Numerical Simulation of Flows in Highly Heterogeneous Porous Media

Transcription

1 Numerical Simulation of Flows in Highly Heterogeneous Porous Media R. Lazarov, Y. Efendiev, J. Galvis, K. Shi, J. Willems The Second International Conference on Engineering and Computational Mathematics (ECM2013) Hong Kong, December 18, 2013 Thanks: NSF, KAUST, IAMCS, deal.ii

2 Outline 1 Motivation and Problem Formulation State of the art 2 Single Grid Approximation of Brinkman equations Subgrid Approximation and Its Performance Numerical Experiments 3

3 Outline Motivation and Problem Formulation State of the art 1 Motivation and Problem Formulation State of the art 2 Single Grid Approximation of Brinkman equations Subgrid Approximation and Its Performance Numerical Experiments 3

4 Motivation and Problem Formulation State of the art Motivation: media at multiple scales Figure: Porous media: real-life scale and macro scale

5 Motivation and Problem Formulation State of the art Motivation: open industrial foams media of low solid fraction Figure: Industrial foams on microstructure scale; porosity over 93% Figure: Trabecular bone: macro- and real life scales

6 Motivation and Problem Formulation State of the art Computer Generated Distribution of the Heterogeneous Permeability Figure: periodic + rand; rand + rand, SPE10 slice Another dimension of difficulty add orthotropic materials.

7 Motivation and Problem Formulation State of the art Computer Generated Distribution of the Heterogeneous Permeability Figure: SPE10 3 dimensional permeability distributions

8 Motivation and Problem Formulation State of the art Reconstruction of micro-structures from CT scans Figure: 3-dimensional micro-structure of cathode consisting of lithium iron phosphate (green) and carbon (black) and its use in a schematic construction of a battery

9 Motivation and Problem Formulation State of the art Modeling of flow in porous media (1) Flows in porous media are modeled by linear Darcy law that relates the macroscopic pressure p and velocity u: p = µκ 1 u, κ permeability, µ viscosity (1) This model is the generally used in hydrology, reservoir modeling, industrial processes. In real problems exhibiting multiscale behavior κ is: 1 highly varying by changing orders of magnitude; 2 highly heterogeneous and often anisotropic; 3 could depend also on a stochastic variable, i.e. κ(x, ω), where ω is in a high dimensional space.

10 Motivation and Problem Formulation State of the art Modeling of flow in porous media (2) To fit better the data for flows near wells a nonlinear correction was proposed by Forcheimer, β > 0, p = µκ 1 u β u u, β > 0 fitting parameter. (2) (3) For flows in highly porous media Brinkman (1947) enhanced Darcy s law by adding dissipative term scaled by viscosity: p = µκ 1 u + µ u. (3)

11 Motivation and Problem Formulation State of the art Modeling of flow in porous media (2) To fit better the data for flows near wells a nonlinear correction was proposed by Forcheimer, β > 0, p = µκ 1 u β u u, β > 0 fitting parameter. (2) (3) For flows in highly porous media Brinkman (1947) enhanced Darcy s law by adding dissipative term scaled by viscosity: p = µκ 1 u + µ u. (3)

12 Motivation and Problem Formulation State of the art Modeling of flow in porous media (4) Another venue for a two-phase flow is a Richards model: p = µκ 1 u, where κ = k(x)λ(x, u) (4) with k(x) heterogeneous function is the intrinsic permeability, while λ(x, u) is a smooth function that varies moderately in both x and u, related to the relative permeability.

13 Motivation and Problem Formulation State of the art Modeling of flow in porous media (cont.) Thus our model is the following boundary value problem: µ u + p + µκ 1 u = f, x Ω u = 0, x Ω u = g, x Γ This system of generalized Stokes is often called Brinkman model. It contains both limits: Darcy: p + µκ 1 u = f and Stokes: µ u + p = f. Following the Russian or French sources this could be considered also as fictitious domain method or penalty formulation.

