3]Transverse or height averaged concentration nationallicence $$c_n /\tilde {c}_n /c_{N,n}$$ : [kg/m3]n-th modal concentration w.r.t. {eigenfunction base of −∂ y (D T (y)∂ y )/cosine Fourier base/eigenfunction base of $${\tilde {\bf D}}_{T,N}$$ } nationallicence $${\bf c}/{\tilde {\bf c}}$$ : [kg/m3] Infinite column vector with modal concentrations $$\{c_{n}/\tilde{c}_{n}\} for n\, = 1,{\ldots},\infty$$ nationallicence $${\tilde {\bf c}}_N/{\bf c}_N$$ : [kg/m3] Finite column vector with modal concentrations $$\{\tilde{c}_{N, n} / {c}_{N, n}\}$$ for n = 1,..., N nationallicence d : [m] Total layer height or transverse width nationallicence D eff : [m2/s] Effective dispersion coefficient nationallicence D L : [m2/s] Longitudinal dispersion coefficient nationallicence D L ,0 : [m2/s] Height or transverse averaged of longitudinal dispersion coefficient nationallicence $${\boldsymbol D}_{L,n}/{\boldsymbol{\tilde D}}_{L,n}$$ : [m2/s] n-th spectral mode of longitudinal dispersion coefficient w.r.t {eigenfunction base of −∂ y (D T (y)∂ y )/cosine Fourier base} nationallicence $${\bf D}_{L} /{\tilde {\bf D}}_{L} /{\bf D}_{L,N}$$ : [m2/s] Modal interaction matrix resulting from spectral decomposition of D L (y)c (x,y,t) w.r.t. {eigenfunction base of −∂ y (D T (y)∂ y )/ cosine Fourier base/eigenfunction base of $${\tilde {\bf D}}_{T,N}$$ } nationallicence $${\bf d}_{L} / {\tilde {\bf d}}_{L}$$ : [m2/s] Infinite column vector with spectral modes of longitudinal dispersion coefficients $$\{{\boldsymbol D}_{L,n}/{\boldsymbol{\tilde D}}_{L,n}\}$$ for n = 1,...,,∞ nationallicence D mol : [m2/s] Molecular diffusion coefficient nationallicence D T ( y ) : [m2/s] Transverse dispersion coefficient nationallicence $${\tilde {\bf D}}_T$$ : [m2/s] Matrix resulting from cosine Fourier decomposition of −∂ y (D T (y)∂ y ) nationallicence $${\boldsymbol{\tilde D}}_{T,N}$$ : [m2/s] $${\boldsymbol{\tilde D}}_{T}$$ truncated at N modes $$(={\boldsymbol{\tilde D}}_{T}(1\ldots N, 1\ldots N))$$ nationallicence E ξ , x , k : k-th non-centred non-normalised spatial moment of quantity ξ nationallicence F : [-] Tortuosity factor nationallicence I : [-] Unity matrix nationallicence J E , n / J E , T : [kg/m2s] Extended {modal dispersive flux/Taylor flux} nationallicence M ξ , x , k : k-th non-centred normalised spatial moment of quantity ξ nationallicence M 0 : [kg] Tracer mass nationallicence t / t r : [s] {Time/Reversal time} nationallicence $${\bf T}_{\cos,\phi}$$ : [-] Transformation matrix from cosine Fourier base to eigenfunction base of −∂ y (D T (y)∂ y ) nationallicence T N : [-] Transformation matrix from cosine Fourier base truncated at n-modes to eigen function base of $${\tilde{\bf D}}_{T,N}$$ nationallicence v ( y ) : [m/s] Longitudinal velocity nationallicence v 0 : [m/s] Height or transverse averaged velocity nationallicence $$v_n /\tilde {v}_n /v_{N,n}$$ : (m/s) n-th modal velocity belonging to {eigenfunction base of −∂ y (D T (y)∂ y )/cosine Fourier base/eigenfunction base of $${\tilde {\bf D}}_{T,N}$$ } nationallicence $${\bf v}/{\tilde{\bf v}}$$ : [m/s] Finite column vector with modal velocities { $${\bf v}_{n}/\tilde{\bf v}_{n}\}$$ for