Pore-scale direct simulation of flow and transport in porous media

Pore-Scale Direct Numerical Simulation of Flow andTransport in Porous Media

Sreejith Pulloor Kuttanikkad

PhD Thesis Defence(Thursday 15 October, 2009)

Interdisciplinary Centre for Scientific Computing (IWR)Faculty of Mathematics and Informatics, University of Heidelberg

Sreejith Kuttanikkad (IWR, University of Heidelberg) Pore-Scale Simulation 15.10.2009 1 / 36

Outline

1 Introduction - Relevance of the topic

2 Thesis Motivation and Objectives

3 Pore-scale Models for Flow and Transport

4 Unfitted Discontinuous Galerkin (UDG) Method for Complex Domains

5 Random Walk Particle Tracking Method

6 Implementation and Validation of Flow and Transport Models

7 Pore-scale simulation of Dispersion in 2D & 3D

8 Summary and Outlook

Introduction - Relevance

Study of flow and transport through porous media has wide practicalapplications

Subsurface/Environmental applications (contaminant transport, nuclear wastedisposal, groundwater remediation, oil/gas/petroleum exploration)Industrial applications (PEM Fuel cells, packed bed reactors, filtration studies)

Studies are being done at various scales using Numerical and Experimentalmethods

Numerical simulations are usually done at the continuum scale which requirethe knowledge of certain parameters (permeability, dispersion coefficients,etc.)

Introduction - Relevance of the topic

Macroscopic simulations fail to explain certain flow and transport behaviours (e.g,tailing of BTC, hysteresis in multiphase flow parameters)

Advection dispersion equation(ADE) is generally used as the toolfor predicting and quantifying solutetransport

∂C

∂t+∇ · (vC)−D.∇2C = 0

The basic assumption of the ADE isthat dispersion follows Fickianbehavior

J = −D∇C

Numerous experiments have shownthat solute spreading does notfollow a Gaussian distribution

Introduction - Relevance

Pore-Scale SimulationsAn alternative and more fundamental approach

Provides link between pore-scale properties of the porous medium and largescale behaviour

Governing flow and transport equations are known at pore-scale

Macroscopic parameters can be obtained using the results of pore-scalesimulations

Pore-Scale MethodsChallenges: require detailed structure of the medium, method should be ableto handle geometry

Pore-scale numerical methods: Pore-Network, LBM, Finite Element, etc.

New and efficient pore-scale simulation methods have its relevance in thiscontext

Motivation and Research Objectives

MotivationLack of fundamental understanding of how pore structure controls flow andtransport behaviour at larger scales!

This work is motivated by the need for better understanding of the physicalprocesses that take place at the pore-scale and to improve the reliability ofnumerical models that describe the flow at larger scales

The objectives set for the study are

To develop a model to simulate single phase flow and solute transportprocesses through porous media at the pore-scale

In particular, to use a new numerical discretisation approach (called UnfittedDiscontinuous Galerkin UDG) for the solution of partial differential equationson the pore-scale geometry

And to predict the macroscopic parameters of porous medium based onpore-scale simulations

Pore-Scale Modelling of Flow and Transport

Present approach involve following steps:

1 Compute the pore-scale velocity field (by Solving Stokes equation)

−µ∇2u +∇p = f ; ∇ · u = 0

By using a new method called Unfitted Discontinuous Galerkin which requiresImplementation of DG finite element discretisation of Stokes equation in theframework of unfitted discontinuous Galerkin method

2 Obtain the flow and transport parameters based on the computed pore-scalevelocity field

Permeability is computed by applying the Darcy’s lawDispersion coefficients are determined by solving the Advection-Diffusionequation posed at the pore-scale by RWPT method

∂C

∂t= −u · ∇C + D∇2C

Much of the challenge in solving Stokes problem (for velocity and pressure) is howto account for the complex pore-scale geometry!

