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- 1. Pore-Scale Direct Numerical Simulation of Flow and Transport 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
- 2. 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
- 3. Introduction - Relevance Study of ﬂow and transport through porous media has wide practical applications Subsurface/Environmental applications (contaminant transport, nuclear waste disposal, 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 Experimental methods Numerical simulations are usually done at the continuum scale which require the knowledge of certain parameters (permeability, dispersion coeﬃcients, etc.)
- 4. 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 tool for predicting and quantifying solute transport ∂C 2 + · (vC) − D. C=0 ∂t The basic assumption of the ADE is that dispersion follows Fickian behavior J = −D C Numerous experiments have shown that solute spreading does not follow a Gaussian distribution
- 5. Introduction - Relevance Pore-Scale Simulations An alternative and more fundamental approach Provides link between pore-scale properties of the porous medium and large scale behaviour Governing ﬂow and transport equations are known at pore-scale Macroscopic parameters can be obtained using the results of pore-scale simulations Pore-Scale Methods Challenges: require detailed structure of the medium, method should be able to handle geometry Pore-scale numerical methods: Pore-Network, LBM, Finite Element, etc. New and eﬃcient pore-scale simulation methods have its relevance in this context
- 6. Motivation and Research Objectives Motivation Lack of fundamental understanding of how pore structure controls ﬂow and transport behaviour at larger scales! This work is motivated by the need for better understanding of the physical processes that take place at the pore-scale and to improve the reliability of numerical 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 transport processes through porous media at the pore-scale In particular, to use a new numerical discretisation approach (called Unﬁtted Discontinuous Galerkin UDG) for the solution of partial diﬀerential equations on the pore-scale geometry And to predict the macroscopic parameters of porous medium based on pore-scale simulations
- 7. Pore-Scale Modelling of Flow and Transport Present approach involve following steps: 1 Compute the pore-scale velocity ﬁeld (by Solving Stokes equation) 2 −µ u+ p = f; ·u=0 By using a new method called Unﬁtted Discontinuous Galerkin which requires Implementation of DG ﬁnite element discretisation of Stokes equation in the framework of unﬁtted discontinuous Galerkin method 2 Obtain the ﬂow and transport parameters based on the computed pore-scale velocity ﬁeld Permeability is computed by applying the Darcy’s law Dispersion coeﬃcients are determined by solving the Advection-Diﬀusion equation posed at the pore-scale by RWPT method ∂C 2 = −u · C +D C ∂t Much of the challenge in solving Stokes problem (for velocity and pressure) is how to account for the complex pore-scale geometry!
- 8. Unﬁtted Discontinuous Galerkin (UDG) Method Method for the solution of the Stokes equation is based on a new numerical approach which has been speciﬁcally developed for applications in complex domains UDG introduced by Engwer and Bastian (2005,2008) Use only a structured grid and based on DG ﬁnite element method with trial and test functions deﬁned on the structured grid Mesh Construction Given the pore geometry, a fundamental structured grid is chosen According to desired accuracy and computational resources Generally a course mesh can be used
