One of such important problems is concerned with the simulation of flow and transport through porous media. Pulloor Kuttanikkad (Heidelberg University).
Pore-scale simulation of Flow and Transport Using an Unfitted Discontinuous Galerkin Finite Element Method Sreejith Pulloor Kuttanikkad Ph.D Student Interdisciplinary Centre for Scientific Computing (IWR) University of Heidelberg & Simulation of Large Systems Department (SGS) IPVS, University of Stuttgart (Joint work with : Christian Engwer, Peter Bastian, Kurt Roth)
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Outline
1
Introduction
2
Pore-scale (Micro-scale) Modelling
3
Unfitted Discontinuous Galerkin Method
4
Discontinuous Galerkin Discretization of Stokes Equation
5
Numerical Experiments
6
A Numerical Upscaling Example
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Introduction and Motivation Numerical simulation of problems in Science and Engineering (simulating physical, biological, chemical, etc. processes) requires handling of complex shaped domains! One of such important problems is concerned with the simulation of flow and transport through porous media
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Introduction and Motivation Numerical simulation of problems in Science and Engineering (simulating physical, biological, chemical, etc. processes) requires handling of complex shaped domains! One of such important problems is concerned with the simulation of flow and transport through porous media Subsurface/Environmental applications: Contaminant transport, nuclear waste disposal, groundwater remediation, oil/gas/petroleum exploration
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Introduction and Motivation Numerical simulation of problems in Science and Engineering (simulating physical, biological, chemical, etc. processes) requires handling of complex shaped domains! One of such important problems is concerned with the simulation of flow and transport through porous media Subsurface/Environmental applications: Contaminant transport, nuclear waste disposal, groundwater remediation, oil/gas/petroleum exploration Industrial applications: Fuel cell, packed bed reactors, filtration studies
Pulloor Kuttanikkad (Heidelberg University)
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Introduction and Motivation Numerical simulation of problems in Science and Engineering (simulating physical, biological, chemical, etc. processes) requires handling of complex shaped domains! One of such important problems is concerned with the simulation of flow and transport through porous media Subsurface/Environmental applications: Contaminant transport, nuclear waste disposal, groundwater remediation, oil/gas/petroleum exploration Industrial applications: Fuel cell, packed bed reactors, filtration studies Biomedical applications: Modelling of flow of blood, nutrients etc. through body tissues
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Introduction and Motivation Numerical simulation of problems in Science and Engineering (simulating physical, biological, chemical, etc. processes) requires handling of complex shaped domains! One of such important problems is concerned with the simulation of flow and transport through porous media Subsurface/Environmental applications: Contaminant transport, nuclear waste disposal, groundwater remediation, oil/gas/petroleum exploration Industrial applications: Fuel cell, packed bed reactors, filtration studies Biomedical applications: Modelling of flow of blood, nutrients etc. through body tissues
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Introduction and Motivation How does pore-scale properties such as pore-scale structure (its geometric features) influence the macroscopic flow and transport parameters ? Despite numerous experimental and numerical studies there is still a lack of fundamental understanding of how pore structure controls subsurface flow and transport!
To determine macro-scale parameters such as permeabilities and dispersion coefficients and study their dependence on pore-scale properties (geometry) using numerical simulation
Reconstructed pore structure (Vogel, Roth; Heidelberg)
Thin section from a loamy-clay soil (Vogel, Roth; Heidelberg)
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Mathematical Modelling Approaches Based on the scale at which porous medium is described Macroscopic Modeling Based on Darcy’s law Neglect geometrical details Need to know some coeffs/parameters Microscopic/Pore-scale Modeling (µm) On a scale where individual pores enter the description Based on rigorous/fundamental physics, No averaging Offers ways to improve the prediction of macroscopic parameters Pulloor Kuttanikkad (Heidelberg University)
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Pore-scale (Micro-scale) Modelling Provides a link between microscopic properties of the porous medium and large scale behaviour Challenge: Simulation of pore-scale processes requires dealing with complicated domains/pore-structure In general, pore-scale modeling is limited by two major factors Pore-scale modelling requires detailed pore structure. Unfortunately detailed pore structure of real materials is rarely obtained Using CT scanning, NMR imaging, etc. we can only obtain structural details for small sample (REV size)
Numerical methods should be able to account for the geometry Handling the pore scale in a numerical simulation is not easy. A good approximation of the geometry is crucial for the simulation.
