Numerical Simulation of Flows in Highly Heterogeneous ... · Single Grid Approximation of Brinkman equations Subgrid Approximation and Its Performance Numerical Experiments 3 Hybridizable

IntroductionNumerical Upscaling of Brinkman equations

Hybridizable discontinuous Galerkin method (HDG)

Numerical Simulation of Flows in HighlyHeterogeneous Porous Media

R. Lazarov,Y. Efendiev, J. Galvis, K. Shi, J. Willems

The Second International Conference on Engineering andComputational Mathematics (ECM2013)

Hong Kong, December 18, 2013

Thanks: NSF, KAUST, IAMCS, deal.II

R. Lazarov, Y. Efendiev, J. Galvis, K. Shi, J. Willems Numercs for Flows in Heterogeneous Media



Outline

1 IntroductionMotivation and Problem FormulationState of the art

2 Numerical Upscaling of Brinkman equationsSingle Grid Approximation of Brinkman equationsSubgrid Approximation and Its PerformanceNumerical Experiments

3 Hybridizable discontinuous Galerkin method (HDG)




Motivation and Problem FormulationState of the art

Outline








Motivation: media at multiple scales

Figure: Porous media: real-life scale and macro scale





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





Computer Generated Distribution of the Heterogeneous Permeability

Figure: periodic + rand; rand + rand, SPE10 slice

Another dimension of difficulty add orthotropic materials.





Computer Generated Distribution of the Heterogeneous Permeability

Figure: SPE10 3 dimensional permeability distributions





Reconstruction of micro-structures from CT scans

Figure: 3-dimensional micro-structure of cathode consisting of lithiumiron phosphate (green) and carbon (black) and its use in a schematicconstruction of a battery





Modeling of flow in porous media

(1) Flows in porous media are modeled by linear Darcy law thatrelates the macroscopic pressure p and velocity u:

∇p = −µκ−1u, κ − permeability, µ − viscosity (1)

This model is the generally used in hydrology, reservoirmodeling, 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.






(2) To fit better the data for flows near wells a nonlinearcorrection was proposed by Forcheimer, β > 0,

∇p = −µκ−1u − β|u|u, β > 0 fitting parameter. (2)

(3) For flows in highly porous media Brinkman (1947) enhancedDarcy’s law by adding dissipative term scaled by viscosity:

∇p = −µκ−1u + µ∆u. (3)






(2) To fit better the data for flows near wells a nonlinearcorrection was proposed by Forcheimer, β > 0,

∇p = −µκ−1u − β|u|u, β > 0 fitting parameter. (2)

(3) For flows in highly porous media Brinkman (1947) enhancedDarcy’s law by adding dissipative term scaled by viscosity:

∇p = −µκ−1u + µ∆u. (3)






(4) Another venue for a two-phase flow is a Richards model:

∇p = −µκ−1u, 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 bothx and u, related to the relative permeability.





Modeling of flow in porous media (cont.)

Thus our model is the following boundary value problem:

−µ∆u +∇p + µκ−1u = f , x ∈ Ω∇ · u = 0, x ∈ Ω

u = g, x ∈ Γ

This system of generalized Stokes is often called Brinkmanmodel. It contains both limits:

Darcy: ∇p + µκ−1u = f and

Stokes: −µ∆u +∇p = f .

Following the Russian or French sources this could beconsidered also as fictitious domain method or penaltyformulation.







−µ∆u +∇p + µκ−1u = f , x ∈ Ω∇ · u = 0, x ∈ Ω

u = g, x ∈ Γ











−µ∆u +∇p + µκ−1u = f , x ∈ Ω∇ · u = 0, x ∈ Ω

u = g, x ∈ Γ









Models of flow in porous media

Darcy and Brinkman equations are introduced as macroscopicequations, without direct link to the underlying microscopicbehavior.Allaire in 1991 studied homogenization of slow viscous fluidflows (with negligible no-slip effects on interface between thefluid and the solid obstacles) for periodic arrangements.

