Simulating Fluid-Fluid Interfacial Area

Simulating Fluid-Fluid Interfacial Area revealed by a pore-network model V. Joekar-Niasar S. M. Hassanizadeh Utrecht University, The Netherlands July 22, 2009

Outline 1 What s a Porous medium 2 Intro to PNMs 3 PNMing technique 4 Two-phase PNMs 5 Applications Case1: idealized porous medium Case2: micromodel Case3: glass-bead column

A porous medium at pore scale: How does it look like? Glass beads vs. volcanic Tuff courtesy of D. Wildenschild

A porous medium at pore scale: How does it look like? Sandstone courtesy of O. Vizika

A porous medium at pore scale: How does it look like? Fibers www.bazylak.mie.utoronto.ca/research/

Characterization of porous media structure Microscopic: Topology, geometry Macroscopic: Porosity, descriptive topology functions, geometry statistics, spatial distribution

Quasti-static vs. dynamic PNMs Quasi-static Computationally, very cheap. No pressure field is solved. Pore-scale geometry and morphology are only important. Used extensively, for two-phase and three-phase flow; P c S w, k r S w, S w a nw, reactive transport, etc. Can be used as a predictive tool. Dynamic Computationally, expensive. Pressure field is solved. Network and fluids properties are important. Notbeenusedasextensivelyas quasi-static ones; P c S w, k r S w, ganglia movement, dynamic pressure field Alongwaytogo!

Investigation tool Pore Network Models? Yes,because: Physical-based models using pore-scale information. Application for static and dynamic processes. Computationally, not so expensive (especially for quasi-static ones). Capability to provide up-scaled information. But: Translation of topology and geometry is inevitable, and not always straight forward! No detail information within a pore (e.g. pressure field). Local laws/rules are inevitable, and devil! be careful!!!!

Definitions Geometry Pore body: large pores among the grains (shape and size) Pore throat: long narrow pores connecting the large pores (shape and size) Topology Coordination number: number of pore bodies connected to a given pore body Fixed coordination number: isotropic, variable coordination number: anisotropic

Pore-network modeling steps Step 1. Topological and geometrical data acquirement Step 2. Pore-network structure Step 3. Defining governing equations and local laws Step 4. Numerical experiments

Step 1. Topological and geometrical data acquirement Step 1.a Topological data acquirement Detailed network acquired by analyzing images of a real porous medium, e.g. 3DMA-Rock software. Hypothetical to idealized pore-networks. They can be isotropic (fixed coordination number) or anisotropic (variable coordination number). Center of pore bodies can be located on lattice points (structured) or random coordinates (unstructured). Step 1.b Geometrical data acquirement Detailed network acquired by analyzing images of a real porous medium, e.g. 3DMA-Rock software. Statistically equivalent or idealized porous medium. Generating based on statistical parameters with/without spatial correlations.

Step 2. Pore-network structure Numbering pore bodies, sequentially. Numbering pore throats, sequentially. Defining connectivity of pore bodies to pore throats (Pb2Pth). Defining connectivity of pore throats to pore bodies (Pth2Pb).

Step 2. Pore-network structure Void-Space Network (Pore-Network)

Step 2. Pore-network structure Pore Bodies Numbering

Step 2. Pore-network structure Pore Throats Numbering

Step 2. Pore-network structure Connection Matrices

Step 3. Governing equations and local laws One-phase or multiphase flow? Drainage or Imbibition process,...? Dynamic process or static process? Probable physical laws: Young-Laplace Equation, Hagen-Poiseuille Equation, Stokes Equation, Conservation, etc. Probable local assumptions: phase trapping assumptions, hydraulic conductivites of pores, etc.

Step 4. Numerical experiments Quasi-Static or dynamic Quasi-static simulations: capillary forces are dominant, no information about time scale of phenomena. Dynamic simulations: solving pressure field, change of saturation with time.

Drainage vs. imbibition Let s watch some movies: courtesy of Fabiano G. Wolf (http://www.lmpt.ufsc.br/ fgwolf/) Drainage First, larger pore will be filled by the nonwetting phase (blue). Imbibition First, smaller pore will be filled by the wetting phase (red). Simulations done by Lattice-Gas Automata Models

Example Quasi-static simulation, drainage process, circular cross section Entry P c, Pi c = 2σ r ij cos θ (for circular cross section). Lower entry pressure: invaded earlier during drainage!