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Pore-scale Modelling: Current Approaches
Pore Network Modelling Lattice Boltzmann based modelling Standard numerical methods (FE, FV, FD, etc.)
A complex pore geometry
Our approach: Use Unfitted Discontinuous Galerkin method ( a new Discontinuous Galerkin FE based approach) Network model
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Standard Numerical methods: Handling Complex Geometry
Using standard finite element Require a mesh resolving the complex domain boundary (realistic 3D meshing is still considered a challenge) High resolution triangulation/meshing results in a large number of DOFs Quality of mesh affects convergence behaviour of iterative solvers, finite element approximation errors
Using various structured grid based methods Fictitious Domain ( Embedded domain), Immersed Boundary, Immersed Interface, Cut Cell Composite Finite Element
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Unfitted Discontinuous Galerkin Method Introduced by Engwer and Bastian (2005,2008) C. Engwer and P. Bastian. (2005) A Discontinuous Galerkin Method for Simulations in Complex Domains. Technical Report 5707, IWR , Universit¨ at Heidelberg, http://www.ub.uni-heidelberg.de/archiv/5707/
Inspired by the unfitted finite elements (Barrett& Elliott 1987; Nitsche 1971) and Some nice features of Discontinuous Galerkin Freedom in the shape of elements Shape functions can be independent of shapes Higher order and Local mass conservation Element wise discontinuous shape functions Easy for parallelisation
What is not so nice thing about DG: Increase in number of unknowns
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Unfitted Discontinuous Galerkin Method Mesh Construction Given the pore geometry A fundamental structured grid is chosen In general fundamental mesh can be chosen according to desired accuracy, computational resource Generally a course mesh can be used
Intersection leads to variety of arbitrary forms
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Unfitted Discontinuous Galerkin Method Intersection Grid intersected by the solids generate arbitrary shaped elements Constrain the support of shape functions to the fluid phase in each element
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Unfitted Discontinuous Galerkin Method Local Triangulation Local triangulation for assembling Elements are subdivided into easily integrable forms Integration using standard quadrature rules
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Unfitted Discontinuous Galerkin Method Local Triangulation procedure: Subdivision of elements into sub-elements which are easily integrable (“Local Triangulation”) - Predefined triangulation rules for a class of similar elements - Reduce number of different classes by appropriate bisection of the element
Use of quadratic transformation for better approximation of curved boundaries Use of standard quadrature rules for the integration over sub-elements Pulloor Kuttanikkad (Heidelberg University)
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Unfitted Discontinuous Galerkin Method Fundamental Mesh and Local Triangulation
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Unfitted Discontinuous Galerkin Method Fundamental Mesh and Local Triangulation
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Unfitted Discontinuous Galerkin Method
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Unfitted Discontinuous Galerkin Method Evaluation of Integrals: Shapefunctions are defind in local coordinates Set Q = {(qi , wi )}, pair of integration points and weights on the reference elements The integral over a globally defined function f on the arbitrary shaped element can be approximated as (from Engwer, Bastian 2005)
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Governing Equations at the pore-scale
In general, Navier–Stokes equations govern the flow dynamics at the pore-scale. However, Flow in porous media (at subsurface) is generally slow (laminar) Viscous dominated and inertial effects are negligible (drop non-linear term in NS equation !) Flow is considered stationary (neglect time dependency) . . . leads to stationary Stokes equation !