O(1)O(1)

Figure: Darcy vs. Brinkman





What is a good method for such problems ?

Our aim is development and study of numerical methods thataddress the following main issues of these classes of problem:

Work well in both limits, Darcy and Brinkman;Work well for both linear and nonlinear highlyheterogeneous problems;Can be used as stand alone numerical upscalingprocedure;Can be used in construction of robust with respect to highvariations of the permeability field preconditioners.





A Remark

These problems are a small subclass of problems in large areaof flows in porous media that include:

single and multi-phase flows in porous mediatransport 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 theWorkshop 2 at this conference dedicated to Professor MaryWheeler.





State of the Art: Contributions of Mary Wheeler

Mary Wheeler has contributed to this field in a way that only fewresearchers have done:(1) She has brought in this filed some of the most innovativeideas and has authored and co-authored some of the mostinfluential papers;(2) Through her industrial affiliates she has introduced the newideas and methods to the petroleum industry and has workedon various applications, in particular in multi-phase flows;(3) She has trained, guided, and and mentored a generation ofnumerical analysts and applied mathematicians whocontributed further to the advancement of this field.





State of the Art: Available Numerical Methods and Tools

Numerical upscaling as subgrid approximation methods:

(1) standard FEM for Darcy: comprehensive exposition inthe 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, Arbogastand Xiao, 2013





State of the Art: Available Numerical Methods and Tools

Preconditioning techniques based on coarse grid spacewith 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 MultigridMethods, Efendiev, Galvis, & Vassielvski, 2011





Objectives of our group led by Y. Efendiev:

1 Derive, study, implement, and test a numerical upscalingprocedure for highly porous media that works well in bothlimits, Brinkman and Darcy;

2 go beyond the polynomial spaces, e.g. utilize spectral localsolutions, snapshots, etc – viewed as numerical modelreduction technique.

3 Design and study of preconditioning techniques for suchproblems for porous media of high contrast with targetedapplications to oil/water reservoirs, filters, insulators, etc




Single Grid Approximation of Brinkman equationsSubgrid Approximation and Its PerformanceNumerical Experiments

Outline








Weak formulation

Let us go back to the Brinkman equations:

−µ∆u +∇p + µκ−1u = f , x ∈ Ω∇ · u = 0, x ∈ Ω

u = 0, x ∈ Γ

Find u ∈ H10 (Ω)n := V and p ∈ L2

0(Ω) := W such that∫Ω

(µ∇u : ∇v +µ

κu · v)dx +

∫Ω

p∇ · vdx =

∫Ω

f · vdx ∀v ∈ V∫Ω

q∇ · udx = 0 ∀q ∈ L20(Ω)

This problem has unique solution (u,p) ∈ V ×W .





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 = P21 + spancurl(x2

1 x2), curl(x1x22;

with dofnormal velocity

pressureH

(VH ,WH) ⊂ (H0(div ; Ω),L20(Ω)) := V ×W ;

Has a natural variant for n = 3.





DG FEM, Wang and Ye (2007) on a Single Grid

Since the tangential derivative along the internal edges will bein general discontinuous, i.e. VH * H1

0 (Ω); therefore we willapply the discontinuous Galerkin method: Find(uH ,pH) ∈ (VH ,WH) such that for all (vH ,qH) ∈ (VH ,WH)

a (uH ,vH) + b (vH ,pH) = F (vH)b (uH ,qH) = 0.

(5)

Because of the nonconformity of the FE spaces, the bilinearform a (uH ,vH) has a special form given below.