14 Motivation and Problem Formulation State of the art Modeling of flow in porous media (cont.) Thus our model is the following boundary value problem: µ u + p + µκ 1 u = f, x Ω u = 0, x Ω u = g, x Γ This system of generalized Stokes is often called Brinkman model. It contains both limits: Darcy: p + µκ 1 u = f and Stokes: µ u + p = f. Following the Russian or French sources this could be considered also as fictitious domain method or penalty formulation.

15 Motivation and Problem Formulation State of the art Modeling of flow in porous media (cont.) Thus our model is the following boundary value problem: µ u + p + µκ 1 u = f, x Ω u = 0, x Ω u = g, x Γ This system of generalized Stokes is often called Brinkman model. It contains both limits: Darcy: p + µκ 1 u = f and Stokes: µ u + p = f. Following the Russian or French sources this could be considered also as fictitious domain method or penalty formulation.

16 Motivation and Problem Formulation State of the art Models of flow in porous media Darcy and Brinkman equations are introduced as macroscopic equations, without direct link to the underlying microscopic behavior. Allaire in 1991 studied homogenization of slow viscous fluid flows (with negligible no-slip effects on interface between the fluid and the solid obstacles) for periodic arrangements. O(1) O(1) Figure: Darcy vs. Brinkman

17 Motivation and Problem Formulation State of the art What is a good method for such problems? Our aim is development and study of numerical methods that address the following main issues of these classes of problem: Work well in both limits, Darcy and Brinkman; Work well for both linear and nonlinear highly heterogeneous problems; Can be used as stand alone numerical upscaling procedure; Can be used in construction of robust with respect to high variations of the permeability field preconditioners.

18 Motivation and Problem Formulation State of the art A Remark These problems are a small subclass of problems in large area of flows in porous media that include: single and multi-phase flows in porous media transport of species and reacting flows; reservoir modeling and simulation; working with multiple scales; share similarities with heat conduction and mass transfer. Some of these complex problems were discussed in the Workshop 2 at this conference dedicated to Professor Mary Wheeler.

19 Motivation and Problem Formulation State of the art State of the Art: Contributions of Mary Wheeler Mary Wheeler has contributed to this field in a way that only few researchers have done: (1) She has brought in this filed some of the most innovative ideas and has authored and co-authored some of the most influential papers; (2) Through her industrial affiliates she has introduced the new ideas and methods to the petroleum industry and has worked on various applications, in particular in multi-phase flows; (3) She has trained, guided, and and mentored a generation of numerical analysts and applied mathematicians who contributed further to the advancement of this field.

20 Motivation and Problem Formulation State of the art State of the Art: Available Numerical Methods and Tools Numerical upscaling as subgrid approximation methods: (1) standard FEM for Darcy: comprehensive exposition in the book of Efendiev & Hou, 2009, (2) mixed methods for Darcy: Aarnes, 2004, Arbogast, 2002, 2004 (with Boyd) Chen & Hou, 2002, (3) DG FEM for Brinkman equation: Arbogast & Brunson, 2007, Willems, 2009, Iliev, Lazarov & Willems, 2010, Juntunen & Stenberg, 2010, Könö & Stenberg, 2011 (4) The pioneering work on Multiscale mortar FEM: Arbogast, Pencheva, Wheeler, and Yotov, 2007, Arbogast and Xiao, 2013

21 Motivation and Problem Formulation State of the art State of the Art: Available Numerical Methods and Tools Preconditioning techniques based on coarse grid space with multiscale finite element functions: (1) FETI and other DD methods: Graham, Klie, Lechner, Pechstein, & Scheichl, 2007, 2008, 2009, 2011 (2) Using energy-minimizing coarse spaces: Xu & Zikatanov, 2004, (3) Spectral Element Agglomerate Algebraic Multigrid Methods, Efendiev, Galvis, & Vassielvski, 2011