The stationary flow of a viscous, incompressible, Newtonian fluid through porous media at the pore-scale can be described by Stokes equation We need to solve Stokes equation on pore-scale complex geometries for the given boundary conditions!
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The Stokes Equation The Stokes problem Let Ω ⊂ Rd be a bounded, open and connected Lipschitz domain Find (u, p) ∈ H 1 (Ω)d × L2 (Ω) such that −µ∆u + ∇p = f
in
Ω
∇·u = 0
in
Ω
with Dirichlet and outflow boundary conditions u = g
on
∂ΩD
∂n u + p = 0
on
∂ΩN
If no Neumann/Outflow BCs applied, additionally we require that Z
p dΩ = 0
Ω
Pulloor Kuttanikkad (Heidelberg University)
Z
and
g.n ds = 0
∂Ω
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The Standard Weak Formulation of Stokes Equation
In abstract form Find (u, p) ∈ V × Q R ∀v ∈ V, µ(A(u, v)) + B(v, p) = Ω f · v dx ∀q ∈ Q, B(u, q) = 0 where A : H01 (Ω)d × H01 (Ω)d → R, A(v, u) = (∇v, ∇u) B : H01 (Ω)d × L02 (Ω) → R,B(u, q) = (∇ · u, q)
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DG (Interior Penalty) Discretization of Stokes Equation Mesh and Function spaces Mesh Let Ω ⊂ Rd be a polygonal computational domain Let Th = {Ei , i ∈ I} be a partition of Ω Function spaces Let Vh ⊂ V,Qh ⊂ Q Vh = {v ∈ L2 (Ω)d : vk ∈ Pk (E)d , E ∈ Th } Let Qh ⊂ Q, Qh = {q ∈ L02 (Ω) : qk ∈ Pk−1 (E), E ∈ Th } where Pk (E) is the space of discontinuous polynomials of degree k on element E Note that polynomial approximation for velocity is one degree higher than pressure (LBB stability condition)
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Definitions ⇒ Define Jump [·] and Average h·i of a function Let Γ = Γint ∪ ΓD set of all faces in the mesh Th γ ⊂ Γint γ ⊂ ΓD
[φ] = φ|e − φ|f
[φ] = (φ − φD )n
1 hφi = (φ|e + φ|f ) 2
hφi = φ
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DG Weak Formulation of Stokes Equation
Let v and q be suitably chosen test functions For {E} ∈ Th Multiplying with test functions and integrating over the elements Z E
(−µ∆u + ∇p) v = Z E
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Z
(∇·u) q = 0
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fv
(1)
E
(2)
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DG Weak Formulation of Stokes Equation
Applying integration by parts and Gauss’s theorem on LHS of equation (1) Z
(−µ∆u + ∇p) v | {z } E
−
Z E Z
µ∆u v dx =
E
Z E
∇p v dx = −
Pulloor Kuttanikkad (Heidelberg University)
µ∇u ∇v dx − Z E
p∇ · v dx +
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Z
µ
Z ∂E
∂u v ds ∂n
pn vds ∂E
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DG Weak Formulation of Stokes Equation
Applying divergence theorem on LHS of equation (2) Z
(∇·u) q | {z } E
Z E
(∇·u) q = −
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Z E
q∇ · u +
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Z
qu · n
∂E
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DG Weak Formulation of Stokes Equation Putting together we have, Z E
(−µ∆u + ∇p) v =
Z
µ∇u ∇v dx −
E
−
Z E
= Z E
Z E
µ∇u ∇v dx −
(∇·u) q = −
Z
p∇ · v dx +
pn vds
Z
f vdx ZE E
µ(∇u · n) v ds −
q∇ · u +
Z
qu · n
∂E
Z E
−
p∇ · v dx + Z E
Pulloor Kuttanikkad (Heidelberg University)
µ(∇u · n) v ds
Z ∂E ∂E
=0
∂E
Z
q∇ · u +
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Z
Z ∂E
pn vds =
Z
f vdx E
q(u · n) = 0
∂E 24 / 56
DG Weak Formulation of Stokes Equation
Summation over all elements Z
∑
Z
µ∇u ∇v dx −
E∈Th E