Discretization of Wang and Ye 2007 on Single Grid

b (vH ,pH) :=

∫Ω

pH∇ · vHdx

F (vH) :=

∫Ω

f · vHdx

τ+en+

en−eT+

τ−eT−

a (uH ,vH) :=∑

T∈TH

∫T

(µ∇uH : ∇vH +µ

κuH · vH)dx

−∑e∈EH

∫eµ

(uH JvHK + vH JuHK︸︷︷︸

symmetrization

− α

|e|JuHK JvHK︸︷︷︸

stabilization

)ds

v|e := 12(n+

e · ∇(v · τ+e )|e+ + n−e · ∇(v · τ−e )|e−)

JvK |e := v |e+ · τ+e + v |e− · τ−e





Stability and Error Estimates: Wang and Ye (2007)

Provided exact solution (u,p) of Brinkman problem is in(H2(Ω),H1(Ω)) we have:

‖p − pH‖L2(Ω) ≤ CH(‖u‖H2(Ω) + ‖p‖H1(Ω))

‖u − uH‖L2(Ω) ≤ CH2(‖u‖H2(Ω) + ‖p‖H1(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.





The Idea of the Multiscale Finite Element Method

The idea is to approximate the boundary value problem byusing two grids – coarse TH and fine Th:

The fine triangulation Th is obtained byrefinement of the coarse TH andresolves the media heterogeneities;The desire is to formulate a globalproblem associated with the coarsemesh only. Ω

TH

Th





Multiscale FEM for second order elliptic problems

The MSFEM for second order problems construct the spaceSH ⊂ H1

0 (Ω) of piecewise polynomials over the coarse mesh THand the spaces Sh ⊂ H1

0 (K ) for each K ∈ Th. Then the fine-gridcomponent uh is eliminated (via say static condensation) andwe end up with a problem on the coarse grid component uH .

Then the multiscale solution has the form uH + uh, whereuH ∈ SH and uh ∈ Sh.

Unfortunately, this simple idea is not going to work for Brinkmansystem. We need to do better.





Two-grid method of Arbogast for saddle points: Fine and Coarse Grid Spaces

The approximation spaces use the two grids Th and TH and areconstructed in the following way:

Wh and Vh consist of bubbles on each T ∈ TH ;the FE spaces on the composite mesh areWH,h = Wh ⊕WH ⊂ L2

0, and VH,h = Vh ⊕ VH ⊂ H0(div)

Crucial properties:

1 ∇ · Vh = Wh and ∇ · VH = WH

2 vh · n = 0 on ∂T , ∀vh ∈ Vh and∂T ∈ TH

3 WH ⊥ WhΩ

TH

Th





Composite Grid Problem

Unique decomposition in components yields: finduH + uh ∈ Wh ⊕WH , and pH + ph ∈ Vh ⊕ VH such that

composite solution

a (uH + uh,vH + vh) + b (vH + vh,pH + ph) = F (vH + vh),

b (uH + uh,qH + qh) = 0,

for all vH + vh ∈ Wh ⊕WH and qH + qh ∈ Vh ⊕ VH .

Our goal is to derive numerical upscaling procedure.





Splitting of the Solution

Equivalently, testing separately for vH ,qH and vh,qh, we get

(∗)

a (uH + uh,vH) + b (vH ,pH +ph) = F (vH) ∀vH ∈ VHb (uH +uh,qH) = 0 ∀qH ∈ WH

(∗∗)

a (uH + uh,vh) + b (vh,pH + ph) = F (vh) ∀vh ∈ Vhb (uH + uh,qh) = 0 ∀qh ∈ Wh

Since

b (v ,q) = (∇ · v ,q), ∇ · Vh = Wh, ∇ · VH = WH , WH ⊥ Wh.

If we knew uH then we could find easily uh from

a (uh,vh) + b (vh,ph) + b (uh,qh) = F (vh)− a (uH ,vh)





a (uh,vh) + b (vh,ph) + b (uh,qh) = F (vh)− a (uH ,vh)

Decompose (∗∗) further into

(∗∗)

a (δu(uH),vh) + b (vh, δp(uH)) = −a (uH ,vh) ∀vh ∈ Vh

b (δu(uH),qh) = 0 ∀qh ∈ Wha(δu,vh

)+ b

(vh, δp

)= F (vh) ∀vh ∈ Vh

b(δu,qh

)= 0 ∀qh ∈ Wh

Note that(δu(uH), δp(uH)) is linear in uH

(δu, δp) and (δu(uH), δp(uH)) can be computed locally





The Upscaled Equation

We have(uh,ph) = (δu + δu(uH), δp + δp(uH)).