22 Motivation and Problem Formulation State of the art Objectives of our group led by Y. Efendiev: 1 Derive, study, implement, and test a numerical upscaling procedure for highly porous media that works well in both limits, Brinkman and Darcy; 2 go beyond the polynomial spaces, e.g. utilize spectral local solutions, snapshots, etc viewed as numerical model reduction technique. 3 Design and study of preconditioning techniques for such problems for porous media of high contrast with targeted applications to oil/water reservoirs, filters, insulators, etc

23 Outline Single Grid Approximation of Brinkman equations Subgrid Approximation and Its Performance Numerical Experiments 1 Motivation and Problem Formulation State of the art 2 Single Grid Approximation of Brinkman equations Subgrid Approximation and Its Performance Numerical Experiments 3

24 Single Grid Approximation of Brinkman equations Subgrid Approximation and Its Performance Numerical Experiments Weak formulation Let us go back to the Brinkman equations: µ u + p + µκ 1 u = f, x Ω u = 0, x Ω u = 0, x Γ Find u H0 1(Ω)n := V and p L 2 0 (Ω) := W such that ( µ u : v + µ Ω κ u v)dx + p vdx = f vdx v V Ω Ω q udx = 0 q L 2 0 (Ω) Ω This problem has unique solution (u, p) V W.

25 Single Grid Approximation of Brinkman equations Subgrid Approximation and Its Performance Numerical Experiments The finite element of Brezzi, Douglas, and Marini of degree 1 For this finite element we have: On a rectangle T the polynomial space is characterized by {v = P span{curl(x 2 1 x 2), curl(x 1 x 2 2 }}; with dof normal velocity H pressure (V H, W H ) (H 0 (div; Ω), L 2 0 (Ω)) := V W ; Has a natural variant for n = 3.

26 Single Grid Approximation of Brinkman equations Subgrid Approximation and Its Performance Numerical Experiments DG FEM, Wang and Ye (2007) on a Single Grid Since the tangential derivative along the internal edges will be in general discontinuous, i.e. V H H0 1 (Ω); therefore we will apply the discontinuous Galerkin method: Find (u H, p H ) (V H, W H ) such that for all (v H, q H ) (V H, W H ) { a (uh, v H ) + b (v H, p H ) = F(v H ) (5) b (u H, q H ) = 0. Because of the nonconformity of the FE spaces, the bilinear form a (u H, v H ) has a special form given below.

27 Single Grid Approximation of Brinkman equations Subgrid Approximation and Its Performance Numerical Experiments Discretization of Wang and Ye 2007 on Single Grid b (v H, p H ) := p H v H dx τ e + Ω n n + e e T + T F(v H ) := f v H dx τe Ω a (u H, v H ) := ( µ u H : v H + µ κ u H v H )dx T T H T ( µ {u H } v H + {v H } u H α ) e E e }{{} e u H v H ds H symmetrization }{{} stabilization {v } e := 1 2 (n+ e (v τ e + ) e + + n e (v τe ) e ) v e := v e + τ + e + v e τ e

28 Single Grid Approximation of Brinkman equations Subgrid Approximation and Its Performance Numerical Experiments Stability and Error Estimates: Wang and Ye (2007) Provided exact solution (u, p) of Brinkman problem is in (H 2 (Ω), H 1 (Ω)) we have: p p H L 2 (Ω) CH( u H 2 (Ω) + p H 1 (Ω) ) u u H L 2 (Ω) CH2 ( u H 2 (Ω) + p H 1 (Ω) ). There are two main issues in this setting: 1 This will be a very large saddle point system, if we want to resolve the finer scale of the heterogeneity by H; 2 The system is very ill-conditioned due to the step-size H and the high variability of the permeability.