Unfitted Discontinuous Galerkin (UDG) MethodMethod for the solution of the Stokes equation is based on a new numericalapproach which has been specifically developed for applications in complexdomains

UDG introduced by Engwer and Bastian (2005,2008)Use only a structured grid and based on DG finite element method with trialand test functions defined on the structured grid

Mesh ConstructionGiven the pore geometry, a fundamental structured grid is chosen

According to desired accuracy and computational resourcesGenerally a course mesh can be used

Unfitted Discontinuous Galerkin Method

Mesh ConstructionGrid intersected by the domain generate arbitrary shaped elements

Support of the trial and test functions are restricted according to the shapeof the elements

Essential boundary conditions are imposed weakly via the DG formulation

Number of dofs is proportional to the number of elements in the grid


Evaluation of surface and volume integrals

Local Triangulation

Local triangulation for assembling

Based on the marching cubealgorithms

Subdivision of elements intosub-elements which are easilyintegrable (“Local Triangulation”)

- Predefined triangulation rules fora class of similar elements

- Reduce number of differentclasses by appropriate bisection ofthe element

Use of quadratic transformation forbetter approximation of curvedboundaries

Use of standard quadrature rules forthe integration over sub-elements


Appealing things

Underlying DGFE discretisation of the PDE model

It has all benefits of standard finite element methodsAdvantages of the DG schemes are naturally incorporated

Allow arbitrary shaped elements

Easy incorporation of the complex geometries via implicit function or level setmethods

Possible to choose the computational grid independent of the pore geometry

Number of unknowns independent of the complex geometry

DG Discretization of the Stokes EquationFind (uh, ph) ∈ Vh × Qh such that(

µ(A(uh,vh) + J0(uh,vh)) + B(vh, ph) = F (vh) ∀vh ∈ Vh

B(uh, qh) = G(qh) ∀qh ∈ Qh

A(uh,vh) =XE∈Th

ZE

∇uh : ∇vhdx−XE∈EI

h

ZE〈∇uh · ne 〉[ vh ]ds+ ε

XE∈EI

h

ZE〈∇vh · ne 〉[ uh ]ds

−XE∈EB

h

ZE

(∇uh · nb)vhds+ εXE∈EB

h

ZE

(∇vh · nb)uhds

J0(uh,vh) =XE∈EI

h

σ

|e|

ZE

[ uh ] · [ vh ]ds+XE∈EB

h

σ

|e|

ZE

uh · vds

B(vh, ph) = −XE

ZE

ph∇ · vhdx+XE∈EI

h

ZE〈 ph 〉[ vh · ne ]ds+

XE∈EB

h

ZEphvh · nbds

F (vh) =XE∈Th

ZE

f · vhdx+ µεXE∈ED

ZE

(∇vh · nb)gds+ µXE∈ED

σ

|e|

ZE

g · vhds

−XE∈EDP

ZEp0vh · nbds

G(q) = −XE∈ED

ZEqhg · nbds

RW Particle Tracking Methods

Lagrangian based numerical approach for the solution of transport problem

Particle distributionThe trajectory of a tracer particle in an external pore velocity field ~u is given as

~Xi(t+ ∆t) = ~Xi(t) + ~S(t)|zAdv. displacement| z ~S(t)=~u(t)∆t

+ ~Z(t)|zDiff. displacement| z ~Z(t)=

√2Dm∆t~ξ

Statistical momentsThe centre of mass of the solute distribution is approximated by the firstmoment as

〈 ~x(t) 〉 =1

Np

NpXi=1

~xi(t)

The spread of mass (spatial variance) around 〈 ~x(t) 〉 is approximated by thesecond moment as

σ2

= var(~x(t)) =1

Np

NpXi=1

~xi(t)2 − 〈 ~x(t) 〉2.

The dispersion coefficient is determined from the spatial variance as

Deff(t) =1

2

dσ2(t)

dt

Code Implementation and Validation

Code implementation was done using the DUNE software framework

Validated by performing standard test problems

Convergence tests for the DG Stokes solutionMass balance checksAnalytical tests, Poiseuille (channel flow), driven cavity and flow aroundcylinder (for flow model)Computation of permeability for ordered sphere packingTaylor-Aris dispersion (for transport model)

Validation of Flow Model

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Y

Velocity

SimulationAnalytical

Poiseuille Flow in a channel

0

0.2

0.4

0.6

0.8

1

−0.4 −0.2 0 0.2 0.4 0.6 0.8 1

Y

Horizontal Velocity

SimulationDonea & Huerta (2003)

Flow in a driven cavity

Validation of The Flow Model

Computation of permeability for ordered packing of spheres

The mean pore velocity is calculated as

u =∫

Ωpudx · |Ωp|−1

The porosity is given as φ = |Ωp||Ω| , where |Ωp| is volume of the pore-space

The xx component of the permeability tensor

κxx = −µ uφ∇p

,

where |Ω| denotes the size/volume of the domain.