Putting all these together we obtain:

Symmetric Form of the Upscaled Equation

Thus we geta (uH + δu(uH),vH + δu(vH)) + b (vH ,pH) =F (vH)− a

(δu,vH

),

b (uH ,qH) =0,

which involves only coarse-grid degrees of freedom.





Sub-grid Algorithm, Willems, 2009

Sub-grid Algorithm (numerical upscaling): Willems, 2009

1 Solve for the fine responses (δu, δp) and (δu(ϕH), δp(ϕH))for coarse basis functions ϕH and each coarse cell.

2 Solve the upscaled equation for (uH ,pH).3 Piece together the solutions to get

(uH,h,pH,h) = (uH ,pH) + (δu(uH), δp(uH)) + (δu, δp).

For pure Darcy this reduces to the method of Arbogast, 2004.

The Question is: DOES IT WORK ?





Sub-grid Algorithm, Willems, 2009

Sub-grid Algorithm (numerical upscaling): Willems, 2009

1 Solve for the fine responses (δu, δp) and (δu(ϕH), δp(ϕH))for coarse basis functions ϕH and each coarse cell.

2 Solve the upscaled equation for (uH ,pH).3 Piece together the solutions to get

(uH,h,pH,h) = (uH ,pH) + (δu(uH), δp(uH)) + (δu, δp).

For pure Darcy this reduces to the method of Arbogast, 2004.

The Question is: DOES IT WORK ?





Sample Geometries

Periodic Vuggy SPE10





Periodic Geometry - Velocity

u = [1,1] on ∂Ω, f = 0, µ = µ = 1, κ−1 =

1e3 in the holes1e−2 elsewhere

Reference velocity Subgrid velocity

Relative L2-error: 3.69e-03.





Periodic Geometry - Pressure

Reference pressure Subgrid pressure

Relative L2-error: 1.77e-02.





SPE10 - Pure Subgrid for Brinkman

u = [1,0] on ∂Ω, f = 0, µ = 1e−2,Th : 1282,κ−1 : ranging from 1e5 in blue to 1e2 in red.

u1

Relative L2-error velocity error: 2.26e-01





SPE10 - Subgrid for Brinkman

u = [1,0] on ∂Ω, f = 0, µ = 1e − 2,κ−1 : as shown on the plot ; Th : 1282.

(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.





Vuggy media - Subgrid Brinkman

u = [1,0] on ∂Ω, f = 0, µ = 1e − 2,κ−1 = 1e3 Th : 1282.

(a) Ref. solution (b) H = 1/16. (c) H = 1/8. (d) H = 1/4.

Figure: Velocity component, u1, for vuggy geometry.





SPE10 benchmark





SPE10 benchmark: convergence of the iterations

Pure Darcy casefor periodic ge-ometry for differ-ent contrasts.

Brinkman case forSPE10 geometryfor different con-trasts.





How to fight the resonance error ?

There are various techniques to overcome the resonance error:

1 Using boundary conditions based on reducing the operatorto the boundaries of the coarse mesh TH ;

2 Oversampling3 Mortar finite element approximation;4 Hybridizable DG FEM;5 WG - weak Galerkin FEM shows great potential




Outline







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 twogrids Th and TH . We consider the mixed formulation of Darcyequation (with α(x) = κ(x)−1):

αu +∇p = 0 x ∈ Ω, ∇ · u = f x ∈ Ω p = g x ∈ ∂Ω.