Quasi-static PNMs Network: Initial Condition

Quasi-static PNMs

Quasi-static example

Quasi-static example P c =2.0, with trapping

Quasi-static example movie

Computational considerations Adaptive pressure update. Gains: a) Calculating the exact P c -S w curve. figure b) Saving computational time especially at the initial and residual saturation of P c -S w curve. Using flags pore those pores which are in contact with the invading phase, but not filled yet. For strongly anisotropic: save Pb2Pth matrix differently to save memory. The best known suggest approach is compact row storage (CRS). Trapping a the most time-consuming. Robust search algorithm to save time such as depth-first search (Cormen, Leiserson, Rivest, 2000).

Computational considerations Compact Row Storage (CRS) for an anisotropic medium or dual porosity medium! Example: dual porosity medium; macro pores connected to micro pores. courtesy of M. Prodanovic

Computational considerations Compact Row Storage (CRS) Yellow colors are indices!

Computational considerations Compact Row Storage (CRS), if pore throats are included in calculations!

Computational considerations Compact Row Storage (CRS), if pore throats are not included in calculations!

Geometrical parameters/issues Geometry Cross section geometry Aspect ratio Shape factor Phenomena Piston-like movement Snap-off Capillary diffusion Cooperative filling,...

Geometrical evolution??? To have corners, or not to have corners: that is the question! To mimic the grain geometry,sometimes this can be also the question!! Wetting phase in the corners will reduce wetting phase trapping during drainage significantly, however may contribute to residual saturation. Wetting phase in the corners can increase snap-off during imbibition. Snap-off happens if wetting film/ corner flow cuts connectivity of the nonwetting fluid. Including corner flow in dynamic models will increase numerical complexity enormously.

Geometrical parameters: cross section geometry (Joekar-Niasar et al,2009) Entry capillary pressure can be calculated for different cross sections using MS-P (Mayer, Stowe and Princen, J. Colloid Interface Sci, 1968, 1969; Ma et al., Colloids Surf, 1996). More complex geometry, more complicated system of equations.

Geometrical parameters: shape factor κ = A/P 2 κ: shape factor, A:cross section area, P:cross section perimeter Pore-network models for glass beads (Joekar-Niasar et al,2009)

Geometrical parameters: aspect ratio Radius of pore body size to pore throat size Significant effect during imbibition: larger aspect ratio, more snap-off! More snap-off, more nonwetting phase trapping (Lenormand and Zarcone, 1983, 1984; Wardlaw and Yu, 1988; Ioannidis et al. 1991) movie.

Geometrical parameters: aspect ratio Overlapping of pore bodies and pore throats size distribution.

A porous medium at pore scale: How does it look like? Glass beads vs. volcanic Tuff courtesy of D. Wildenschild

A porous medium at pore scale: How does it look like? Sandstone courtesy of O. Vizika

A porous medium at pore scale: How does it look like? Fibers www.bazylak.mie.utoronto.ca/research/

What we want: We are interested in investigating extended theories of two-phase flow.

Objectives 1. Insights into pore-scale dynamics of two-phase flow 2. Studying validity of extended two-phase flow equations

Extended theories of two-phase flow n S α + q α = 0 t q α = K α ( p α ρ α g σ αa a wn σ αs S α) a wn t + (a wn w wn )=r wn (a wn, S w ) w wn = K wn (a wn γ wn ) p n p w = p c τ S w t, pc = f (S w, a wn )

Extended theories of two-phase flow in porous media Capillary pressure p c = f (S w, a wn )=(p n p w ) equil Difference in fluid s pressure are equal to static capillary pressure but only at equilibrium. Under non-equilibrium conditions, assuming linearity: p n p w = p c τ S w t, pc = f (S w, a wn )

Case1: idealized porous medium Joekar-Niasar, V., Hassanizadeh, S.M., Leijnse A., 2008, Insights into the relationships among capillary pressure, saturation, interfacial area and relative permeability using pore-network modeling, Transport in Porous Media, 74:201219, doi 10.1007/s11242-007-9191-7

Idealized pore-network Specifications Topology: structured, isotropic, lattice network with coordination number of 6 Geometry: sphere pore bodies, circular cross-section pore throats Governing equation: P c = 2σ r cos θ TiPM Joekar-Niasar et al. 2008,

Filling process Drainage: Starting from the largest pore throats. If a pore throat is invaded, the connected pore body will be fully filled. Imbibition: Starting from the smallest pore throats. If a pore throat is invaded, the connected pore body will be partially filled.