- 9. Unﬁtted Discontinuous Galerkin Method Mesh Construction Grid intersected by the domain generate arbitrary shaped elements Support of the trial and test functions are restricted according to the shape of 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
- 10. Unﬁtted Discontinuous Galerkin Method Evaluation of surface and volume integrals Local Triangulation Local triangulation for assembling 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 Based on the marching cube Use of standard quadrature rules for algorithms the integration over sub-elements
- 11. Unﬁtted Discontinuous Galerkin Method Appealing things Underlying DGFE discretisation of the PDE model It has all beneﬁts of standard ﬁnite element methods Advantages of the DG schemes are naturally incorporated Allow arbitrary shaped elements Easy incorporation of the complex geometries via implicit function or level set methods Possible to choose the computational grid independent of the pore geometry Number of unknowns independent of the complex geometry
- 12. DG Discretization of the Stokes Equation Find (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 X Z X Z X Z A(uh , vh ) = uh : vh dx − uh · ne [ vh ]ds + vh · ne [ uh ]ds E∈Th E E E E∈E I E∈E I h h X Z X Z − ( uh · nb )vh ds + ( vh · nb )uh ds E E E∈E B E∈E B h h σ σ X Z X Z J0 (uh , vh ) = [ uh ] · [ vh ]ds + uh · vds |e| E |e| E E∈E I E∈E B h h XZ X Z X Z B(vh , ph ) = − ph · vh dx + ph [ vh · ne ]ds + ph vh · nb ds E E E E E∈E I E∈E B h h X Z X Z σ X Z F (vh ) = f · vh dx + µ ( vh · nb )gds + µ g · vh ds E∈Th E E∈ED E E∈ED |e| E X Z − p0 vh · nb ds E∈EDP E X Z G(q) = − qh g · nb ds E∈ED E
- 13. RW Particle Tracking Methods Lagrangian based numerical approach for the solution of transport problem Particle distribution The trajectory of a tracer particle in an external pore velocity ﬁeld u is given as Xi (t + ∆t) = Xi (t) + S(t) + Z(t) |{z} | {z } Adv. displacement Diﬀ. displacement | {z } | {z } √ S(t)=u(t)∆t Z(t)= 2Dm ∆tξ Statistical moments The centre of mass of the solute distribution is approximated by the ﬁrst moment as Np 1 X x(t) = xi (t) Np i=1 The spread of mass (spatial variance) around x(t) is approximated by the second moment as Np 2 1 X 2 2 σ = var(x(t)) = xi (t) − x(t) . Np i=1 The dispersion coeﬃcient is determined from the spatial variance as 1 dσ 2 (t) Deﬀ (t) = 2 dt
- 14. 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 solution Mass balance checks Analytical tests, Poiseuille (channel ﬂow), driven cavity and ﬂow around cylinder (for ﬂow model) Computation of permeability for ordered sphere packing Taylor-Aris dispersion (for transport model)
- 15. Validation of Flow Model 1 1 Simulation Analytical 0.8 0.8 0.6 0.6 Y Y 0.4 0.4 0.2 0.2 0 Simulation 0 0.2 0.4 0.6 0.8 1 Donea & Huerta (2003) 0 Velocity −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Horizontal Velocity Poiseuille Flow in a channel Flow in a driven cavity
- 16. Validation of The Flow Model Computation of permeability for ordered packing of spheres The mean pore velocity is calculated as ¯ u= udx · |Ωp |−1 Ωp |Ωp | The porosity is given as φ = |Ω| , where |Ωp | is volume of the pore-space The xx component of the permeability tensor ¯ uφ κxx = −µ , p where |Ω| denotes the size/volume of the domain.
- 17. Validation of Flow Model 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-05 SC 0.476 2.52e-03 2.51e-03
- 18. Validation of Flow Model Permeability of SC, for various porosity’s: Scaled vol- Porosity κeﬀ (Sangani and κeﬀ (Com- Relative Error ume fraction Acrivos 1982) puted) (%) (ψ) 0.1 0.99951 0.91107 1.0169 11.61 0.2 0.99587 0.38219 0.40158 5.07 0.4 0.96661 0.12327 0.12578 2.03 0.6 0.88709 0.044501 0.04488 0.85 0.8 0.73 0.013197 0.01320 0.08 1.0 0.478 0.0025203 0.002516 0.17 1e+01 Sangani&Acrivos (1982) Computed 1e+00 Permeability, κxx 1e-01 1e-02 1e-03 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
- 19. Validation of Flow Model Grid Convergence: Permeability computed for FCC converging to the anlytical value on a relatively coarser grid 1e-02 FCC (φ = 0.26) 1e-03 Permeability, κxx 1e-04 1e-05 1 1/2 1/4 1/8 1/16 1/32 h
- 20. Permeability for an artiﬁcial porous medium (sphere pack) Grid Convergence 1e-01 φ = 0.768 1e-02 Permeability, κxx 1e-03 1e-04 1 1/2 1/4 1/8 1/16 1/32 h Permeability of the artiﬁcial porous medium computed on various grid levels Artiﬁcial porous medium made of randomly packed spheres
- 21. Porosity Vs Permeability Varied 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-01 h = 1/16 h = 1/32 1e-02 Permeability, κxx 1e-03 1e-04 1e-05 0.4 0.5 0.6 0.7 0.8 0.9 1,0 Porosity, φ
- 22. Validation of Transport Model 30 Dm = 0.35 t=0.0 um = 0.8326 y Pe = 71.365 Taylor-Aris Dispersion 0 −10 0 10 20 x 30 40 50 30 t=43.5 C 1 x − um t y = erfc C0 2 2(Deﬀ · t)1/2 0 0 20 40 x 60 80 100 30 1 t=217.0 15000 y 10000 0.9 5000 1000 0 0 50 100 150 200 250 300 350 400 Analytical 0.8 x 30 0.7 t=433.7 y 0.6 0 0 100 200 300 400 500 600 700 800 C0 0.5 C x 30 0.4 t=867.55 y 0.3 0 0 200 400 600 800 1000 1200 1400 0.2 x 30 0.1 t=1301.3 y 0 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 0 0 500 1000 1500 2000 t∗ = t u/L x Cumulative breakthrough curve (for various number of particles) for the 30 t=3036.4 Taylor-Aris dispersion compared to the analytical solution. y Gaussian 0 0 500 1000 1500 2000 2500 3000 3500 4000 x
- 23. Validation of Transport Model Variation of the longitudinal dispersion coeﬃcient Deﬀ with the P´clet e Number for the Taylor-Aris dispersion Deﬀ Pe2 Analytical Solution: =1+ Dm 210 102 Analytical Computed Fit 101 eﬀ Dm D 100 10−1 100 101 102 Peclet Number (Pe)
- 24. Pore-scale simulation of Transport in 2D
- 25. Pore-scale simulation of Transport in 2D This work (RWPT) Based on Eulerian method (DGFE) by Fahlke (2008) Sreejith Kuttanikkad (IWR, University of Heidelberg) Pore-Scale Simulation 15.10.2009 25 / 36
- 26. Pore-scale simulation of Transport in 2D This work (RWPT) Based on Eulerian method (DGFE) by Fahlke (2008) Sreejith Kuttanikkad (IWR, University of Heidelberg) Pore-Scale Simulation 15.10.2009 25 / 36
- 27. Pore-scale simulation of Transport in 2D This work (RWPT) Based on Eulerian method (DGFE) by Fahlke (2008) Sreejith Kuttanikkad (IWR, University of Heidelberg) Pore-Scale Simulation 15.10.2009 25 / 36
- 28. Pore-scale simulation of Transport in 2D This work (RWPT) Based on Eulerian method (DGFE) by Fahlke (2008) Sreejith Kuttanikkad (IWR, University of Heidelberg) Pore-Scale Simulation 15.10.2009 25 / 36
- 29. Pore-scale simulation of Transport in 2D Concentration breakthrough curve 0.25 0.25 10,000 Present (RWPT) 25,000 Fahlke, 2008 (DGFEM) 50,000 100,000 0.2 0.2 Normalised Concentration 0.15 Concentration 0.15 0.1 0.1 0.05 0.05 0 0 −0.05 0 5 10 15 20 25 0 5 10 15 20 25 Time Time Breakthrough curve plotted for diﬀerent number of Breakthrough curve compared with the result of an solute particles Eulerian scheme
