Capillarity in Porous Media

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World Class Training Solutions
Capillarity in Porous Media
Professor Majid Hassanizadeh
30.11.2020
www.petro-teach.com

12/1/2020
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Professor
Majid Hassanizadeh
PetroTeach
Distingushed Instructor
• Professor Hassanizade has more than 40 years experience in
theoretical, experimental, and computational studies of flow and
transport in porous media.
• He has worked at RIVM Institute, Delft University of Technology,
and Utrecht University, where he is currently emeritus professor
of hydrogeology.
• He has more than 300 publications on theories of flow and
transport in porous media, pore network modeling and
experimental studies of two-phase flow including studies of low-
salinity effect.
• He has been (associated) editor of major journals: Advances in
Water Resources, Water Resources Research (2004-2009), and
Transport in Porous Media.
• He is co-founder and Managing Director of International Society
for Porous Media (InterPore).
• He is elected Fellow of American Geophysical Union (2002) and
American Association for Advancement of Science (2007).
• He was selected as 2012 Darcy Lecturer by the US National
Groundwater Association. He received the Royal Medal of
Honor, Knight in the Order of the Netherlands Lion, in 2015, and
Robert Horton Medal of American Geophysical Union in 2019.
3Capillarity in Porous Media
www.petro-teach.com
Capillarity in porous media at different scales
S. Majid Hassanizadeh
Stuttgart Center for Simulation Science,
Stuttgart University, Germany
Multiscale Porous Media Lab; Dept. of Earth Sciences
Utrecht University; The Netherlands

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Capillarity in porous media at different scales
Overview
Processes underlying capillarity phenomenon
Capillarity at pore scale;
static and dynamic conidtions
Capillarity at macroscale; basic concepts
Capillarity at macroscale; advanced theories
Capillarity at macroscale; computational and
experimental studies
Two categories of interfaces
Short course

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Interfacial phenomena; Origin of surface tension
Molecular view
Typical molecular picture:
http://labman.phys.utk.edu/phys221core/modules/m7/surface_tension.html
https://www.google.com/search?q=Surface+tension&sxsrf=ACYBGNSHT0ZSmAhcuZ5bp02NeTNMqJLIyg:1570386
980046&source=lnms&tbm=isch&sa=X&ved=0ahUKEwiN2-
uwo4jlAhWMLVAKHW7jBewQ_AUIEigB&biw=1368&bih=721#imgrc=1mIl8AdqscVEPM:
Wikipedia Picture:
https://en.wikipedia.org/wiki/Surface_tension
Molecular view
There are two types of intermolecular forces:

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Origin of fluid pressure; Molecular view
Molecules interact with each other in two ways:
1. Attraction/repulsion forces
In liquids In vapor
2. Momentum transfer
Ono, S., and Kondo, S., "Molecular Theory of Surface Tension in Liquids," pp. 134-304
in. Handbuch der Physik, Vol. 10, E. Flügge (Ed.), Chap. 2. (Fluid interfaces and capillarity)
Molecular distance
vx
Piston
area A
L
Volume = LA
2 21
v v
N
x x
i
P m
V
 
Pressure is the resultant of collisions between molecules across a surface
Pressure = Force/Area = [Momentum change ]/Area
There are two collisions over a period of
t = 2 L/vx
Momentum change is: Px = 2 m vx
22 v 1 1
v
2 / v
x
i x
x
m
P m
L A V
 
2 2
v v
N
x x
i
PV m N m 
Sum for N molecules:
Origin of fluid pressure; Molecular view
Short course

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Origin of surface tension; Molecular view
liquid
vapor
Ono, S., and Kondo, S., "Molecular Theory of Surface Tension in Liquids," pp. 134-304
in. Handbuch der Physik, Vol. 10, E. Flügge (Ed.), Chap. 2. (Fluid interfaces and capillarity)
In liquids In vapor
Molecular distance
Molecular view
Ono, S., and Kondo, S., "Molecular Theory of Surface Tension in Liquids," pp. 134-304 in. Handbuch der
Physik, Vol. 10, E. Flügge (Ed.), Chap. 2. (FLUID INTERFACES AND CAPILLARIT)

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Surface tension is interfacial energy: energy per unit area
Spontaneous events
Forced events
Surface tension is force per unit length
See video

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Interfacial phenomena; Wettability Concept
Emptying one glass into another glass via a thread
(at 4:20 min)
Hydrophilic and Hydrophobic Solids
For a hydrophilic solid,
interfacial energy of solid-air interface
is larger than
interfacial energy of solid-water interface
The reverse is true for a hydrophobic solid

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Why does water spread over a hydrophilic surface?
Experiments on wetting of paper
IMPORTANT: Temperature is everywhere the same
Aslannejad, et al. “Occurrence of temperature spikes at a wetting front
during spontaneous imbibition”, Scientific Reports, Vol. 7, 2017.

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Experimental setup
Four pieces of papers were mounted on blackened perspex

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Coated paper, 85 micron thick
Water goes up against gravity and viscosity!
Even heat is generated!
Short course
What is the difference between
interfacial tension and surface tension?

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Experimental Surface Tension Values
Liquid in contact with
air
Temperature
(degrees C)
Surface Tension
(mN/m, or dyn/cm)
Benzene 20 28.9
Carbon tetrachloride 20 26.8
Ethanol 20 22.3
Glycerin 20 63.1
Mercury 20 465.0
Olive oil 20 32.0
Soap solution 20 25.0
Water 0 75.6
Water 20 72.8
Water 60 66.2
Water 100 58.9
Liquid Oxygen -193 15.7
Liquid Neon -247 5.15
Liquid Helium -269 0.12
Table of common surface tension values
for solid surfaces
Material Surface Energy (mN/m)
Glass 83
Gypsum 370
Copper 1650
Magnesium oxide 1200
Calcium fluoride 450
Lithium fluoride 340
Calcium carbonate 230
Sodium chloride 300
Sodium chloride 400
Potassium chloride 110
Barium fluoride 280
Silicon 1240

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Relationship between interfacial tension and surface tensions
Antanow’s Law
Short course
Some interfacial tension values
Interfacial tension for water-mercury: 415 mN/m
Interfacial tension for Benzene-water: 35 mN/m
Interfacial tension for water-glass: 10 mN/m

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Wetting of “different” liquids.
A has a large contact angle, and C has a small contact angle.
This is all controlled by surface energy.
IT IS ALLABOUT ENERGY!!
What happens if we put a drop of liquid on a flat solid surface?
Wettability Concept ; Contact Line
γ 𝐿𝐺
𝐶𝑜𝑠𝜃 = γ 𝐺𝑆
− γ 𝐿𝑆
G
L
S
Young’s Equation
Can we predict what happens to a droplet?

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https://web.mit.edu/16.unified/www/FALL/fluids/Lectures/surf_tension.pdf
Orders of Wettability
Interfacial tension for water-air: 72 mN/m
Interfacial tension for Benzene-water: 35 mN/m
Interfacial tension for Benzene-air: 30 mN/m
Short course

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contact angle 90o
  contact angle 90o
 
What happens when a capillary tube in inserted into a liquid?
Microscopic capillarity
Short course
Young-Laplace Equation for a general interface

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Young-Laplace Equation for a general interface
Multiscale Porous Media Laboratory Utrecht
c n w
p p p 
The 2nd equation is a definition, which is valid under
all conditions, even if the interface moves and mean
curvature κM varies with time.
What is the definition of Capillary Pressure at
microscale?
The first equation comes from a balance of forces.
It is an approximation that is not always valid!
κM is the interface mean curvature,
γnw is the interfacial tension
𝑝 𝑐
= 2𝛾 𝑛𝑤
𝜅 𝑀

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is interfacial mass density
is interfacial velocity vector
is surface viscous stress tensor
, are pressures of nonwetting and wetting phases
, is fluid viscous stress tens
wn
wn
wn
n w
n n
p p

v
τ
τ τ or
is interfacial tensionwn

The balance of forces at a fluid-fluid interface*
Capillary pressure at microscale
     
wn
wn wn wn s n n w w wnD
p p
Dt

              
v
τ I N τ I τ I g
is surface divergence operator
is surface unit tensor (not a constant in curvilinear coordiantes)
is unit vector normal to the interface
s


I
N
* Thermodynamics of an interface; GP Moeckel; Archive for Rational Mechanics and Analysis, 1975
Under dynamic conditions (neglect surface viscosity):
Non-equilibrium capillarity pressure at microscale
2 ( ) ( )wn n w n w
M p p       N τ τ N
is interfacial tension
is fluid pressures
is fluid viscous stress tensor
is vector normal to the interface
wn
p


τ
N

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Consider two-phase flow in a single tube (Sheng and Zhou, 1992):
Microscale Dynamic Effects
Nonwetting phase
Wetting phase
2n w wn
Mp p  
Local circulations at an air-water interface
Particles of
6 microns in size.
Conc. optimized
Experiments by Aslandnejad, Utrecht, 2016

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Silicone oil flow
(0.07 ml/h)
Silicone oil flow
(0.50 ml/h)
Local circulations at an fluid-fluid interface
Experiments by de Winter, Utrecht, 2016
Microscale Dynamic Effects
2
2
A
n w nw q
p p B
r

 
 
    
 
Nonwetting phaseWetting phase
When there is flow (Sheng and Zhou, 1992):

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Macroscale Capillarity
under equilibrium (quasi-static) conditions
𝑃 𝑛 − 𝑃 𝑤 = 𝑃 𝑐 = 𝑓(𝑆 𝑤)
Macroscale capillary pressure:
Can we derive this relationship?
irreducible saturation Swir
Consider a capillary tube with radius r, containing a meniscus with
contact angle θ;
Young-Laplace eq. gives:
We can also write:
So:
Averaging this equation will not result in the macroscale capillary
pressure being (only) a function of saturation.
2
cos
wn
c
p
r


 
2
4 / cos
2
wn
A r

 
 
  
 
 4 / 2 /c wn wn
p A   

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𝑝 𝑛 − 𝑝 𝑤 = 𝑝 𝑐 = 2𝛾 𝑛𝑤 𝜅 𝑀
At an interface, we have:
2 2wn wn
M M n w n wp p p p       
Under no-flow conditions, fluid pressures are everywhere
constant within each phase (neglecting gravity).
Microscale capillary pressure is everywhere constant.
There is no distinction between micro-and macro-scale:
( )wf S?

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x
P
2 cosO w
c meniscus meniscus
t
p p p
R
 
  
pc
oil
L
water
Rt = Radius
Results from a bundle-of-capillary-tubes model
Dahle, Celia, Hassanizadeh, TiPM, 2005
under non-equilibrium (flow) conditions
Consider flow in a simple single-tube model:
 o w
P P
x
P
pc
oil
L
water Rt = Radius
Tube-scale Capillarity
Results from a bundle-of-capillary-tubes model
Dahle, Celia, Hassanizadeh, TiPM, 2005

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l
L
2
8 ( )
o w cdl r
P P p
dt l L
      
 1 o w
w
c
S
P P P
t 
     
  
Upscaling:
Capillary pressure at Macroscale
From Hagen-Poiseulle Formula, we have:
Capillary pressure-saturation curves
irreducible saturation Swir

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Capillary pressure-saturation hysteresis
Scanning curves
Morrow, 1968
Capillary pressure-saturation relationship is not really a curve,
or a set of curves; it is a collection of equilibrium data points.
Capillarypressurehead,Pc/g
Water content
Water content
Imbibition Drainage