∑
µ[(∇u · n) v] ds −
γ∈Γint γ
−
Z
∑
E∈Th E
−
∑
Z
∑
[pn v]ds +
γ∈Γint γ
q∇ · u +
E∈Th
µ(∇u · n)v
γ∈ΓD γ
p∇ · v dx +
∑
Z
Z
∑
[q(u · n)] +
γ∈Γint γ
Z
∑
pvnds =
Z
γ∈ΓD γ
Z
∑
fv Ω
q(u · n) = 0
γ∈ΓD γ
where [.] is the discontinuity or jump of the function
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DG Weak Formulation of Stokes Equation Note the property [φψ] = [φ]hψi + [ψ]hφi Apply the property on internal face terms −
Z
∑
µ[(∇u · n) v] ds = −
γ∈Γint γ
Z
∑
µh(∇u · n)i[v] −
γ∈Γint γ
Z
∑
[pn v]ds =
γ∈Γint γ
Z
∑
[q(u · n)] =
γ∈Γint γ
Pulloor Kuttanikkad (Heidelberg University)
Z
∑
Z
∑
γ∈Γint γ
µ [(∇u · n)]hvi | {z } =0
hpi[vn]ds
γ∈Γint γ
Z
∑
hqi[(u · n)]
γ∈Γint γ
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SIPG Formulation In order to have symmetry, add Z
ε
∑
µh(∇v · n)i[u]
γ∈Γint γ
In order to have good stability, apply penalty/jump term σ γ∈Γint |e|
J(u, v) =
Z
∑
[u] · [v]
γ
Both the above terms vanish for smooth solution Apply Dirichlet BC’s weakly Z
ε
∑
µ(∇v · n)(u − g)
γ∈ΓD γ
σ ∑ |e| γ∈ΓD
Z
(u − g) · [v]
γ
where ε = ±1 and σ ≥ 0 are some parameters Pulloor Kuttanikkad (Heidelberg University)
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Different Interior Penalty DG Schemes
If ε = −1 and σ ≥ 1 , SIPG If ε = +1 and σ ≥ 1 , NIPG If ε = +1 and σ = 0 , OBB
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DG Formulation of Stokes Equaton: Global view! Find (u, p) ∈ V × Q ∀v ∈ V, µ (A(u, v) + J(u, v)) + B(v, p) = F(v) ∀q ∈ Q, B(u, q) = G(q) A(u, v) =
Z
∑ E
∇u · ∇v −
Ω
∑
(∇u · n)v + ε (∇v · n)u
∑
F(v) =
Z
∑ E
G(q) =
[u] · [v] ,
B(v, p) = − ∑
γ
q∇ · u +
Z
∑
Z
p∇ · v +
Ω
Z
∑
hpi[v · n]
γ∈Γint γ
hqi[u · n]
γ∈Γint γ
Ω
f · v + µε
Ω
Z
∑
γ
Z
σ J(u, u) = ∑ γ∈Γint |e| Z
γ
Z
γ∈ΓD γ
B(u, q) = − ∑
Z
h∇u · ni[v] + ε h∇v · ni[u]
γ∈Γint γ
Z
−
Z
Z
∑ γ∈ΓD
σ (∇v · n)g + |e| γ
Z
g·v
γ
q(g · n)
γ∈ΓD
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Implicit Geometry Representation UDG approach to complex geometry representation Using the idea of implicit geometry representation via level set functions Complex domain: Ω = {x : φ(x) > 0} Domain boundary: ∂Ω = {x : φ(x) = 0} (iso-surface, zero level-set)
Provides a general and flexible way to represent complex geometry as image data (scalar function) Surface extraction from CAD and image data (e.g. medical, porous media, etc. image using CT, NMR etc.) The surface is then represented on a Cartesian grid with implicit functions Enable direct simulation in irregular microscale geometries (e.g., porous media)
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Marching Cube/Tetrahedron
Well known algorithm for polygonizing a scalar field Polygonization routine, which provides a 3D mesh description of the implicit surface =⇒ Input an implicit function, I(x, y, z) = 0 =⇒ Apply marching cube/tetrahedron method =⇒ Marching cubes algorithm solves equation I(x, y, z) = 0 to provide mesh description of the unknown surface =⇒ Generate 2D/3D mesh description
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Random Sphere Packing (Idealized porous medium in 3D) Sphere Packing (number of spheres, centres & radii) is generated using an open source code (http://atom.princeton.edu/Packing/) The implicit function used to represent the sphere packing: φ(x) = min(|x − xk |2 − rk2 ) where xk centre of kth sphere, rk radius of rth k
sphere and x : φ(x) = 0 gives boundary