HDG

Let EH , Eh denote the set of all edges of TH , Th, respectively.We defineE0

h := eh ∈ Eh| eH 6∈ ∂KH – the internal edges of Th∂Th := ∪Kh∈Th∂Kh – the edges the fine mesh Th∂TH := ∪KH∈TH∂KH – the edges of the coarse mesh TH




HDG

the HDG generates an approximation (ph,uh, ph,H) to(p,u,p|Eh ) on the finite dimensional (often piece-wisepolynomial but not necessarily) spaces defined on each K ∈ Th:

Vh := v ∈ L2(Th) : v |K ∈ V (K ),

Wh := w ∈ L2(Th) : w |K ∈W (K ),

and Mh,H := M0h ⊕MH , with

Mh := µ ∈ L2(Eh) : µ|Fh ∈ Mh(F ),Fh ∈ Eh,

M0h := µ ∈ Mh : µ|Fh = 0,Fh ⊂ FH ∈ EH,

MH := µ ∈ L2(EH) : µ|FH ∈ MH(FH), FH ∈ EH.




HDG

Then we seek (ph,uh, ph,H) ∈Wh × Vh ×Mh,H as the onlysolution of the following weak problem:

−(ph , ∇ · v)Th + (αuh , v)Th + 〈ph,H , v · n〉∂Th = 0, ∀v ∈ Vh

−(uh , ∇w)Th + 〈uh,H · n , w〉∂Th = (f , w)Th , ∀w ∈Wh

〈uh,H · n, µ〉∂Th = 0, ∀µ ∈ M0h,H

〈uh,H , λ〉∂D = 〈g, λ〉∂D, ∀λ ∈ M∂h,H

Here M0h,H = µ ∈ Mh,H : µ|∂D = 0, M∂

h,H = Mh,H |∂D,(η , ζ)Th :=

∑K∈Th

(η, ζ)K and

uh,H · n = uh · n + τ(ph − ph,H) on ∂Th.




HDG

We split into two sets of equations by testing separately for µ inM0

h and M0H := µ ∈ MH : µ|∂D = 0:

〈uh,H · n, µ〉∂Th = 0, ∀µ ∈ M0h & 〈uh,H · n, µ〉∂TH = 0, ∀µ ∈ M0

H .

On each cell K ∈ TH , given the boundary data of ph,H = ξH forξH ∈ MH(F ),F ∈ ∂K , we find (uh,ph, ph,H) on K = KH :

−(ph , ∇ · v)K + (αuh , v)K + 〈ph,H , v · n〉∂K = 0,−(uh , ∇w)K + 〈uh,H · n , w〉∂K = (f , w)K ,

〈uh,H · n, µ〉∂K = 0

for all (w ,v , µ) ∈Wh|K × Vh|K ×M0h |K . In fact the above local

system is the regular HDG methods defined on a coarse cell K .




HDG

The numerical upscaling method

This HDG method defines a global mapping from MH toWh ×Vh ×M0

h . This implies that one possible implementation ofthe method would be to solve the local problems for all basisfunctions of M0

H and then solve the global system on the coarsemesh:

a(ξH , µ) := 〈uh,H(ξH) , µ〉∂TH = 0, ∀µ ∈ M0H ,

〈ξH , λ〉∂D = 〈g · n , λ〉∂D ∀λ ∈ M∂H .

(6)




HDG

This is quite similar to the mortar MS FEM, e.g. T Arbogast, GPencheva, MF Wheeler, I Yotov, 2007.The main difference is that instead of special choice of themortar space the stability is provided by the stabilizationparameter (or often called operator) τ in the choice of thenumerical flux

uh,H · n = uh · n + τ(ph − ph,H) on ∂Th.

This choice often generates superconvergent schemes orallows more flexible choices of the approximation spaces,including locally generated solutions of some spectralproblems.




HDG – current and future work

Now we are working in various directions that include:

An implementation of this partGenerating the numerical trace space by solving localspectral problems or traces of snapshot spacesBrinkman/Stokes problem and its applications;Theoretical justification of the method and optimal errorestimates;Efficient solution methods for the resulting systems oflinear equations;

Most important: this is a framework for novel model reductiontechnique for parameter dependent problems.




Thank you

Thank you for your attention !!!


Documents

Numerical Simulation of Flows in Highly Heterogeneous ... · Single Grid Approximation of Brinkman equations Subgrid Approximation and Its Performance Numerical Experiments 3 Hybridizable