Compared simulated permeability with analytical values given in (Sangani & Acrivos, Int. J.

Multiphase Flow,1982) for different ordered sphere packings (FCC, SC)

Simple Cubic (SC) Face Centered Cubic (FCC)

Type φ κanalytical κsimulated

FCC 0.259 8.68e-05 8.69e-05SC 0.476 2.52e-03 2.51e-03

Validation of Flow ModelPermeability of SC, for various porosity’s:

Scaled vol-ume fraction(ψ)

Porosity κeff (Sangani andAcrivos 1982)

κeff (Com-puted)

Relative Error(%)

0.1 0.99951 0.91107 1.0169 11.610.2 0.99587 0.38219 0.40158 5.070.4 0.96661 0.12327 0.12578 2.030.6 0.88709 0.044501 0.04488 0.850.8 0.73 0.013197 0.01320 0.081.0 0.478 0.0025203 0.002516 0.17

1e-03

1e-02

1e-01

1e+00

1e+01

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Per

mea

bility,

κxx

Volume Fraction

Sangani&Acrivos (1982)Computed


Grid Convergence: Permeability computed for FCC converging to theanlytical value on a relatively coarser grid

1e-05

1e-04

1e-03

1e-02

1 1/2 1/4 1/8 1/16 1/32

Per

mea

bility,

κxx

h

FCC (φ = 0.26)

Permeability for an artificial porous medium (sphere pack)

Grid Convergence

Artificial porous medium made of randomly packedspheres

1e-04

1e-03

1e-02

1e-01

1 1/2 1/4 1/8 1/16 1/32

Per

mea

bility,

κxx

h

φ = 0.768

Permeability of the artificial porous medium computedon various grid levels

Porosity Vs PermeabilityVaried the radius r of the spheres to change the porosity Φ

r 0.0318 0.0530 0.0742 0.0954 0.1060 0.1166

Φ 0.9886 0.9432 0.8437 0.6732 0.5534 0.4161

1e-05

1e-04

1e-03

1e-02

1e-01

0.4 0.5 0.6 0.7 0.8 0.9 1,0

Perm

eability,κ

xx

Porosity, φ

h = 1/16h = 1/32

Validation of Transport Model

Taylor-Aris Dispersion

C

C0=

12erfc

[x− umt

2(Deff · t)1/2

]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

C C0

t∗ = t u/L

15000

10000

5000

1000

Analytical

Cumulative breakthrough curve (for various number of particles) for theTaylor-Aris dispersion compared to the analytical solution.

0

30

−10 0 10 20 30 40 50

y

x

Dm = 0.35

um = 0.8326

Pe = 71.365

t=0.0

0

30

0 20 40 60 80 100

y

x

t=43.5

0

30

0 50 100 150 200 250 300 350 400

y

x

t=217.0

0

30

0 100 200 300 400 500 600 700 800

y

x

t=433.7

0

30

0 200 400 600 800 1000 1200 1400y

x

t=867.55

0

30

0 500 1000 1500 2000

y

x

t=1301.3

0

30

0 500 1000 1500 2000 2500 3000 3500 4000

y

x

t=3036.4

Gaussian

Validation of Transport Model

Variation of the longitudinal dispersion coefficient Deff with the PecletNumber for the Taylor-Aris dispersion

Analytical Solution:Deff

Dm= 1 +

Pe2

210

10−1

100

101

102

100 101 102

Deff

Dm

Peclet Number (Pe)

AnalyticalComputed

Fit

Pore-scale simulation of Transport in 2D


This work (RWPT) Based on Eulerian method (DGFE) by Fahlke (2008)












Concentration breakthrough curve

0

0.05

0.1

0.15

0.2

0.25

0 5 10 15 20 25

Conce

ntr

ation

Time

10,00025,00050,000

100,000

Breakthrough curve plotted for different number ofsolute particles

−0.05

0

0.05

0.1

0.15

0.2

0.25

0 5 10 15 20 25

Norm

alise

dC

once

ntr

ation

Time

Present (RWPT)Fahlke, 2008 (DGFEM)