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Implementation and Testing Numerical implementation of the algorithm is done using the DUNE frame work DUNE, the Distributed and Unified Numerics Environment (http://www.dune-project.org/index.html) A new frame work for the numerical solution of partial differential equations designed with following principles - Flexibility - Efficiency - Legacy Code
ISTL (Iterative Solver Template Library in DUNE) for the solution of linear system Second order monomial basis function for velocity and first order basis for pressure No-slip boundary conditions on the surface of internal obstacles OBB scheme Visualization with VTK and Paraview Pulloor Kuttanikkad (Heidelberg University)
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Grid Convergence: Stokes Equation (2D) Pressure around a circular obstacle
Grid Refinement Level = 1
Grid Refinement Level = 2
Grid Refinement Level = 4
Grid Refinement Level = 5
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Grid Refinement Level = 3
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Grid Convergence: Stokes Equation (2D) Velocity around a circular obstacle
Grid Refinement Level = 1
Grid Refinement Level = 2
Grid Refinement Level = 4
Grid Refinement Level = 5
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Grid Refinement Level = 3
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Stokes flow through an artificial porous medium An artificially generated pore structure is given as a scalar function on mesh with mesh size hg = 1/64 100 Spheres of radius r = 0.106 fill a domain of size H 3 with H = 1.
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Stokes Simulation, h = 1/8
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Stokes Simulation, h = 1/16
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Stokes Simulation, h = 1/32
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Stokes Simulation, h = 1/8
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Stokes Simulation, h = 1/16
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Stokes Simulation, h = 1/32
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Stokes Simulation, h = 1/32 Zoom on a velocity field computed on a grid with mesh size h = 1/32 and a geometry given on a grid with hg = 1/64
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Stokes Simulation Stream tracer
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Stokes Simulation Stream tracer
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Numerical Upscaling Steps: A macroscopic pressure gradient is imposed along the x axis. The boundary conditions: The pressure is prescribed at the inlet and outlet No-flow condition is applied on other boundaries No-Slip condition On the grain/sphere surfaces
Solve the Stokes equation on the pore-scale. Obtain the pore-scale pressure and velocity.
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Numerical Upscaling: Prscribed pressure formulation The flow (stationary) of a viscous, incompressible, Newtonian fluid through porous media can be described by Stokes equation. Let Ω ⊂ IR3 be a bounded, open and connected Lipschitz domain. For any prescribed constants p0 , find velocity and pressure (u,p) ∈ H 1 (Ω)d × L2 (Ω) such that −µ∆u + ∇p = f
in
Ω ⊂ IR3
∇·u = 0
in
Ω
u=0
on
Γ0 ⊆ ∂Ω
∂n u + p = p0
on
ΓP .
where µ > 0 is the dynamic viscosity of the fluid. On the surface of the pore structure no-slip boundary condition is applied.
