Nonlocal models of transport in multiscale porous media:something old and som. something borrowed...

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1 Nonlocal models of transport in multiscale porous media: something old and something new, something borrowed... Department of Mathematics Oregon State University

2 Outline 1 Flow and transport in subsurface, nomenclature Multiscale flow and transport in subsurface, experimental results 2 Literature review Double porosity models, diffusion+advection 3 Ideas and steps Computational experiments Construct affine approximations Model calculations Computational experiments with elements of upscaled model NSF Model adaptivity in porous media, DOE Modeling, Analysis, and Simulation of Multiscale Preferential Flow. Also, see presentations at NSF-CMBS Nevada 5/20-25, DOE Multiscale workshop Tacoma, 5/25-30 (links from my webpage)

3 Flow and transport in subsurface, nomenclature Multiscale flow and transport in subsurface, experimental results Flow coupled to transport F(Θ) = 0 with Θ = (u, p, c) Flow Diffusive-dispersive transport u = K p, u = 0 φ c t + (uc D(u) c) = 0 Definitions D(u) := diffusion + dispersion := d mol I + u (d long E(u) + d transv (I E(u))). E(u) = 1 u 2 u iu j D(u) d mol I + d long u E(u)

4 Flow and transport in subsurface, nomenclature Multiscale flow and transport in subsurface, experimental results Multiscale flow and transport, set-up Model F(Θ) = 0 with Θ = (p, u, c) φ c t u = K p, u = 0 + (uc D(u) c) = 0 K fast, K slow give u fast, u slow give D fast, D slow

5 Flow and transport in subsurface, nomenclature Multiscale flow and transport in subsurface, experimental results Advection+diffusion in multiscale media: tailing Breakthrough curves = total concentration at outlet MOVIE

6 Flow and transport in subsurface, nomenclature Multiscale flow and transport in subsurface, experimental results Advection+diffusion in multiscale media: tailing Breakthrough curves = total concentration at outlet MOVIE

7 Flow and transport in subsurface, nomenclature Multiscale flow and transport in subsurface, experimental results Experimental visualization by Haggerty et al Presentation at SIAM Annual 2004 by Haggerty [ZMH + 04] Brendan Zinn, Lucy C. Meigs, Charles F. Harvey, Roy Haggerty, Williams J. Peplinski, and Claudius Freiherr von Schwerin, Experimental visualization of solute transport and mass transfer processes in two-dimensional conductivity fields with connected regions of high conductivity, Environ Sci. Technol. 38 (2004),

8 Flow and transport in subsurface, nomenclature Multiscale flow and transport in subsurface, experimental results Experimental breakthrough curves Results from [ZMH + 04]

9 Flow and transport in subsurface, nomenclature Multiscale flow and transport in subsurface, experimental results Challenge in view of the experimental results Not-well separated scales: double porosity diffusion model does not fit in low/intermediate contrast regime ɛ 0 > 0 is fixed (perhaps the homogenized model not good enough? need a corrector?) K fast K slow small, moderate, intermediate, or large Evidence of advection-diffusion-dispersion in Ω slow and advection-dispersion in Ω fast Related project (Wood, Haggerty, Waymire, Thomann, Ramirez, OSU) on Taylor-Aris dispersion/skew diffusion models other results on tailing [HG95, HMM00, HFMM01] Formidable challenge: find an upscaled model similar to double-porosity which can capture all of the above

10 Literature review Double porosity models, diffusion+advection Todd Arbogast, Jim Douglas, Jr., and Ulrich Hornung, Derivation of the double porosity model of single phase flow via homogenization theory, SIAM J. Math. Anal. 21 (1990), no. 4, MR 91d:76074 Todd Arbogast, Analysis of the simulation of single phase flow through a naturally fractured reservoir, SIAM J. Numer. Anal. 26 (1989), no. 1, MR 90e:76122, On the simulation of incompressible, miscible displacement in a naturally fractured petroleum reservoir, RAIRO Modél. Math. Anal. Numér. 23 (1989), no. 1, MR MR (91d:76073), Computational aspects of dual-porosity models, Homogenization and porous media (U. Hornung, ed.), Interdiscip. Appl. Math., vol. 6, Springer, New York, 1997, pp MR John D. Cook and R. E. Showalter, Microstructure diffusion models with secondary flux, J. Math. Anal. Appl. 189 (1995), no. 3, MR 96j:76138 J. Jr. Douglas and T. Arbogast, Dual-porosity models for flow in naturally fractured reservoirs, Dynamics of Fluids in Hierarchical Porous Media (J. H. Cushman, ed.), Academic Press, 1990, pp J. Douglas, Jr., M. Peszyńska, and R. E. Showalter, Single phase flow in partially fissured media, Transp. Porous Media 28 (1997), Jim Douglas, Jr. and Anna M. Spagnuolo, The transport of nuclear contamination in fractured porous media, J. Korean Math. Soc. 38 (2001), no. 4, , Mathematics in the new millennium (Seoul, 2000). MR MR (2002g:76111) Mark N. Goltz and Paul Roberts, Using the method of moments to analyze three-dimensional diffusion-limited solute transport from temporal and spatial perspectives, Water Res. Research 23 (1987), no. 8, R. Haggerty, S.W. Fleming, L.C. Meigs, and S.A. McKenna, Tracer tests in a fractured dolomite 2. analysis of mass transfer in single-well injection-withdrawal tests, Water Resources Research 37 (2001), no. 5,

11 Literature review Double porosity models, diffusion+advection R. Haggerty and S.M. Gorelick, Multiple-rate mass transfer for modeling diffusion and surface reactions in media with pore-scale heterogeneity, Water Resources Research 31 (1995), no. 10, R. Haggerty, S.A. McKenna, and L.C. Meigs, On the late-time behavior of tracer test breakthrough curves, Water Resources Research 36 (2000), no. 12, Ulrich Hornung and Ralph E. Showalter, Diffusion models for fractured media, J. Math. Anal. Appl. 147 (1990), no. 1, MR MR (91d:76072) Małgorzata Peszyńska, Fluid flow through fissured media. mathematical analysis and numerical approach, Ph.D. thesis, University of Augsburg, Augsburg, Germany, Małgorzata Peszyńska, Analysis of an integro-differential equation arising from modelling of flows with fading memory through fissured media, J. Partial Differential Equations 8 (1995), no. 2, MR MR (96a:45007), Finite element approximation of diffusion equations with convolution terms, Math. Comp. 65 (1996), no. 215, MR MR (96j:65104) M. Peszyńska, Mortar adaptivity in mixed methods for flow in porous media, International Journal of Numerical Analysis and Modeling 2 (2005), no. 3, M. Peszyńska, M. F. Wheeler, and I. Yotov, Mortar upscaling for multiphase flow in porous media, Computational Geosciences 6 (2002), Sanchez-Villa X. and J. Carrera, On the striking similarity between the moments of breakthrough curves for a heterogeneous medium and a homogeneous medium with a matrix diffusion term, Journal of Hydrology 294 (2004), Brendan Zinn, Lucy C. Meigs, Charles F. Harvey, Roy Haggerty, Williams J. Peplinski, and Claudius Freiherr von Schwerin, Experimental visualization of solute transport and mass transfer processes in two-dimensional conductivity fields with connected regions of high conductivity, Environ Sci. Technol. 38 (2004),

12 Literature review Double porosity models, diffusion+advection Notation Ω = i ˆΩ i, Ω slow = i=1 Ω i, Ω slow Ω fast i Γ i Ω = Ω slow Ω fast i Γ i ˆΩ i ɛ 0

13 Literature review Double porosity models, diffusion+advection Averaged (single porosity) model Exact model at microscale replaced by homogenized model D = D slow, D fast with D Compute homogenized coefficients D D jk = 1 D jk (y)(δ jk + k ω j (y))da Ω 0 Ω 0 { D ωj (y) = (De j ), y Ω 0 ω j Ω 0 periodic

14 Literature review Double porosity models, diffusion+advection Averaged (single porosity) model Exact model at microscale replaced by homogenized model D = D slow, D fast with D Compute homogenized coefficients D D jk = 1 D jk (y)(δ jk + k ω j (y))da Ω 0 Ω 0 { D ωj (y) = (De j ), y Ω 0 ω j Ω 0 periodic But this doesn t work very well for time-dependent problems with large contrast D fast /D slow

15 Literature review Double porosity models, diffusion+advection Double porosity model: main idea I Exact model at microscale replaced by homogenized model with two sheets D = D slow, D fast with D plus cell model Compute homogenized coefficients D 1 D jk = D jk (y)(δ jk + k ω j (y))da Ω 0 Ω { 0 D ωj (y) = (De j ), y Ω 0,fast ω j Ω 0 periodic

16 Literature review Double porosity models, diffusion+advection Double porosity model: main idea I Exact model at microscale replaced by homogenized model with two sheets D = D slow, D fast with D plus cell model Compute homogenized coefficients D 1 D jk = D jk (y)(δ jk + k ω j (y))da Ω 0 Ω { 0 D ωj (y) = (De j ), y Ω 0,fast ω j Ω 0 periodic This formulation introduces nonlocal effects and works very well for time-dependent problems with large contrast D fast /D slow

17 Literature review Double porosity models, diffusion+advection Double porosity model: main idea II Exact model at microscale replaced by homogenized model with two sheets D = D slow, D fast Global equation, x Ω with D plus cell model Cell problem, x Ω i φ c t + i χ i q i (t) D c = 0 q i (t) = Π 0,i (Π 0,i( c)) φ slow c i t D slow c i = 0 c i Γi = Π 0,i ( c) This formulation works well for single & multi-phase multicomponent problems and has been implemented in commercial reservoir simulators

18 Literature review Double porosity models, diffusion+advection Recall double porosity models for diffusion Exact ɛ 0 model F ɛ0 (Θ ɛ0 ) = 0 φ α c α t D α c α = 0, x Ω α, α = fast, slow plus interface conditions on Ω slow Ω fast : Approximate microstructure model [Arb89a, Arb97] F ɛ0 ( Θ ɛ0 ) = 0 φ c t + i c fast = c slow, D fast c fast ν = D slow c slow ν χ i q i (t) D c = 0 q i (t) = Π 0,i (Π 0,i( c)) also for multiphase problems[da90] Homogenized model [ADH90, HS90, Pes92] F ɛ (Θ ɛ ) F 0 (Θ 0 ) = 0 φ c c +τ t t D c = 0, analysis and convergence computational approach[pes95, Pes96, DPS97]

19 Literature review Double porosity models, diffusion+advection Local (cell) problem and averages Π 0, Π 0 Local averages Π 0,i, Π 0,i 1 Π 0,i ξ := ξ(x)da ˆΩ i ˆΩ i Π 0,i γ := 1 c i (γ) D slow c i (γ)(x, t) νds = Π 0,i (φ slow ) ˆΩ i Γ i t where c i = c i (γ) solves the local (cell) problem φ slow c i t D slow c i = 0, x Ω i, c i = γ(x, t), x Ω i

20 Literature review Double porosity models, diffusion+advection Double porosity models for diffusion-advection Exact ɛ 0 model F ɛ0 (Θ ɛ0 ) = 0 φ c α t (D α c α u α c α ) = 0, x Ω α, α = fast, slow plus interface conditions on Ω slow Ω fast Approximate microstructure model [Arb89b] F ɛ0 ( Θ ɛ0 ) = 0 φ c t + i χ i q i ( D c ũ c) = 0, q i (t) = Π 1,i (Π 1,i( c)) Π 1 =local L 2 projections onto linears, Π 1 its dual. Numerical model only. Limit ɛ 0 model[ds01] F 0 (Θ 0 ) = 0 φ c t +φ c slow t ( D c ũ c) = 0 φ slow c t Π 0,i (Π 1,i( c)) Π 1 =local Taylor. Cell problem: u slow 0, symmetry exploited.