n=1,...,∞ nationallicence $${\tilde {\bf v}}_N /{\bf v}_N$$ : [m/s] Infinite column vector with model velocities $$\{{\tilde {\bf v}}_n/{\bf v}_{N,n}\}$$ for n = 1,...,N nationallicence x / y : [m] {Longitudinal/transverse} co-ordinate nationallicence x 0 : [m] Initial longitudinal position nationallicence Greek nationallicence α L /α T : [m] {Longitudinal/Transverse} dispersivity nationallicence $$\phi _n (y)$$ : [-] n-th eigenfunction of operator −∂ y (D T (y )∂ y ) nationallicence λ n /λ N , n : [s−1] n-th eigenvalue of {operator −∂ y (D T (y)∂ y ) / matrix $${\tilde {\bf D}}_{T,N}$$ } nationallicence $${\Lambda}/{\Lambda}_N$$ : [s−1] (Diagonal) matrix with eigenvalues of {−∂ y (D T (y)∂ y ) / $${\tilde {\bf D}}_{T,N}$$ } nationallicence μ : Mean nationallicence σ : Co-variance or variance nationallicence σ c , x 2 : [m2] Spatial variance belonging to concentration nationallicence $$\Delta \sigma _{\rm uni}$$ : [m2] Asymptotic deviation of variance from Fickian behaviour for particle distributions that are initially distributed uniform over the height nationallicence τ n /τ N , n : [s] $$=(\lambda _n^{-1} /\lambda _{N,n}^{-1} )$$ n-th modal relaxation time nationallicence τ : [s] Relaxation time nationallicence τeff : [s] Effective relaxation time nationallicence Subscript nationallicence c : Concentration nationallicence eff : Effective nationallicence L : Longitudinal nationallicence N : Truncated at N modes nationallicence n : n-th nationallicence n , m : With respect to the interaction of the n-th and m-th base functions (n,m > 0) nationallicence rev : After flow reversal nationallicence T : Transverse nationallicence TEL : Generalised Telegraph equation nationallicence v : Velocity nationallicence x : Spatial nationallicence ∞ : For asymptotic long times nationallicence overbars nationallicence ~ : [-] W.r.t cosine Fourier base nationallicence Berentsen C. Department of Geotechnology, Delft University of Technology, Mijnbouwstraat 120, 2628 RX, Delft, The Netherlands aut van Kruijsdijk C. Department of Geotechnology, Delft University of Technology, Mijnbouwstraat 120, 2628 RX, Delft, The Netherlands aut Verlaan M. Department of Geotechnology, Delft University of Technology, Mijnbouwstraat 120, 2628 RX, Delft, The Netherlands aut Transport in Porous Media Kluwer Academic Publishers 68/2(2007-06-01), 187-218 0169-3913 68:2<187 2007 68 11242 https://doi.org/10.1007/s11242-006-9040-0 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s11242-006-9040-0 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Berentsen C. Department of Geotechnology, Delft University of Technology, Mijnbouwstraat 120, 2628 RX, Delft, The Netherlands aut NATIONALLICENCE 700 1- van Kruijsdijk C. Department of Geotechnology, Delft University of Technology, Mijnbouwstraat 120, 2628 RX, Delft, The Netherlands aut NATIONALLICENCE 700 1- Verlaan M. Department of Geotechnology, Delft University of Technology, Mijnbouwstraat 120, 2628 RX, Delft, The Netherlands aut NATIONALLICENCE 773 0- Transport in Porous Media Kluwer Academic Publishers 68/2(2007-06-01), 187-218 0169-3913 68:2<187 2007 68 11242 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer