Table of Contents

Cover

Introduction

I.1. Introduction
I.2. Organization of the text
I.3. Objectives, contents, and readership
I.4. Acknowledgments

1 Fluids, Porous Media and REV: Basic Concepts

1.1. Geologic porous media: basic concepts
1.2. Porous media: basic concepts, porosity and specific area
1.3. Single-phase flow and Darcy’s law: basic concepts
1.4. The Darcy–Buckingham law and the Richards equation: basic concepts of unsaturated flow
1.5. Capillarity and two-phase flow systems at different scales: basic concepts
1.6. A basic approach to pore scale two-phase flow
1.7. A basic approach for continuum scale description of two-phase flow in porous media: the Darcy–Muskat model
1.8. Other issues: capillarity vs. gravity and viscosity, heterogeneity and upscaling

2 Two-Phase Physics: Surface Tension, Interfaces, Capillary Liquid/Vapor Equilibria

2.1. Summary and objectives
2.2. Physics of capillarity and surface tension at equilibrium
2.3. Dimensionless groups (characteristic forces, length scales, timescales)
2.4. Thermodynamics, Gibbs energy, pressure, suction
2.5. Kelvin’s liquid/vapor relation (suction vs. air humidity)

3 Capillary Equilibria in Pores, Tubes and Joints

3.1. Introduction and summary
3.2. Capillary equilibrium in a single tube or planar joint of constant diameter or aperture
3.3. Capillary equilibria in variable tubes and joints (a(x))
3.4. Capillary equilibrium in a random set of tubes: calculation of water retention curve θ(ψ)
3.5. Capillary equilibrium of soap films: minimal area surfaces and Euler-Lagrange equations
3.6. Case study of soap film equilibrium between two circular rings: minimal area surface (catenoid)
3.7. Additional topic: the equilibrium depth of a bubble

4 Pore-Scale Capillary Flows (Tubes, Joints)

4.1. Introduction and summary: pore-scale flow in capillary tubes and planar joints (steady and transient)
4.2. Single-phase steady flow in tubes: Poiseuille, Darcy, Kozeny-Carman permeability
4.3. Unsaturated and two-phase steady flow in sets of planar joints: equivalent mesoscale quantities (porosity ϕ, permeability k, capillary length λ_cap)
4.4. Transient two-phase visco-capillary dynamics: interface motion X(t) in axially uniform or variable tubes/joints
4.5. Two-dimensional two-phase dynamics: transient drainage in a planar joint with randomly variable aperture field a(x,y)
4.6. Other transient capillary phenomena in fluid dynamics: waves, bubbles, etc. (brief indications)

5 Darcy-Scale Capillary Flows in Heterogeneous or Statistical Continua (Richards and Muskat)

5.1. Introduction, objectives and applications
5.2. Concepts: porous media, Darcy scale and REV (revisited)
5.3. Single-phase Darcy-scale continuum flow equations (Navier–Stokes, Poiseuille, Darcy)
5.4. Richards equation for unsaturated water flow with fixed air pressure in the porous medium
5.5. Philip’s theory of infiltration – vertical unsaturated flow
5.6. Darcy–Muskat equations for immiscible two-phase flow

Conclusion to Volume 1 and Outline of Volume 2

References

Index

End User License Agreement

List of Tables

2 Two-Phase Physics: Surface Tension, Interfaces, Capillary Liquid/Vapor Equilibria

Table 2.1. Recap of dimensionless numbers in multiphase fluid systems (synoptic table). Note: most of the dimensionless groups are presented as ratios of forces, although they could also be presented as ratios of times, or length scales, etc.
Table 2.2. Typical values of relative air humidity and suction, per Kelvin’s law

3 Capillary Equilibria in Pores, Tubes and Joints

Table 3.1. Notations for equilibrium calculations of two-phase fluid systems in tubes and joints under the influence of gravitational and capillary forces
Table 3.2. Typical values of capillarity, viscosity and density parameters for a water/oil system (liquid water + light oil) (Cohen and Ababou 2003, Parameter Set No.2)

4 Pore-Scale Capillary Flows (Tubes, Joints)

Table 4.1. Dimensionless “shape coefficient” c_F for various geometries of flowing objects (“pores”)
Table 4.2. Interface displacement for a horizontal planar joint with exponentially convergent aperture (-b < 0) or exponentially divergent aperture (+b > 0), where b is a positive parameter (the case of constant aperture b = 0 is treated in Table 4.6)

5 Darcy-Scale Capillary Flows in Heterogeneous or Statistical Continua (Richards and Muskat)

Table 5.1. Some popular functional models for relative hydraulic conductivity versus effective saturation Kr(Se) in unsaturated materials (soils)
Table 5.2. Relative conductivity–saturation curves K_r (S_e ) obtained by inserting two different saturation curves S_e (h) into the three functional models Kr{Se} shown in the previous table
Table 5.3. Relative conductivity–pressure curves Kr(h) obtained by inserting saturation curves S_e (h) into the three functional models K_r {S_e } shown in previous tables
Table 5.5. Hydrodynamic properties of the soil used in the numerical infiltration experiments, for comparison with Philip’s infiltration theory
Table 5.5. Recapitulation of two-phase flow variables, parameters and nonlinear properties (coefficients)