21 Literature review Double porosity models, diffusion+advection Why these are not enough... and other related results Approximate microstructure model [Arb89b] F ɛ0 ( Θ ɛ0 ) = 0 Numerical model only. Want to have F ɛ0 ( Θ ɛ0 ) = 0 Limit ɛ 0 model[ds01] F 0 (Θ 0 ) = 0 Cell problem: u slow 0. Use Π 0 for flux. constructed with global (upscaled) flavor (akin diffusion model φ c c t +τ t ( D c ũ c) = 0,) or secondary diffusion as in [CS95] account for (lack of) separation of scales ɛ 0 > 0 and advection-dispersion track transition between different regimes of phenomena

22 Literature review Double porosity models, diffusion+advection Why these are not enough... and other related results Approximate microstructure model [Arb89b] F ɛ0 ( Θ ɛ0 ) = 0 Numerical model only. Want to have F ɛ0 ( Θ ɛ0 ) = 0 Limit ɛ 0 model[ds01] F 0 (Θ 0 ) = 0 Cell problem: u slow 0. Use Π 0 for flux. constructed with global (upscaled) flavor (akin diffusion model φ c c t +τ t ( D c ũ c) = 0,) or secondary diffusion as in [CS95] account for (lack of) separation of scales ɛ 0 > 0 and advection-dispersion track transition between different regimes of phenomena Other models known in hydrology/ applied math and geosciences Gerke van Genuchten 1993 (for Richards equation) nonlocal models of dispersion (Cushman et al)

23 Ideas and steps Computational experiments Construct affine approximations Model calculations Computational experiments with elements of upscaled model Building the upscaled model F ɛ0 ( Θ ɛ0 ) = 0 Want to have F ɛ0 ( Θ ɛ0 ) = 0 constructed with global flavor. Computational experiments on microscale Building the model use the model á la [Arb89b] but with different Π 1, Π 1, construct convolution approximations of all terms á la [Pes92] with a family of kernels simulate the upscaled nonlocal model for a continuum of regimes of phenomena kernels reflect the regimes

24 Ideas and steps Computational experiments Construct affine approximations Model calculations Computational experiments with elements of upscaled model Computational experiments at microscale GOAL: reproduce qualitatively experimental results, understand significance of different regimes of flow and trasport MOVIES

25 Ideas and steps Computational experiments Construct affine approximations Model calculations Computational experiments with elements of upscaled model Choice of ffine approximations Π 1 Recall Π 0 f := 1 Ω 0 Ω 0 f (x)da, assume here Ω 0 = 1. Denote x C - center of mass of Ω 0. General affine approximation f (x) Π 1 f := m + n x, x Ω 0 Choice of m, n Taylor (f C 1 (Ω 0 )) about midpoint f (x) f (x C ) + f (x C )(x x C ) L 2 (Ω 0 )-projection onto affines that is: (f, v) Ω0 = (m + n x, v) Ω0, affine v H 1 (Ω 0 ) projection: f (x) Π 1 f := Π 0 f + Π 0 f (x x C ) Basis functions not necessarily orthogonal.

26 Ideas and steps Computational experiments Construct affine approximations Model calculations Computational experiments with elements of upscaled model Dual affine approximations Π 1 to Π 1 L 2 (Ω 0 )-projection onto affines, use an orthonormal basis (φ 0, φ 1, φ 2 ) Π 1 f (x) = f k φ k (x) k Flux calculations Π 1q = k q k φ k (x), q k = Π 0 (qφ k ) H 1 (Ω 0 ) projection: f (x) Π 0 f + Π 0 f (x x C ) Note (1, (x x C ) 1, (x x C ) 2 ) are not necessarily orthogonal! Π 1q = q 0 ξ 0 (x) + q 1 ξ 1 (x) + q 2 ξ 2 (x) We use H 1 (Ω i )-projection denoted Π 1,i Π i and Pi1,i Π i