29 Single Grid Approximation of Brinkman equations Subgrid Approximation and Its Performance Numerical Experiments The Idea of the Multiscale Finite Element Method The idea is to approximate the boundary value problem by using two grids coarse T H and fine T h : The fine triangulation T h is obtained by refinement of the coarse T H and resolves the media heterogeneities; The desire is to formulate a global problem associated with the coarse mesh only. Ω T h T H

30 Single Grid Approximation of Brinkman equations Subgrid Approximation and Its Performance Numerical Experiments Multiscale FEM for second order elliptic problems The MSFEM for second order problems construct the space S H H 1 0 (Ω) of piecewise polynomials over the coarse mesh T H and the spaces S h H 1 0 (K ) for each K T h. Then the fine-grid component u h is eliminated (via say static condensation) and we end up with a problem on the coarse grid component u H. Then the multiscale solution has the form u H + u h, where u H S H and u h S h. Unfortunately, this simple idea is not going to work for Brinkman system. We need to do better.

31 Single Grid Approximation of Brinkman equations Subgrid Approximation and Its Performance Numerical Experiments Two-grid method of Arbogast for saddle points: Fine and Coarse Grid Spaces The approximation spaces use the two grids T h and T H and are constructed in the following way: W h and V h consist of bubbles on each T T H ; the FE spaces on the composite mesh are W H,h = W h W H L 2 0, and V H,h = V h V H H 0 (div) Crucial properties: 1 V h = W h and V H = W H 2 v h n = 0 on T, v h V h and T h T T H T H 3 W H W h Ω

32 Single Grid Approximation of Brinkman equations Subgrid Approximation and Its Performance Numerical Experiments Composite Grid Problem Unique decomposition in components yields: find u H + u h W h W H, and p H + p h V h V H such that composite solution a (u H + u h, v H + v h ) + b (v H + v h, p H + p h ) = F(v H + v h ), b (u H + u h, q H + q h ) = 0, for all v H + v h W h W H and q H + q h V h V H. Our goal is to derive numerical upscaling procedure.

33 Single Grid Approximation of Brinkman equations Subgrid Approximation and Its Performance Numerical Experiments Splitting of the Solution Equivalently, testing separately for v H, q H and v h, q h, we get { a (uh + u ( ) h, v H ) + b (v H, p H + p h ) = F(v H ) v H V H b (u H + u h, q H ) = 0 q H W H { a (uh + u ( ) h, v h ) + b (v h, p H + p h ) = F(v h ) v h V h b ( u H + u h, q h ) = 0 q h W h Since b (v, q) = ( v, q), V h = W h, V H = W H, W H W h. If we knew u H then we could find easily u h from a (u h, v h ) + b (v h, p h ) + b (u h, q h ) = F(v h ) a (u H, v h )

34 Single Grid Approximation of Brinkman equations Subgrid Approximation and Its Performance Numerical Experiments a (u h, v h ) + b (v h, p h ) + b (u h, q h ) = F(v h ) a (u H, v h ) Decompose ( ) further into ( ) Note that { a (δu (u H ), v h ) + b (v h, δ p (u H )) = a (u H, v h ) v h V h b (δ u (u H ), q h ) = 0 q h W h { a ( δu, v h ) + b ( v h, δ p ) = F(v h ) v h V h b ( δ u, q h ) = 0 qh W h (δ u (u H ), δ p (u H )) is linear in u H (δ u, δ p ) and (δ u (u H ), δ p (u H )) can be computed locally

35 Single Grid Approximation of Brinkman equations Subgrid Approximation and Its Performance Numerical Experiments The Upscaled Equation We have (u h, p h ) = (δ u + δ u (u H ), δ p + δ p (u H )). Putting all these together we obtain: Symmetric Form of the Upscaled Equation Thus we get a (u H + δ u (u H ), v H + δ u (v H )) + b (v H, p H ) =F(v H ) a ( δ u, v H ), b (u H, q H ) =0, which involves only coarse-grid degrees of freedom.

36 Single Grid Approximation of Brinkman equations Subgrid Approximation and Its Performance Numerical Experiments Sub-grid Algorithm, Willems, 2009 Sub-grid Algorithm (numerical upscaling): Willems, Solve for the fine responses (δ u, δ p ) and (δ u (ϕ H ), δ p (ϕ H )) for coarse basis functions ϕ H and each coarse cell. 2 Solve the upscaled equation for (u H, p H ). 3 Piece together the solutions to get (u H,h, p H,h ) = (u H, p H ) + (δ u (u H ), δ p (u H )) + (δ u, δ p ). For pure Darcy this reduces to the method of Arbogast, The Question is: DOES IT WORK?