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Capillary pressure-saturation relationship is not really a curve; all points in the red
domain are possible equilibrium points. This domain is the projection of a three-
dimensional surface on Pc-S plane.
Imbibition Drainage
Water content Water content

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There may be a unique relationship among interfacial area, capillary
pressure, and saturation in the form of a three-dimensional surface
(Held and Celia, 2001)
Capillary pressure-saturation points from micromodel
experiments
Karadimitriou et al., 2012

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Specificinterfacialarea
Karadimitriou
et al., 2012
Capillary pressure-saturation-interfacial area Surface
Fitted to drainage points – Micromodel experiments
 , , 0c nw w
af P S 
Capillary pressure-saturation-interfacial area Surface
Fitted to imbibition points – Micromodel experiments
Specificinterfacialarea
Karadimitriou
et al., 2012
 , , 0c nw w
af P S 

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Capillary pressure-saturation-interfacial area
Surface
The average difference between the surface for
drainage and the surface with all the data points is
9.7%.
The average difference between the surface for
imbibition and the surface with all the data points
is -5.77%.
32
1 (1 ) *wn
ca S S P
 
Karadimitriou et al., 2012
Capillary pressure-saturation-interfacial area
Surface
Can we provide a more fundamental derivation of
Pc-awn-Sw relationship and capillarity?

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General (non-equilibrium) capillarity theory
(Hassanizadeh and Gray, Adv. in Water Resources, 1990)
Equilibrium conditions:
 ,n w c w
P P P S a
 
Linear non-equilibrium capillarity:
 , ( , )
w
n w c w w S
P P P S a S a
t
 


  

where τ is a damping coefficient.
Infiltration experiments;
Rezanejad, 2002
Infiltration fingers durinng pentration of water into
(almost) dry soil

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Multiscale Porous Media Laboratory UtrechtExperiments by Rezanejad, 2002
Vertical infiltration of water in (almost) dry soil
S
S
S
Development of vertical wetting fingers in dry soil
Stability analysis by Dautov et al. (2002) has proven that:
Sharp Front Richards model is unconditionally unstable.
It produces a monotonically increasing saturation profile
toward the front and an abrupt drop to the initial
saturation.
Modified Richards equation (with dynamic effect) is
conditionally unstable.
It is able to produce gravity wetting fingers.
Richards equation is unconditionally stable.
It does not produce any fingers, but a monotoniclly
decreasing saturation profile toward the wetting front.

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Dautov et al. (2002)
Development of vertical wetting fingers in dry soil;
Simulations based on new capillarity theory
Thank You
Any Questions ?
64

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Capillarity in Porous Media (online)
8 – 10 June 2021
4 – 6 October 2021
Register@petro-teach.com
Course Overview:
First, underlying mechanisms of capillarity at various scales are explained. Concepts of immiscibility, fluid-fluid interface, surface tension, surface
energy, hydrophobicity, wettability, and pressure are introduced based on molecular phenomena. Next, capillary pressure is defined at the pore
scale. It is shown that Young-Laplace equation holds under static conditions. Extension to flow conditions is provided. Then, capillary pressure at
the core scale (Darcy scale) is introduced. Methods of measurements of capillary pressure-saturation curves are explained and causes of capillary
hysteresis are discussed.An advanced theory of capillarity is introduced and its effects on the modelling of moisture transport in soils and two-
phase flow processes are discussed. Computational and experimental studies are presented that investigate new generalized equations.
Learning Objectives:
• Origin of properties such as wettability and capillarity
• Basic formulas for capillarity at the pore scale
• Capillarity at the core scale
• Laboratory measurement of capillary pressure-saturation relationship
• Capillarity at the reservoir scale and the link to laboratory measurements
• Advanced theories of capillarity and their consequences for modelling two-phase flow through porous media
20% Discounts available for Ph.D. students, Group (≥ 3 person) and early bird registrants (4 weeks before).
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