27 Ideas and steps Computational experiments Construct affine approximations Model calculations Computational experiments with elements of upscaled model Calculate Π i and Π i Recall Π i : H 1 (Ω) H 1 (ˆΩ i ) ( Π i (w)(x) 1 w(y) da + ˆΩ i ˆΩ i 2 [ ] ) k w(y) da (x k (ˆx C i ) k ) ˆΩ i k=1 Dual Π i : (H1 (ˆΩ i )) (H 1 (Ω)) affine approximation of flux q H 1/2 (Γ i ) Π i (q), w = q, Π i(w), w C0 (Ω) with q, v := i Γ i q(s)v(s)ds uses moments Mi 0,M 1 i q, Π i (w) = 1 [ ] q(s) wda + (s x c i ˆΩ i ) wds Γ i ˆΩ i ˆΩ i = χ i (x) 1 q(s)dsw(x)da+ χ i (x) 1 q(s)(s x C )ds w(x)da Ω ˆΩ i Γ i Ω ˆΩ i Γ i = χ i (x)mi 0 (q)w(x)da ( χ i (x)m 1 i (q)) w(x)da = Π i (q), w Ω Ω

28 Calculations Π i and Π i summary Ideas and steps Computational experiments Construct affine approximations Model calculations Computational experiments with elements of upscaled model Affine approximation Π i : H 1 (Ω) H 1 (ˆΩ i ) Π i (w)(x) Π 0 w + Π 0 ( w) (x x C ) Its dual Π i : H 1 (ˆΩ i ) H 1 (Ω) pointwise Π i (q)(x) = χ i(x)m 0 i (q) χ i (x)m 1 i (q) Note the last term is a scaled line source!

29 Ideas and steps Computational experiments Construct affine approximations Model calculations Computational experiments with elements of upscaled model Application of Green s theorem to moments For any smooth region D, smooth v = (v 1, v 2 ) and ˆx C D, ( v)(x k (ˆx C ) k )da = v ν(x k (ˆx C ) k )ds D hence for the flux from the cell q(s) = (D i c i (s) v i c i (s)) ν D D v k da M 1 i (q) ( ) = (D i c i (y, t) v i c i (y, t)) (y ˆx C i ) + D i c i (y, t) v i c i (y, t) da. Ω i = c i χ i (x) (φ i i Ω i t (y, t)(y ˆxC i ) + D i c i (y, t) v i c i (y, t)) da

30 Cell problem: elementary solutions Cell problem solved for u j i, j = 0, 1, 2 Ideas and steps Computational experiments Construct affine approximations Model calculations Computational experiments with elements of upscaled model φ slow u j i t (D slow u j i u slow u j i ) = 0, x Ω i, u j i (x, 0) = 0, x Ω i ui 0 Γi = 1, u 1 i Γi = (x x C i ) 1, ui 2 Γi = (x x C i ) 2 Represent the solution to the cell problem φ slow c i t (D slow c i u slow c i ) = 0, x Ω i, c i (x, 0) = 0, x Ω i c i Γi = Π 1,i (c )(x, t) A 0 i (t) + (A1 i, A2 i ) (x xc i ), By linearity c i (x, t) = t 2 u j i 0 j=0 t (x, t s)aj i (s)ds = 2 u j i (x, ) j=0 t A j i

31 Putting it together Ideas and steps Computational experiments Construct affine approximations Model calculations Computational experiments with elements of upscaled model Solution to the cell problem c i (x, t) = 2 u j i (x, ) j=0 t A j i c i Γi c i φ slow (D slow c i u slow c i ) t = 0, x Ω i, c i (x, 0) = 0, x Ω i = Π 1,i (c )(x, t) A 0 i (t) + (A1 i, A2 C i ) (x ˆx i ), x Γ i Use u j i and A j so that Π 1,i (c )(x, t) A 0 i (t) + (A1 i, A2 i Compute the normal flux q(s) (D slow c i u slow c i ) η, s Γ i ) (x ˆx C i )... and its affine approximations Π 1,i q using A j, u j i... and the moments M 0 i (q), M 1 i (q).