37 Single Grid Approximation of Brinkman equations Subgrid Approximation and Its Performance Numerical Experiments Sub-grid Algorithm, Willems, 2009 Sub-grid Algorithm (numerical upscaling): Willems, Solve for the fine responses (δ u, δ p ) and (δ u (ϕ H ), δ p (ϕ H )) for coarse basis functions ϕ H and each coarse cell. 2 Solve the upscaled equation for (u H, p H ). 3 Piece together the solutions to get (u H,h, p H,h ) = (u H, p H ) + (δ u (u H ), δ p (u H )) + (δ u, δ p ). For pure Darcy this reduces to the method of Arbogast, The Question is: DOES IT WORK?

38 Single Grid Approximation of Brinkman equations Subgrid Approximation and Its Performance Numerical Experiments Sample Geometries Periodic Vuggy SPE10

39 Single Grid Approximation of Brinkman equations Subgrid Approximation and Its Performance Numerical Experiments Periodic Geometry - Velocity u = [1, 1] on Ω, f = 0, µ e = µ = 1, κ 1 = Reference velocity n 1e3 in the holes 1e 2 elsewhere Subgrid velocity Relative L2 -error: 3.69e-03. R. Lazarov, Y. Efendiev, J. Galvis, K. Shi, J. Willems Numercs for Flows in Heterogeneous Media

40 Single Grid Approximation of Brinkman equations Subgrid Approximation and Its Performance Numerical Experiments Periodic Geometry - Pressure Reference pressure Subgrid pressure Relative L 2 -error: 1.77e-02.

41 Single Grid Approximation of Brinkman equations Subgrid Approximation and Its Performance Numerical Experiments SPE10 - Pure Subgrid for Brinkman u = [1, 0] on Ω, f = 0, µ = 1e 2,T h : 128 2, κ 1 : ranging from 1e5 in blue to 1e2 in red. u 1 Relative L 2 -error velocity error: 2.26e-01

42 Single Grid Approximation of Brinkman equations Subgrid Approximation and Its Performance Numerical Experiments SPE10 - Subgrid for Brinkman u = [1, 0] on Ω, f = 0, µ = 1e 2, κ 1 : as shown on the plot ; Th : (a) Ref. solution (b) H = 1/16. (c) H = 1/8. (d) H = 1/4. Figure: Velocity component, u1, for SPE10 on 3 coarse grids. R. Lazarov, Y. Efendiev, J. Galvis, K. Shi, J. Willems Numercs for Flows in Heterogeneous Media

43 Single Grid Approximation of Brinkman equations Subgrid Approximation and Its Performance Numerical Experiments Vuggy media - Subgrid Brinkman u = [1, 0] on Ω, f = 0, µ = 1e 2, κ 1 = 1e3 T h : (a) Ref. solution (b) H = 1/16. (c) H = 1/8. (d) H = 1/4. Figure: Velocity component, u 1, for vuggy geometry.

44 Single Grid Approximation of Brinkman equations Subgrid Approximation and Its Performance Numerical Experiments SPE10 benchmark R. Lazarov, Y. Efendiev, J. Galvis, K. Shi, J. Willems Numercs for Flows in Heterogeneous Media

45 Single Grid Approximation of Brinkman equations Subgrid Approximation and Its Performance Numerical Experiments SPE10 benchmark: convergence of the iterations Pure Darcy case for periodic geometry for different contrasts. Brinkman case for SPE10 geometry for different contrasts.

46 Single Grid Approximation of Brinkman equations Subgrid Approximation and Its Performance Numerical Experiments How to fight the resonance error? There are various techniques to overcome the resonance error: 1 Using boundary conditions based on reducing the operator to the boundaries of the coarse mesh T H ; 2 Oversampling 3 Mortar finite element approximation; 4 Hybridizable DG FEM; 5 WG - weak Galerkin FEM shows great potential

47 Outline 1 Motivation and Problem Formulation State of the art 2 Single Grid Approximation of Brinkman equations Subgrid Approximation and Its Performance Numerical Experiments 3

48 How to fight the resonance error? This is the subject of our recent research with Efendiev and Shi. First, we discuss HDG for Darcy flow. Recall that we have two grids T h and T H. We consider the mixed formulation of Darcy equation (with α(x) = κ(x) 1 ): αu + p = 0 x Ω, u = f x Ω p = g x Ω.

49 HDG Let E H, E h denote the set of all edges of T H, T h, respectively. We define Eh 0 := {e h E h e H K H } the internal edges of T h T h := Kh T h K h the edges the fine mesh T h T H := KH T H K H the edges of the coarse mesh T H

50 HDG the HDG generates an approximation (p h, u h, p h,h ) to (p, u, p Eh ) on the finite dimensional (often piece-wise polynomial but not necessarily) spaces defined on each K T h : V h := {v L 2 (T h ) : v K V (K )}, W h := {w L 2 (T h ) : w K W (K )}, and M h,h := M 0 h M H, with M h := {µ L 2 (E h ) : µ Fh M h (F ), F h E h }, M 0 h := {µ M h : µ Fh = 0, F h F H E H }, M H := {µ L 2 (E H ) : µ FH M H (F H ), F H E H }.

51 HDG Then we seek (p h, u h, p h,h ) W h V h M h,h as the only solution of the following weak problem: (p h, v) Th + (αu h, v) Th + p h,h, v n Th = 0, v V h (u h, w) Th + û h,h n, w Th = (f, w) Th, w W h û h,h n, µ Th = 0, µ M 0 h,h û h,h, λ D = g, λ D, λ M h,h Here M 0 h,h = {µ M h,h : µ D = 0}, M h,h = M h,h D, (η, ζ) Th := K T h (η, ζ) K and û h,h n = u h n + τ(p h p h,h ) on T h.

52 HDG We split into two sets of equations by testing separately for µ in M 0 h and M0 H := {µ M H : µ D = 0}: û h,h n, µ Th = 0, µ M 0 h & û h,h n, µ TH = 0, µ M 0 H. On each cell K T H, given the boundary data of p h,h = ξ H for ξ H M H (F), F K, we find (u h, p h, p h,h ) on K = K H : (p h, v) K + (αu h, v) K + p h,h, v n K = 0, (u h, w) K + û h,h n, w K = (f, w) K, û h,h n, µ K = 0 for all (w, v, µ) W h K V h K M 0 h K. In fact the above local system is the regular HDG methods defined on a coarse cell K.

53 HDG The numerical upscaling method This HDG method defines a global mapping from M H to W h V h Mh 0. This implies that one possible implementation of the method would be to solve the local problems for all basis functions of MH 0 and then solve the global system on the coarse mesh: a(ξ H, µ) := û h,h (ξ H ), µ TH = 0, µ M 0 H, ξ H, λ D = g n, λ D λ M H. (6)

54 HDG This is quite similar to the mortar MS FEM, e.g. T Arbogast, G Pencheva, MF Wheeler, I Yotov, The main difference is that instead of special choice of the mortar space the stability is provided by the stabilization parameter (or often called operator) τ in the choice of the numerical flux û h,h n = u h n + τ(p h p h,h ) on T h. This choice often generates superconvergent schemes or allows more flexible choices of the approximation spaces, including locally generated solutions of some spectral problems.

55 HDG current and future work Now we are working in various directions that include: An implementation of this part Generating the numerical trace space by solving local spectral problems or traces of snapshot spaces Brinkman/Stokes problem and its applications; Theoretical justification of the method and optimal error estimates; Efficient solution methods for the resulting systems of linear equations; Most important: this is a framework for novel model reduction technique for parameter dependent problems.

56 Thank you Thank you for your attention!!!