Multiscale Computation for Incompressible Flow and Transport Problems

Multiscale Computation for Incompressible Flow and Transport Problems Thomas Y. Hou Applied Mathematics, Caltech Collaborators: Y. Efenidev (TAMU), V. Ginting (Colorado), T. Strinopolous (Caltech), Danping Yang (Shandong Univ.), H. Ran (Caltech) Multiscale Computation for Incompressible Flow and Transport Problems p.1/66

Introduction Subsurface flows and transport are affected by heterogeneities at multiple scales (pore scale, core scale, field scale). Because of wide range of scales direct numerical simulations are not affordable. Upscaling of flow and transport parameters is commonly used in practice. Multiscale approaches are developed as an alternative to perform upscaling in the solution space. Multiscale Computation for Incompressible Flow and Transport Problems p.2/66

Outline Multiscale finite element methods (MsFEM) as a subgrid modeling technique; Applications of multiscale finite element methods to porous media flows; Upscaling of two-phase flow in flow-based coordinate system; Multiscale computation for the Navier-Stokes equations; Multiscale Computation for Incompressible Flow and Transport Problems p.3/66

Darcy s law and permeability Darcy s empirical law, 1856: The volumetric flux u(x, t) (Darcy velocity) is proportional to the pressure gradient u = k p = K p, µ where k(x) is the measured permeability of the rock, µ is the fluid viscosity, p(x) is the fluid pressure, u(x) is the Darcy velocity. We obtain the second order elliptic system u = K p in Q Darcy s Law div(u) = f in Q conservation Sand Water 000000000000000000000000000 111111111111111111111111111 000000000000000000000000000 111111111111111111111111111 000000000000000000000000000 111111111111111111111111111 000000000000000000000000000 111111111111111111111111111 000000000000000000000000000 111111111111111111111111111 000000000000000000000000000 111111111111111111111111111 000000000000000000000000000 111111111111111111111111111 000000000000000000000000000 111111111111111111111111111 000000000000000000000000000 111111111111111111111111111 000000000000000000000000000 111111111111111111111111111 000000000000000000000000000 111111111111111111111111111 000000000000000000000000000 111111111111111111111111111 000000000000000000000000000 111111111111111111111111111 000000000000000000000000000 111111111111111111111111111 Multiscale Computation for Incompressible Flow and Transport Problems p.4/66

A natural porous media 000000000000000 1111111111111110000000000000 000000000000000 1111111111111110000000000000 000000000000000 111111111111111 Rock Facies 2 00000000000000 11111111111111 000000000000000 111111111111111 00000000000000 11111111111111 000000000000000 111111111111111 00000000000000 11111111111111 000000000000000 111111111111111 (1 1000m) 00000000000000 11111111111111 000000000000000 111111111111111 00000000000000 11111111111111 000000000000000 111111111111111 00000000000000 11111111111111 Fault (1e 3m) 000000000000000 111111111111111 00000000000000 11111111111111 1111111111111100000000000000 1111111111111100000000000000 000000000000000 111111111111111 01 00000000000000 00000000000000 11111111111111 11111111111111 000000000000000 111111111111111 01 00000000000000 11111111111111 000000000000000 111111111111111 01 00000000000000 11111111111111 000000000000000 111111111111111 01 00000000000000 11111111111111 Vugs (0.01m) 000000000000000 111111111111111 00000000000000 11111111111111 000000000000000 111111111111111 00000000000000 11111111111111 Fracture (1e 4 m) 000000000000000 111111111111111 0000000000000 1111111111111 0000000000000000 1111111111111111 01 0000000000000 1111111111111 0000000000000000 1111111111111111 01 0000000000000 1111111111111 0000000000000000 1111111111111111 0000000000000 1111111111111 0000000000000000 1111111111111111 0000000000000 1111111111111 0000000000000000 1111111111111111 0000000000000 1111111111111 0000000000000000 1111111111111111 0000000000000 1111111111111 0000000000000000 1111111111111111 Rock Facies 1 000000000000 111111111111 Shale (0.01m) 00000000000000000 11111111111111111 000000000000 111111111111 00000000000000000 11111111111111111 000000000000 111111111111 00000000000000000 11111111111111111 (1 1000m) 000000000000 111111111111 00000000000000000 11111111111111111 000000000000 111111111111 00000000000000000 11111111111111111 1111111111100000000000000000 11111111111111111 00 11 01 00 11 01 01 01 01 00 11 01 01 01 00 11 00 11 00 11 00 11 00 11 01 01 00 1101 01 00 11 01 01 01 00 11 01 00 11 01 01 01 00 11 00 11 00 11 00 11 01 00 11 01 00 11 01 00 11 01 00 11 01 00 11 01 01 01 01 00 11 01 01 00 1101 00 11 01 01 01 Rock Grains and Voids (1e 5 m) Multiscale Computation for Incompressible Flow and Transport Problems p.5/66

Requirements/Challenges Accuracy and Robustness Valid for different types of subsurface heterogeneity 3 2 9 8 7 8 6 1 0 6 5 4 4 2 1 3 2 0 2 1 0 2 3 1 4 6 Left: two-point variogram based. Middle: synthetic channelized. Right: channelized (North Sea). Applicable for varying flow scenarios Multiscale Computation for Incompressible Flow and Transport Problems p.6/66

Requirements/Challenges Accuracy and Robustness Valid for different types of subsurface heterogeneity 3 2 9 8 7 8 6 1 0 6 5 4 4 2 1 2 3 3 2 1 0 1 0 2 4 6 Left: two-point variogram based. Middle: synthetic channelized. Right: channelized (North Sea). Applicable for varying flow scenarios Multiscale Computation for Incompressible Flow and Transport Problems p.6/66

Requirements/Challenges Accuracy and Robustness Valid for different types of subsurface heterogeneity 3 9 8 8 2 7 6 1 6 5 4 0 4 2 1 3 2 0 2 3 1 0 1 2 4 6 Left: two-point variogram based. Middle: synthetic channelized. Right: channelized (North Sea). Applicable for varying flow scenarios Multiscale Computation for Incompressible Flow and Transport Problems p.6/66

Multiscale Finite Element Methods Babuska-Osborn (83,94), Hou and Wu (1997). Consider div(k ǫ (x) p ǫ ) = f, where ǫ is a small parameter. The central idea is to incorporate the small scale information into the finite element bases Basis functions are constructed by solving the leading order homogeneous equation in an element K (coarse grid or Representative Elementary Volume (RVE)) div(k ǫ (x) φ i ) = 0 in K It is through the basis functions that we capture the local small scale information of the differential operator. Multiscale Computation for Incompressible Flow and Transport Problems p.7/66

Multiscale Finite Element Methods Boundary conditions? φ i = linear function on K, φ i (x j ) = δ ij Coarse grid Fine grid Multiscale Computation for Incompressible Flow and Transport Problems p.8/66

Basis functions 40*(2.1+sin(60*(x y))+sin(60*y)) 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 K 0.2 0.2 0.1 0.1 bilinear 0 φ i = φ i 0 on K, where φi 0 are standard bilinear basis functions. Multiscale Computation for Incompressible Flow and Transport Problems p.9/66

Multiscale Finite Element Methods Except for the multiscale basis functions, MsFEM is the same as the traditional FEM (finite element method). Find p h ǫ V h = {φ i } such that k(p h ǫ, vh ) = f(v h ) v h V h, where k(u, v) = Z Q kij ǫ u v (x) dx, f(v) = x i x j Z Q fvdx The coupling of the small scales is through the variational formulation Solution has the form pǫ p 0 (x) + ǫn k (x/ǫ) x k p 0 (x), if k(x/ǫ) is periodic. If standard finite element method is used (linear basis functions): p ǫ p h ǫ H 1 (Q) Chα p ǫ H 1+α (Q) = O(hα ǫ α ). Thus, h ǫ, which is not affordable in practice. Multiscale Computation for Incompressible Flow and Transport Problems p.10/66

Relation to other approaches Subgrid modeling (by I. Babuska, T. Hughes, G. Allaire, T. Arbogast, and others) Subgrid stabilization (by F. Brezzi, L Franco, J.L. Guermond, T. Hughes, A. Russo, and others). Multiscale finite element methods (Hou and Wu, Allaire and Brizzi,...) Multiscale finite volume methods (Jenny, Lee and Tchelepi) Multiscale finite element methods based on two-scale convergence (C. Schwab,...) Variational multiscale method and residual free bubble methods (T. Hughes, F. Brezzi, T. Arbogast, G. Sangalli,...) Partition of Unity Methods (Babuska, Osborn, Fish,...) Wavelet based homogenization (Dorobantu, Engquist,...) Heterogeneous multiscale methods (E and Engquist...) Multiscale Computation for Incompressible Flow and Transport Problems p.11/66

Convergence property of MsFEM Consider k ǫ (x) = k(x/ǫ), where k(y) is periodic in y. h - computational mesh size. Theorem Denote p h ǫ the numerical solution obtained by MsFEM, and p ǫ the solution of the original problem. Then, If h >> ǫ, r ǫ p ǫ p h ǫ 1,Q C(h + h ) This theorem shows that MsFEM converges to the correct solution as ǫ 0 The ratio ǫ/h reflects two intrinsic scales. We call ǫ/h the resonance error The theorem shows that there is a scale resonance when h ǫ. Numerical experiments confirm the scale resonance. Multiscale Computation for Incompressible Flow and Transport Problems p.12/66

Resonance errors For problems with scale separation, we can choose h ǫ in order to avoid the resonance, but for problems with continuous spectrum of scales, we cannot avoid this resonance. To demonstrate the influence of the boundary condition of the basis function on the overall accuracy of the method we perform multiscale expansion of φ i Multiscale expansion of φ i φ i = φ 0 (x) + ǫφ 1 (x, x/ǫ) + ǫθ ǫ +..., φ1 (x, x/ǫ) = N k (x/ǫ) x k φ 0, where N k (x/ǫ) is a periodic function which depends on k(x/ǫ). θǫ satisfies div(k ǫ θ ǫ ) = 0 in K, θ i ǫ = φ 1(x, x/ǫ) + (φ i φ 0 )/ǫ on K Oscillations near the boundaries (in ǫ vicinity) of θ i ǫ lead to the resonance error Multiscale Computation for Incompressible Flow and Transport Problems p.13/66

Illustration of θ ǫ θ (without oversampling) 4 θ 0 y x Multiscale Computation for Incompressible Flow and Transport Problems p.14/66

Oversampling technique To capture more accurately the small scale information of the problem, the effect of θ ǫ needs to be moderated Since the boundary layer of θǫ is thin (O(ǫ)) we can sample in a domain with size larger than h + ǫ and use only interior sampled information to construct the basis functions. Let ψ k be the functions in the domain S, div(k ǫ (x) ψ k ) = 0 in S, ψ k = linear function on S, ψ k (s i ) = δ ik. S K Oversampled domain Coarse grid Fine grid Multiscale Computation for Incompressible Flow and Transport Problems p.15/66

Oversampling technique The base functions in a domain K S constructed as φ i K = X c ij ψ j K, φ i (x k ) = δ ik The method is non-conforming. The derivation of the convergence rate uses the homogenization method combined with the techniques of non-conforming finite element method (Efendiev et al., SIAM Num. Anal. 1999) By a correct choice of the boundary condition of the basis functions we can reduce the effects of the boundary layer in θ. Multiscale Computation for Incompressible Flow and Transport Problems p.16/66

Illustration of θ with oversampling θ (with oversampling) 2.8 θ 1.8 y x Multiscale Computation for Incompressible Flow and Transport Problems p.17/66

Numerical Results U h ǫ U h 0 l 2, ǫ/h = 0.64 h MsFEM MsFEM-os Resolved FEM l 2 rate l 2 rate h fine l 2 1/16 3.54e-4 7.78e-5 1/256 1.34e-4 1/32 3.90e-4-0.14 3.38e-5 1.02 1/512 1.34e-4 1/64 4.00e-4-0.05 1.97e-5 0.96 1/1024 1.34e-4 1/128 4.10e-4-0.02 1.03e-5 0.95 1/2048 1.34e-4 Multiscale Computation for Incompressible Flow and Transport Problems p.18/66

MsFEM for problems with scale separation For periodic problems or problems with scale separation, multiscale finite element methods can take an advantage of scale separation. Local problems can be solved in RVE div(k φ i ) = 0 φ i = φ i 0 on RV E. Basis functions can be also approximated φ i = φ i 0 + N ǫ φ i 0, where φ 0 i is linear basis functions and N is the periodic solution of auxiliary problem in ǫ-size period div(k ǫ (x)( N + I)) = 0. (cf. Durlofsky 1981, etc.). Note, the above procedure works when homogenization by periodization is applicable (e.g., random homogeneous case). Multiscale Computation for Incompressible Flow and Transport Problems p.19/66

Various generalizations Once basis functions are constructed, various global formulation (mixed, control volume finite element, DG and etc) can be used to couple the subgrid effects. Control volume finite element: Find ph V h such that Z V z k(x) p h n dl = Z V z q dx V z Q, where V z is control volume. Mixed finite element: In each coarse block K, we construct basis functions for the velocity field div(k(x) wi K ) = 1 in K K ( 1 k(x) wi K on nk = e K e K i i 0 else. For the pressure, the basis functions are taken to be constants. Multiscale Computation for Incompressible Flow and Transport Problems p.20/66

Applications of MsFEM to subsurface flow simulations Two-phase flow model. Darcy s law for each phase v i = kk i(s i ) µ i p i, i=1,2. Here k - permeability field representing the heterogeneities (micro-level information), p i - the pressure, v i - velocity, k i - relative permeability, S i -saturation, µ i - viscosity phase 1 phase 2 Multiscale Computation for Incompressible Flow and Transport Problems p.21/66

Applications of MsFEM At least two way one can apply MsFEM 1) Solve the pressure equation on the coarse-grid and solve the saturation equation on the fine-grid div(λ(s)k p) = 0 S + v f(s) = 0, t where v = λ(s)k p. Basis functions are updated only near sharp fronts. 0000000 1111111000000 0000000 1111111000000 0000000 1111111000000 0000000 1111111000000 0000000 1111111000000 0000000 1111111000000 0000000 1111111000000 0000000 1111111000000 0000000 1111111000000 0000000 1111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 Coarse grid Fine grid 2) The coarse-scale equation for the transport is obtained and coupled with MsFEM for pressure equation. Multiscale Computation for Incompressible Flow and Transport Problems p.22/66

MsFVEM applied to two-phase flow problem Given S 0, for n = 1,2,3,, do the following: find p n 1 h V h such that Z (IM)plicit (P)ressure (E)xplicit (S)aturation: V z λ(s n 1 )k(x) p n 1 h n dl = compute v n 1 = λ(s n 1 )k(x) p n 1 h time march on the saturation equation: Z V z q dx V z Q Z `Sn S n 1 Z Z dx + t n 1 f(s n 1 )v n 1 n dl = t n 1 c z c z c z q S dx Multiscale Computation for Incompressible Flow and Transport Problems p.23/66

Numerical Setting Rectangular domain is considered. The permeability field is generated using geostatistical libraries. The boundary conditions: no flow on top and bottom boundaries, a fixed pressure and saturation (S = 1) at the inlet (left edge), fixed pressure at the outlet (right edge). The production rate F = q0 /q, where q 0 the volumetric flow rate of oil produced at the outlet edge and q the volumetric flow rate of the total fluid produced at the outlet edge. The dimensionless time is defined as PV I = qt/v p, where t is time, V p is the total pore volume of the system. F(t) no flow 000000000000000000000000 111111111111111111111111 S=1 p=1 p=0 000000000000000000000000 111111111111111111111111 no flow Multiscale Computation for Incompressible Flow and Transport Problems p.24/66

A random porosity field with layered structure Multiscale Computation for Incompressible Flow and Transport Problems p.25/66

Recovery of fine scale velocity Left: Fine grid horizontal velocity field, N = 1024 1024. Right: Recovered horizontal velocity field from the coarse grid N = 64 64 using multiscale bases. Multiscale Computation for Incompressible Flow and Transport Problems p.26/66

Recovery of fine scale saturation Left: Fine grid saturation, N = 1024 1024. Right: Saturation computed from the recovered fine grid velocity field. Multiscale Computation for Incompressible Flow and Transport Problems p.27/66

Fractional flow and total flow curves 1 0.9 0.8 fine modified MsFVM standard MsFVM 11 10 fine modified MsFVM standard MsFVM 0.7 9 0.6 8 F 0.5 0.4 0.3 0.2 0.1 0 0 0.5 1 1.5 2 PVI Q 7 6 5 4 3 0 0.5 1 1.5 2 PVI Fractional flow and total flow for a realization of permeability field with exponential variogram and l x = 0.4, l z = 0.02, σ = 1.5. Multiscale Computation for Incompressible Flow and Transport Problems p.28/66

Channelized permeability fields Benchmark tests: SPE 10 Comparative Project 8 6 4 2 0 2 4 6 Multiscale Computation for Incompressible Flow and Transport Problems p.29/66

Channelized reservoir 1 fine scale model standard MsFVEM 350 fine scale model standard MsFVEM 0.8 300 0.6 250 F Q 200 0.4 150 0.2 100 0 0 0.5 1 1.5 2 PVI 50 0 0.5 1 1.5 2 PVI Comparison of upscaled quantities (Layer 43) Multiscale Computation for Incompressible Flow and Transport Problems p.30/66

Channelized reservoir 1 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 z 0.5 z 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 x 0 x 0 Comparison of saturation profile at PVI=0.5: (left) fine-scale model, (right) standard MsFVEM Multiscale Computation for Incompressible Flow and Transport Problems p.31/66

MsFVEM utilizing limited global information The numerical tests using strongly channelized permeability fields (such as SPE 10 Comparative) show that local basis functions can not accurately capture the long-range information. There is a need to incorporate a global information. The main idea is to use the solution of the fine-scale problem at time zero, p 0, to determine the boundary conditions for the multiscale basis formulation. φ ( x )=1 i i x i φ ( x i+1 )=0 i x i+1 φ ( x)=0 i x i 1 φ ( x i 1 )=0 i φ i ( x)=0 These approach is different from oversampling technique. Previous related work: J. Aarnes; L. Durlofsky et al. Multiscale Computation for Incompressible Flow and Transport Problems p.32/66

Channelized reservoir 1 0.8 fine scale model modified MsFVEM standard MsFVEM 350 300 fine scale model modified MsFVEM standard MsFVEM 0.6 250 F Q 200 0.4 150 0.2 100 0 0 0.5 1 1.5 2 PVI 50 0 0.5 1 1.5 2 PVI Comparison of upscaled quantities Multiscale Computation for Incompressible Flow and Transport Problems p.33/66

Channelized reservoir 1 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 z 0.5 z 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 x 0 x 0 Comparison of saturation profile at PVI=0.5: (left) fine-scale model, (right) modified MsFVEM Multiscale Computation for Incompressible Flow and Transport Problems p.34/66

Channelized reservoir 1 0.9 0.8 0.7 0.6 fine modified MsFVM 200 150 fine modified MsFVM F 0.5 Q 0.4 100 0.3 0.2 50 0.1 0 0 0.5 1 1.5 2 PVI 0 0 0.5 1 1.5 2 PVI Comparison of upscaled quantities (Layer 43, changing boundary conditions) Multiscale Computation for Incompressible Flow and Transport Problems p.35/66

Channelized reservoir fine scale saturation plot at PVI=0.7 1 saturation plot at PVI=0.5 using modified MsFVEM 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 Comparison of saturation profile at PVI=0.5: (left) fine-scale model (right) modified MsFVEM (changing boundary condition) Multiscale Computation for Incompressible Flow and Transport Problems p.36/66

Brief Analysis Main goal is to show that time-varying pressure is strongly influenced by the initial pressure field. Use the streamline-pressure coordinates: ψ/ x 1 = v 2, ψ/ x 2 = v 1 Set η = ψ(x, t = 0) and ζ = p(x, t = 0) and transform as follows: y 01 000000 111111 00000000 11111111 00000000 11111111 0000000 1111111 000000 111111 00000 11111 00000 11111 000000 111111 0000 1111 00000 11111 000000 111111 0000 1111 00000 11111 0000 1111 000 111 00 11 x η 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 ζ Ο(δ) Ο(δ) 000 111 000 111 000 111 000 111 000 111 high flow channel Multiscale Computation for Incompressible Flow and Transport Problems p.37/66

Brief Analysis The transformed pressure equation: η k 2 λ(s) p «+ η ζ λ(s) p «ζ = 0 The transformed saturation equation: S t + (v η) f(s) η + (v ζ) f(s) ζ = 0 k 2 λ(s) = k 0 2 λ 0 (ζ, t)1 Q1 δ + k 1 2 λ 1 (η, ζ, t)1 Qδ, λ(s) = λ 0 (ζ, t)1 Q1 δ + λ 1 (η, ζ, t)1 Qδ. The pressure has the following expansion: p(η, ζ, t) = p 0 (ζ, t) + δp 1 (η, ζ, t) +..., λ ζ 0 (ζ, t) p 0 = 0. ζ Modified basis functions can exactly recover the initial pressure. Multiscale Computation for Incompressible Flow and Transport Problems p.38/66

Analysis Assumption G. There exists a sufficiently smooth scalar valued function G(η) (G C 3 ), such that p G(p sp ) 1,Q Cδ, where δ is sufficiently small. Under Assumption G and p sp W 1,s (Q) (s > 2), multiscale finite element method converges with the rate given by p p h 1,Q Cδ + Ch 1 2/s p sp W 1,s (Q) Cδ + Ch1 2/s. Multiscale Computation for Incompressible Flow and Transport Problems p.39/66

Extensions Assume there exists a sufficiently smooth scalar valued function G(η), η R N (G C 3 ), such that p G(u 1,..., u N ) 1,Q Cδ, where δ is sufficiently small. Let ω i be a patch, and define φ 0 i to be piecewise linear basis function in patch ω i, such that φ 0 i (x j) = δ ij. For simplicity of notation, denote u 1 = 1. Then, the multiscale finite element method for each patch ω i is constructed by ψ ij = φ 0 i u j where j = 1,.., N and i is the index of nodes. First, we note that in each K, P n i=1 ψ ij = u j is the desired single-phase flow solution. Theorem. Assume u i W 1,s (Q), s > 2, i = 1,..., N. Then p p h 1,Q Cδ + Ch 1 2/s. Multiscale Computation for Incompressible Flow and Transport Problems p.40/66

An approach to general 2nd order pdes (H. Owhadi and L. Zhang, CPAM, 2006) div(λ(x)k(x) p) = 0, where λ(x) is smooth function, while k(x) is rough (e.g., k(x) = k(x/ǫ)). Take u 1 and u 2 that satisfy div(k(x) u i ) = 0 in Q, u i = x i on Q. Then, p(u 1, u 2 ) W 2,p because it satisfies a ij 2 p u i u j 0. (e.g., p = p 0 + ǫ 2 N(x/ǫ)...). Owhadi and Zhang showed that the non-conforming method with basis functions that span u 1 and u 2 converge. Multiscale Computation for Incompressible Flow and Transport Problems p.41/66

Two-phase flow equations in flow-based coor. ψ k 2 λ(s) P «+ ψ p λ(s) P p «= 0. S t + (v ψ) f(s) ψ + (v p) f(s) p = 0. Consider λ(s) = 1. Homogenization of hyperbolic equations. S ǫ t + vǫ 0 f(sǫ ) p = 0 S(p, ψ, t = 0) = S 0, v ǫ 0 (p) = v 0(p, p ǫ ). Multiscale Computation for Incompressible Flow and Transport Problems p.42/66

Homogenization of transport Then, for each ψ, it can be shown that S ǫ (p, ψ, t) S(p, ψ, t) in L 1 ((0,1) (0, T)), where S satisfies S t + ṽ 0 f( S) p = 0, where ṽ 0 is harmonic average of v ǫ 0, i.e., 1 v ǫ 0 1 ṽ 0 weak in L (0,1). Theorem. S ǫ S n Gǫ 1/n. Note. S can be considered as an upscaled S ǫ along streamlines. Can we average across streamlines? Multiscale Computation for Incompressible Flow and Transport Problems p.43/66

Numerical results 3 9 8 8 2 7 6 1 6 5 4 0 1 4 3 2 2 0 2 3 1 0 1 2 4 Permeability fields used in the simulations. Left - permeability field with exponential variogram, middle - synthetic channelized permeability field, right - layer 36 of SPE comparative project Multiscale Computation for Incompressible Flow and Transport Problems p.44/66

Numerical results 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 Saturation snapshots for variogram based permeability field (top) and synthetic channelized permeability field (bottom). Linear flux is used. Left figures represent the upscaled saturation plots and the right figures represent the fine-scale saturation plots. Multiscale Computation for Incompressible Flow and Transport Problems p.45/66

Numerical results 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 Saturation snapshots for variogram based permeability field (top) and synthetic channelized permeability field (bottom). Nonlinear flux is used. Left figures represent the upscaled saturation plots and the right figures represent the fine-scale saturation plots. Multiscale Computation for Incompressible Flow and Transport Problems p.46/66

Numerical results Upscaling error for permeability generated using two-point geostatistics LINEAR FLUX 25x25 50x50 100x100 200x200 L 1 error of S 0.0021 6.57e 4 2.15e 4 8.75e 5 L 1 error of S with macrodispersion 0.115 0.0696 0.0364 0.0135 L 1 error of S fine without macrodispersion 0.1843 0.0997 0.0505 0.0191 NONLINEAR FLUX 25x25 50x50 100x100 200x200 L 1 error of S 0.0023 8.05e 4 2.89e 4 1.29e 4 L 1 error of S with macrodispersion 0.116 0.0665 0.0433 0.0177 L 1 error of S fine without macrodispersion 0.151 0.0805 0.0432 0.0186 Multiscale Computation for Incompressible Flow and Transport Problems p.47/66

Numerical results Upscaling error for for synthetic channelized permeability field LINEAR FLUX 25x25 50x50 100x100 200x200 L 1 error of S 0.0222 0.0171 0.0122 0.0053 L 1 error of S with macrodispersion 0.0819 0.0534 0.0333 0.0178 L 1 error of S fine without macrodispersion 0.123 0.0834 0.0486 0.0209 NONLINEAR FLUX 25x25 50x50 100x100 200x200 L 1 error of S 0.0147 0.0105 0.0075 0.0040 L 1 error of S with macrodispersion 0.0842 0.0658 0.0371 0.0207 L 1 error of S fine without macrodispersion 0.119 0.0744 0.0424 0.0214 Multiscale Computation for Incompressible Flow and Transport Problems p.48/66

Numerical results Upscaling error for SPE 10, layer 36 LINEAR FLUX 25x25 50x50 100x100 200x200 L 1 error of S 0.0128 0.0093 0.0072 0.0042 L 1 error of S with macrodispersion 0.0554 0.0435 0.0307 0.0176 L 1 error of S fine without macrodispersion 0.123 0.0798 0.0484 0.0258 NONLINEAR FLUX 25x25 50x50 100x100 200x200 L 1 error of S 0.0089 0.0064 0.0054 0.0033 L 1 error of S with macrodispersion 0.0743 0.0538 0.0348 0.0189 L 1 error of S fine without macrodispersion 0.0924 0.0602 0.0395 0.0202 Multiscale Computation for Incompressible Flow and Transport Problems p.49/66

Numerical results Computational cost fine x.y fine p, ψ S S layered, linear flux 5648 257 9 1 layered, nonlinear flux 14543 945 28 4 percolation, linear flux 8812 552 12 1 percolation, nonlinear flux 23466 579 12 1 SPE10 36, linear flux 40586 1835 34 2 SPE10 36, nonlinear flux 118364 7644 25 2 Multiscale Computation for Incompressible Flow and Transport Problems p.50/66

Numerical results Left: Pressure and streamline function at time t = 0.4 in Cartesian frame. Right: pressure and streamline function at time t = 0.4 in initial pressure-streamline frame. For two-phase flow, equations are upscaled on flow-based coordinate system. MsFEM using limited global information is equivalent to standard MsFEM. Multiscale Computation for Incompressible Flow and Transport Problems p.51/66

Multiscale analysis for Navier-Stokes Equations t u ǫ + (u ǫ )u ǫ = p ǫ + ν u ǫ + f, u ǫ = 0, u ǫ t=0 = U(x) + W(x, x ǫ ). Motivated by a multiscale analysis for the Euler equations, we look for (u ǫ, p ǫ ) of form as u ǫ = ū(t,x, τ) + eu(t, θ, τ,z), p ǫ = p(t,x, τ) + ep(t, θ, τ,z), where eu and ep are periodic in z with zero mean, z = θ/ǫ. Multiscale Computation for Incompressible Flow and Transport Problems p.52/66

Main Results. The main results can be summarized as follows: Effective equation for ū: t ū + (ū x )ū + x w w = x p + ν ū + f, x ū = 0, ū t=0 = U(x). Leading order small scale velocity field w: τ w + z w x θw + x θ z q = ν ǫ z ( x θ x θ z w), with ` x θ z w = 0 and, with w τ=t=0 = W. Effective equation for flow map θ: t θ + (ū x ) θ + ǫ x Θ w = 0, θ t=0 = x. t Θ + (I + z Θ) x θw = 0, with Θ t=0 = 0. Multiscale Computation for Incompressible Flow and Transport Problems p.53/66

Numerical Experiments for 2D NSE We use direct numerical simulation (DNS) to check the accuracy of our multiscale analysis. In computation, we choose ν = 10 4. The DNS is performed using 512 by 512 mesh. The multiscale computations use 64 by 64 on the coarse grid, and 32 by 32 for the subgrid. The initial vorticity is generated in Fourier space, satisfying bω k O( k 1 ) with random phases. Our main objective is to test how accuate we can capture the inverse cascade of 2D turbulence. Multiscale Computation for Incompressible Flow and Transport Problems p.54/66

Numerical results in 2D 6 5 4 3 2 1 0 0 1 2 3 4 5 6 Vorticity contour at t = 0. Multiscale Computation for Incompressible Flow and Transport Problems p.55/66

Numerical results in 2D Vorticity contour, DNS (left) and Homogenization (right) at t=5.0. Multiscale Computation for Incompressible Flow and Transport Problems p.56/66

Numerical results in 2D Vorticity contour, DNS (left) and Homogenization (right) at t=10.0. Multiscale Computation for Incompressible Flow and Transport Problems p.57/66

Numerical results in 2D Vorticity contour, DNS (left) and Homogenization (right) at t=20.0. Multiscale Computation for Incompressible Flow and Transport Problems p.58/66

Numerical results in 2D 10-1 10-2 10-3 10-4 10-5 10 1 10 2 Spectrum of velocity component u 1.(dashed), t = 0.0; (solid), DNS, t = 20.0; (chndot), homogenization, t = 20.0. Multiscale Computation for Incompressible Flow and Transport Problems p.59/66

Numerical Experiments for 3D forced NSE DNS is performed using 512 3 mesh in a periodic box to check accuracy of the multiscale method. The Taylor Reynolds number is 223 in our computation (equvilent viscosity is 0.0005). The specific form of the forcing is given by ˆf k = δ N û k pûk û k (-12) N is the number of wave modes that are forced. The above form of forcing ensures that the energy injection rate, P ˆf û, is a constant which is equal to δ. We chose δ = 0.1 for all of our runs. Multiscale Computation for Incompressible Flow and Transport Problems p.60/66

3D Results 1.4 1.2 1 e k 0.8 0.6 0.4 0.2 0 5 10 15 20 t Temporal evolution of total kinetic energy. Multiscale Computation for Incompressible Flow and Transport Problems p.61/66

3D Results Iso-surface plot of vorticity. Multiscale Computation for Incompressible Flow and Transport Problems p.62/66

3D Results 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 0 10 1 10 2 10 3 Comparison of energy spectrum at t = 30.0. (blue), DNS. (red), N=128; (green), N=64; (yellow), N=32. The computations are performed for 30 eddy turn-over times. Multiscale Computation for Incompressible Flow and Transport Problems p.63/66

3D Results 4 3.5 3.5 3 3 2.5 2.5 2 2 1.5 1.5 1 0.5 1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 PDF of ω e 1 (solid), ω e 2 (dashed line), ω e 3 (dotted line). DNS (left) vs LES (right), at t=20. Multiscale Computation for Incompressible Flow and Transport Problems p.64/66

3D Results 3 3.5 2.5 3 2 2.5 2 1.5 1.5 1 1 0.5 0.5 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 PDF of e 3 t 1 (solid), e 2 t 2 (dashed line), e 3 t 1 (dotted line). DNS (left) vs LES (right), at t=20. Here t j are eigenvectors of the SGS tensor. Multiscale Computation for Incompressible Flow and Transport Problems p.65/66

Conclusions Multiscale finite element methods provide an effective multiscale computational method for flow and transport in heterogeneous porous media. Resonance error can be effectively removed by the over-sampling technique. Long range scale interaction can be captured by incorporating limited global information into the multiscale bases. Flow-based adaptive coordinates offer a natural physical coordinate to capture the long range interaction in a channelized media. The hybrid Eulerian-Lagrangian coordinates offer a natural multiscale frame to capture the nonlinear nonlocal scale interactions for the Navier-Stokes equations. Multiscale Computation for Incompressible Flow and Transport Problems p.66/66