32 Convolution kernels Ideas and steps Computational experiments Construct affine approximations Model calculations Computational experiments with elements of upscaled model Write the moments M 0 i (q), M 1 i (q) in terms of A k and u j i Define the kernels for each i and each function u j i, j = 0, 1, 2 by S j0 i (t) S jk i (t) T j i (t) (T j1 i j ui φ i (x, t) da, 0 j 2. Ω i t j ui φ i Ω i t (x, t)(x k (ˆx C i ) k ) da, 1 k 2., T j2 i ) (D i v i ) u j i (x, t) da. Ω i t Together we have 15 scalar kernels, some of which will be zero due to symmetry/lack of thereof

33 Computational experiments Ideas and steps Computational experiments Construct affine approximations Model calculations Computational experiments with elements of upscaled model Model summary (suppress i, χ i etc.) t {φ c + S 00 c + (S 01, S 02 ) c (S 10 c + (S 11, S 12 ) c )} {D c v c + T 0 c + (T 1, T 2 ) c } = 0 Convolution kernels for different regimes of diffusion vs advection no advection with advection with signficant advection Upscaled problem with nonlocal terms Comparison between exact model and upscaled model with nonlocal terms and computed kernels

34 Ideas and steps Computational experiments Construct affine approximations Model calculations Computational experiments with elements of upscaled model Convolution kernels: regimes of Pe = advection diffusion Solution u j and the associated kernels S j0, S j1, T j1 j = 0 No advection With advection Large advection j = 1

35 Upscaled model: numerical treatment Ideas and steps Computational experiments Construct affine approximations Model calculations Computational experiments with elements of upscaled model nonlocal diffusion (FE+time-stepping on c t ) [Pes95] stable, convergence O( t + h 2 ), singular kernels nonlocal diffusion with secondary diffusion terms (as in viscoelasticity) ([Thomee,Lin,Ewing 91-01]) with nonsingular kernels nonlocal advection (FD+time-stepping): stable, convergent O( t + h) [P06], singular kernels nonlocal advection+diffusion+secondary diffusion+secondary advection: issues of memory storage, need adaptive treatment relative importance of the terms c, 2 c : adaptivity a must pore-scale modeling (with K. Augustson) unsaturated flow models use experimental results by Wildenschild et al

36 Ideas and steps Computational experiments Construct affine approximations Model calculations Computational experiments with elements of upscaled model

37 Ideas and steps Computational experiments Construct affine approximations Model calculations Computational experiments with elements of upscaled model Connection to mortar upscaling [PWY02, Pes05]

38 Ideas and steps Computational experiments Construct affine approximations Model calculations Computational experiments with elements of upscaled model

39 Summary: the upscaled model Ideas and steps Computational experiments Construct affine approximations Model calculations Computational experiments with elements of upscaled model ( N t φ c (x, t) + χ i (x) Si 00 (t τ) 1 c (y, τ) da dτ t i=1 0 ˆΩ i ˆΩ i N t 2 + χ i (x) S j0 i (t τ) 1 j c (y, τ) da dτ ˆΩ i ˆΩ i + N i=1 N i=1 i=1 t χ i (x) 0 t χ i (x) 0 2 j=0 0 (S j1 i j=1 (t τ), S j2 i (t τ)) 1 j c (y, τ) da dτ ˆΩ i ˆΩ i ( D c (x, t) v c (x, t) 2 (T j1 i (t τ), T j2 i (t τ)) 1 j c (y, τ) da dτ ˆΩ i ˆΩ i j=0 = 0, x Ω, t > 0.

40 Ideas and steps Computational experiments Construct affine approximations Model calculations Computational experiments with elements of upscaled model