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DG Discretisation for prescribed pressure formulation The DG discretization of the Stokes equation using the non-symmetric scheme introduced by Oden-Babuska-Baumann reads: (i) 3
(i)
Find u ∈ Vk , p ∈ Vk−1 such that µa(u, v) + b(v, p) = l(v) b(u, q) = 0 a(u, v) =
Z
∑ E
Ω
b(u, q) = − ∑ l(v) = −
∇u : ∇v −
∑
q∇ · u +
Ω
∑
. Z
∑
h∇v · ni[u]
γef ∈Γint γef
hqi[u · n]
γef ∈Γint γef
Z
p0
γp ∈ΓP
Z
,
(i) ∈ Vk−1
h∇u · ni[v] +
γef ∈Γint γef
Z
∑
∀q Z
(i) 3
∀ v ∈ Vk
v · nds
γp
where [ · ] and h · i denote jump and average of the function on an element boundary. Pulloor Kuttanikkad (Heidelberg University)
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Numerical Upscaling The mean velocity is calculated as u¯ =
Z Ωp
udx · |Ωp |−1
p
| p The porosity is given as φ = |Ω |Ω| , where |Ω | is volume of the pore-space and |Ω| denotes the volume of the domain.
Assuming no preferred orientation, permeability tensor to be an isotropic diagonal tensor The xx component of the permeability tensor κxx = −µ
¯ uφ , ∇p
For water, µwater = 1 · 10−3 Pa.s. Pulloor Kuttanikkad (Heidelberg University)
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Numerical Upscaling: Comparison with analytical results
Computing permeability from pore-scale simulation
Compared simulated permeability with analytical values for different ordered sphere packings (FCC,SC,BCC) (Sangani & Acrivos, Int. J. Multiphase Flow,1982) Type FCC SC
Φ 0.259 0.476
κAnal 8.68e-05 2.52e-03
Simple Cubic Packing (SC)
κSim 8.69e-05 2.51e-03 Face Centered Cubic Packing (FCC)
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Numerical Upscaling: Comparison with analytical results Grid Convergence: κ for different mesh size h
κeff [m2 ]
0.01
Φ = 0.26
0.001
0.0001
1e-05
1
2
H/h
3
4
Permeability computed for FCC converging to the analytical value (Sangani & Acrivos, 1982) on a relatively coarser grid
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Upscaling: Finding permeability of an artificial porous medium An artificial porous medium is created by a sphere packing and the packing geometry is given as a scalar function on mesh with mesh size hg = 1/64 100 Spheres of radius r = 0.106 fill a domain of size H 3 with H = 1.
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Upscaling: Grid Convergence for the artificial porous medium Grid Convergence: κ for different mesh size h
κeff [m2 ]
0.1
Φ = 0.768
0.01
0.001
0.0001
1
2
3
H/h
4
Permeability computed with grid refinements on the artificial porous medium Pulloor Kuttanikkad (Heidelberg University) From Micro to Macro
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Upscaling: Porosity Vs Permeability Varied the radius r of the spheres to change the porosity Φ r Φ
0.0318 0.9886
0.0530 0.9432
0.0742 0.8437
0.0954 0.6732
0.1060 0.5534
0.1166 0.4161
Φ Vs κ for two different mesh size h 0.1
h = 1/16 h = 1/32
κeff [m2 ]
0.01 0.001 0.0001 1e-05 Pulloor Kuttanikkad (Heidelberg University)
0.5
0.7
P orosity
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Φ
0.9 54 / 56
Summary The DG based scheme is applied for solving PDEs on complex domain, which uses a cartesian grid Exploited the nice properties of DG schemes Method offers a direct discretization of the PDE’s on pore-scale Retain the benefits of the standard finite element methods, offers higher flexibility in the mesh Minimum number of unknowns even for complicated domains An implicit geometrical representation fits nicely with this approach Applicability of the method is shown through numerical upscaling examples Outlook A real comparison (performance, accuracy etc.) with other methods (standard FEM, pore-network models) Apply on real geometries obtained from images (porous medium images from CT scan) Extend the approach to two-phase flow Pulloor Kuttanikkad (Heidelberg University)
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Thank You!