Breakthrough curve compared with the result of anEulerian scheme


Artificial porous medium and the computational grid


Computed pore-scale velocity field


Calculated concentration profiles along the porous medium

0

0.005

0.01

0.015

0.02

0.025

0e+00 2e+04 4e+04 6e+04 8e+04 1e+05

Norm

alised

Concentr

ation

Time


Dependence of dispersion coefficients on Peclet number

A standard way of describing longitudinal dispersion coefficient as a function of Pein the laminar flow condition is by using

DL

Dm=

1Fφ︸︷︷︸

molecular diffusion

+ αPe︸︷︷︸mechanical dispersion

+ βPeδ︸︷︷︸boundary layer diffusion

+ γPe2︸︷︷︸hold-up dispersion


Dependence of dispersion coefficient (DL) on Peclet number

10−1

100

101

102

103

104

10−2 10−1 100 101 102 103 104

DL

Dm

Peclet Number (Pe)

Pfannkuch, 1963Perkins and Johnston, 1963

Seymour and Callaghan, 1997Maier et al. (2000),LBM+RWPT

Kandhai et al.(2002),NMRKhrapitchev and Callaghan, 2003

Stohr (2003), PLIFBijeljic et al.(2004), Pore-network+RWPT

Freund et al.(2005), LBM+RWPTUDG+RWPT


Dependence of longitudinal dispersion coefficient on Peclet number inthe power law regime

10−1

100

101

102

103

101 102

DL

Dm

Peclet Number (Pe)



Kandhai et al.(2002),NMRStohr (2003), PLIF

Bijeljic et al.(2004), Pore-network+RWPTFreund et al.(2005), LBM+RWPT

UDG+RWPTFit

Simulated longitudinal dispersion coefficients in a random sphere packing compared to datareported in literature in the power law regime (3 < Pe < 300). The line corresponds

to the fit of the data toDLDm

= βPeδ with β=0.214003 and δ=1.20331.

Reference β δ

Pfannkuch(1963)

- 1.2

Gist et al.(1990)

0.46 - 3.9 0.93 - 1.2

Dullien(1992)

- 1.2

Coelho etal. (1997)

0.26 1.29

Manz et al.(1999)

- 1.12

Stoehr(2003)

0.77 1.18

Bijeljic etal. (2004)

0.45 1.19

Freund etal. (2005)

0.303 1.21

This work 0.214 1.2033


Least square fit of the simulated DL

DL

Dm=

1

Fφ+ αPe+ βPe

δ+ γPe

2

10−1

100

101

102

103

10−2 10−1 100 101 102 103

DL

Dm

Peclet Number (Pe)



Kandhai et al.(2002),NMRStohr (2003), PLIF

Bijeljic et al.(2004), Pore-network+RWPTFreund et al.(2005), LBM+RWPT

UDG+RWPTFit

The values of the parameters obtained by fitting are τ = 1Fφ

=0.79, β= 0.214, δ=1.203 and γ=1.241e-5.


Pe vs Transverse dispersion coefficients

10−1

100

101

102

10−3 10−2 10−1 100 101 102 103

DT

Dm

Peclet Number (Pe)

Maier et al. (2000), LBM+RWPTFreund et al (2005), LBM+RWPTBijeljic et al.(2007), Pore-network

UDG+RWPT, DTy

UDG+RWPT, DTz

Simulated transverse dispersion coefficients are compared to data reported in literature

Summary

Summary

New numerical method has been used for pore-scale simulation

Method offers a direct discretization of the PDE’s on pore-scale

Retain benefits of the standard finite element methods, offers higherflexibility in the mesh

Easy incorporation of complex geometries

Studied the dependence of permeability and dispersion coefficients on porestructure

OutlookUDG is Computationally demanding, a parallel implementation is necessary

A quantitative comparison with other well known approaches

Application to more realistic geometry

Extension to multiphase flows at pore-scale

Thank You!

Pore-scale direct simulation of flow and transport in porous media

Documents

porescale properties

pore geometry

complex porescale geometry

porescale modelling

objectives3 porescale

results of pore

continuum scale

scale simulation15