P c -S w and a nw -S w curves: imbibition

P c -S w and a nw -S w curves: drainage

What s a Porous medium Intro to PNMs PNMing technique Two-phase PNMs Applications P c -S w -anw surfaces Drainage vs. imbibition a y b y y

Relative permeabilities Flux passing through the pore throat ij based on Hagen-Poiseuille s law q ij = πr ij 4 8μl ij (P i P j )

Relative permeabilities Flux passing through the pore throat ij based on Hagen-Poiseuille s law q ij = πr ij 4 8μl ij (P i P j ) Based on volumetric conservation law for all pore bodies: Ni j=1 q ij =0,N i number of pore throats connected to pore body i. Linear system of equations will be resulted AP=B. A is a symmetric banded sparse matrix made of conductivities of pore throats.

k r -S w -a nw surfaces c Drainage and imbibition

Conclusions We can have unique surfaces for P c -S w -a nw as well as k r -S w -a nw for drainage and imbibition using our idealized pore-network model. Increasing aspect ratio decreases uniqueness of the surface. Further developments should be done; such as including angular cross sections to allow different mechanisms during imbibition and implementing for an unstructured anisotropic cross sections. Continuing for a real porous medium!

Case2: micromodel Joekar-Niasar, V., Hassanizadeh, S. M., Pyrak-Nolte, L. J., Berentsen, C., Simulating drainage and imbibition experiments in a high-porosity micro-model using an unstructured pore-network model, 2009, Water Resources Research, 45: W02430, doi:10.1029/2007wr006641.

Specifications Topology: unstructured, anisotropic. Geometry: square cross-section Joekar-Niasar et al.2008,wrr

Challenge How can we use the concept of pore-network modelling for a high-porosity medium? Pore-body and pore throat concept does not hold!!!

Challenge How can we use the concept of pore-network modelling for a high-porosity medium? Pore-body and pore throat concept does not hold!!! Skeletonization using medial axis: defining pixels and bottle necks

Medial axis transform

Governing equation P e = σ nw (a + b)cosθ + (a + b) 2 cos 2 θ +4ab( π 4 θ 2cos( π 4 + θ)cosθ) 4( π 4 θ 2cos( π 4 + θ)cosθ)

Another challenge! How can we simulate imbibition process? High porosity and small aspect ration causes significant amount of cooperative filling. No snap-off occurs during imbibition.

Another challenge! How can we simulate imbibition process? High porosity and small aspect ration causes significant amount of cooperative filling. No snap-off occurs during imbibition. Defining a rule for no-trapping during imbibition

Cooperative filling

Simulations vs. experiments

Conclusions Given geometry, pore-network model can be used to simulate not only P c -S w but also a nw -S w curves. P c -S w -a w surfaces resulted from simulation and experiments showed good agreement. This predictive tool can be useful for assessing the micromodel before manufacturing. It is very hard to simulate imbibition process in high porosity media.

Case3: glass-bead column Joekar-Niasar, V., Prodanovic, M., Wildenschild, D., Hassanizadeh, M., Translation of a Granular Porous Medium into a Pore-Network Model: Application for Capillary Pressure-Saturation and Interfacial Area-Saturation Relationships

Experiment Air-water system in glass beads. Image obtained through CT scanner.

Specifications Topology: unstructured, anisotropic Geometry: mixed hyperbolic polygonal cross sections for pore throats and prolate spheroid for pore bodies. Image analysis and data acquisition using 3DMA-Rock: skeleton, shape factor, pore unit volume, mean inscribed circle at pore throats andporebodiesaswellasmeanradiusofporebodies.

Pore throats geometry: irregular hyperbolic For a desired value of cross section area (or R) andκ, onecan calculate radii R 1, R 2, R 3,andanglesα, β, andγ.

Pore throats geometry: regular hyperbolic For a desired value of cross section area (or R) andκ, onecan calculate radius R and corner angle ϕ.

Entry pressure: irregular hyperbolic

Entry pressure: regular hyperbolic

Pore throats determination Determination of pore throats cross section based on shape factor distribution.

Numerical simulation Determination of cross sections types. Determining pore body geometries based on two min. and mean inscribed radii. Solving a nonlinear system to determine geometry of a cross section based on area and shape factor. Solving a nonlinear system to determine entry capillary pressure. Determining fluid occupancy during drainage and imbibition based on imposed capillary pressure.

Results:grain size distribution

Results: P c -S w curves

Results: a nw -S w curves

Conclusions Given geometry, pore-network model can be used to simulate not only the P c -S w but also a nw -S w curves. At least for granular porous medium, including shape factor to generate hyperbolic polygonal cross sections works well. There is enough consistency between grain size distribution and cross sections of pore throats.