- 30. Pore-scale simulation of Transport in 3D Artiﬁcial porous medium and the computational grid
- 31. Pore-scale simulation of Transport in 3D Computed pore-scale velocity ﬁeld
- 32. Pore-scale simulation of Transport in 3D Calculated concentration proﬁles along the porous medium 0.025 0.02 Normalised Concentration 0.015 0.01 0.005 0 0e+00 2e+04 4e+04 6e+04 8e+04 1e+05 Time
- 33. Pore-scale simulation of Transport in 3D Dependence of dispersion coeﬃcients on Peclet number A standard way of describing longitudinal dispersion coeﬃcient as a function of Pe in the laminar ﬂow condition is by using DL 1 = + αPe + βPeδ + γPe2 Dm Fφ mechanical dispersion boundary layer diﬀusion hold-up dispersion molecular diﬀusion
- 34. Pore-scale simulation of Transport in 3D Dependence of dispersion coeﬃcient (DL ) on Peclet number 104 103 102 Dm DL 101 Pfannkuch, 1963 Perkins and Johnston, 1963 Seymour and Callaghan, 1997 Maier et al. (2000),LBM+RWPT 100 Kandhai et al.(2002),NMR Khrapitchev and Callaghan, 2003 St¨hr (2003), PLIF o Bijeljic et al.(2004), Pore-network+RWPT Freund et al.(2005), LBM+RWPT UDG+RWPT 10−1 10−2 10−1 100 101 102 103 104 P´clet Number (Pe) e
- 35. Pore-scale simulation of Transport in 3D Dependence of longitudinal dispersion coeﬃcient on Peclet number in the power law regime 103 Reference β δ Pfannkuch - 1.2 102 (1963) Gist et al. 0.46 - 3.9 0.93 - 1.2 (1990) Dullien - 1.2 (1992) Dm 101 DL Coelho et 0.26 1.29 al. (1997) Pfannkuch, 1963 Manz et al. - 1.12 Perkins and Johnston, 1963 Seymour and Callaghan, 1997 (1999) 100 Maier et al. (2000),LBM+RWPT Kandhai et al.(2002),NMR Stoehr 0.77 1.18 St¨hr (2003), PLIF o (2003) Bijeljic et al.(2004), Pore-network+RWPT Freund et al.(2005), LBM+RWPT Bijeljic et 0.45 1.19 UDG+RWPT Fit al. (2004) 10−1 Freund et 0.303 1.21 101 102 P´clet Number (Pe) e al. (2005) Simulated longitudinal dispersion coeﬃcients in a random sphere packing compared to data This work 0.214 1.2033 reported in literature in the power law regime (3 < P e < 300). The line corresponds DL to the ﬁt of the data to = βPeδ with β=0.214003 and δ=1.20331. Dm
- 36. Pore-scale simulation of Transport in 3D Least square ﬁt of the simulated DL DL 1 δ 2 = + αPe + βPe + γPe Dm Fφ 103 102 Dm 101 DL Pfannkuch, 1963 Perkins and Johnston, 1963 Seymour and Callaghan, 1997 100 Maier et al. (2000),LBM+RWPT Kandhai et al.(2002),NMR St¨hr (2003), PLIF o Bijeljic et al.(2004), Pore-network+RWPT Freund et al.(2005), LBM+RWPT UDG+RWPT Fit 10−1 10−2 10−1 100 101 102 103 P´clet Number (Pe) e The values of the parameters obtained by ﬁtting are τ = 1 =0.79, β= 0.214, δ=1.203 and γ=1.241e-5. Fφ
- 37. Pore-scale simulation of Transport in 3D Pe vs Transverse dispersion coeﬃcients 102 101 Dm DT 100 Maier et al. (2000), LBM+RWPT Freund et al (2005), LBM+RWPT Bijeljic et al.(2007), Pore-network UDG+RWPT, DT y UDG+RWPT, DT z 10−1 10−3 10−2 10−1 100 101 102 103 Peclet Number (Pe) Simulated transverse dispersion coeﬃcients are compared to data reported in literature
- 38. 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 higher ﬂexibility in the mesh Easy incorporation of complex geometries Studied the dependence of permeability and dispersion coeﬃcients on pore structure Outlook UDG 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
- 39. Thank You!