List of Illustrations

1 Fluids, Porous Media and REV: Basic Concepts

Figure 1.1. Sand/gravel fluvio-glacial deposits, illustrating stratification and cross-bedding in a sedimentary formation. For a color version of this figure, see www.iste.co.uk/ababou/capillary1.zip
Figure 1.2. This saprolite – a highly altered fractured gneiss – lies beneath a layer of red clayey soil, overtopping the less altered fractured gneiss rock at depth (not visible). The vertical scale is a fraction of a meter. Formation of the Moole Hole Basin, Kerala, India. For a color version of this figure, see www.iste.co.uk/ababou/capillary1.zip
Figure 1.3. A dye injection experiment in a physical model of groundwater flow in a heterogeneous aquifer system (the vertical slab) containing two sand layers (fine and coarse), a bottom clay layer, and an upper clay lens. This laboratory scale model was designed in 2006 at IMFT, Toulouse, by the author and colleagues including E. Milnes and J. Langot. It is used for the qualitative demonstration of groundwater flow paths and contaminant migration. For a color version of this figure, see www.iste.co.uk/ababou/capillary1.zip

2 Two-Phase Physics: Surface Tension, Interfaces, Capillary Liquid/Vapor Equilibria

Figure 2.1. Capillarity effects in nature. The photograph shows two six-legged water striders standing on water (Gerris Remigis insects from the Gerridae family): these two water striders are mating on the water surface without sinking
Figure 2.2. (a) Top: a single droplet of water on a hydrophobic lotus leave (source: Michael Apel). (b) Bottom: close-up of a droplet resting on a hydrophobic lotus leaf. The contact angle was measured approximately on this grayscale image (which is a 2D projection of the 3D spherical cap). The value found was apparently 146.9° on the left side and 146.4° on the right side (source: http://en.wikipedia.org/wiki/Hydrophobe, 2013)
Figure 2.3. Photograph of three falling droplets after they were exuded from the stomata of a bamboo leaf: after a growth phase during exudation, the water droplets gain weight and finally fall due to gravity, as shown in this particular photograph (source: Anonymous and Kuribayashi 1993) (exposure time: 1/6,000 second)
Figure 2.4. Small and large drops of liquid on a smooth solid surface (source: Willem van Aken, CSIRO, Australia)
Figure 2.5. These illustrate the rise of water and its geometric configuration in granular porous media (natural sand, artificial glass beads, etc.) due to capillary forces. Here the equilibrium state is shown after capillary rise, schematically and experimentally. (a) Top left: schematic view of a vertical column in a soil-aquifer system, showing, from bottom to top: the saturated zone, water table, capillary fringe, unsaturated zone, and soil surface (modified from Musy 2006). (b) Top right and (c) bottom: photograph of an experimental column of hydrophilic glass beads during capillary rise enhanced by evaporation (personal communication: P. Duru and LinGuo, Institut de Mécanique des Fluides de Toulouse/Porous Media group, 2015)
Figure 2.6. Schematic showing the configuration of intermolecular forces. These forces are equidistributed, for those molecules located far from the interface (at the bottom of the schematic). However, the forces become non-equidistributed near the interface. (Note: this schematic does not show the molecules of the non-wetting fluid located above the interface; it only shows the molecules of the wetting fluid located below the interface)
Figure 2.7. Representation of the fluid/fluid interface at intermediate scale. Left: perspective view of the interfacial transition zone between the two fluids, also showing an orthogonal section. Right: pressure/tension profile orthogonally to the interfacial zone. Note: in this schematic, the interface is horizontal, and the orthogonal direction is the vertical direction “z”
Figure 2.8. Representation of the fluid/fluid interface, and of the vertical pressure/tension profile, at a coarser scale of analysis. The transition zone has become infinitesimal (vanishing thickness), and it is represented by a surface, the interface. The vertical pressure/tension profile (shown within the 3D transition zone in Figure 2.7) is replaced here by an interfacial tension σ_S tangent to the surface separating the two fluids
Figure 2.9. (a) Top: diagram showing the contact angle and the 3 interphase tensions between 3 phases (gas, liquid, solid). The liquid droplet rests on a solid surface, and is surrounded by gas. The contact angle θ_C is the angle formed by the liquid at the 3-phase boundary where liquid, gas, and solid intersect. (b) Below: diagram illustrating the contact angle for a vertical solid wall in contact with an open interface separating two immiscible fluids (non-wetting fluid 1, wetting fluid 2)
Figure 2.10. The ring tensiometer, an apparatus for measuring surface tension. Left: photograph of a ring tensiometer apparatus. A torsion dynamometer is used to measure force F, in Newtons, from which surface tension σ is calculated in Newtons/m (note that the scale of the dynanometer is in milli-Newtons mN). In this particular device, the level of the liquid is allowed to diminish (see the outlet tube at left) until the ring becomes detached from the liquid. The force F to be recorded should be the maximal force measured during this process (source: PHYWE 2014). Right: photograph showing the metal ring dipped at the surface of a wetting liquid, in the classical ring method of Du Noüy for measuring surface tension. The force F needed to lift this ring from the liquid is measured with a dynamometer (not shown in this photo) (source: Ptirgyuru 2014)
Figure 2.11. Schematic illustration (left) and photograph (right) of a Tempe pressure plate apparatus for imposing a given suction in a soil sample, the purpose being to measure the water retention curve relating water content (θ) to suction (Ψ) for a broad range of suctions or capillary pressures up to 15 bars (if the air entry pressure of the pressure plate disk is 15 bars) (source: Rahardjo and Nio 2017)
Figure 2.12. Schematic representation of a thermodynamic system: the system is defined by a control volume; it can be either closed or open to thermodynamic fluxes through its boundary
Figure 2.13. Plot of the curve H_R(Ψ), relative air humidity versus pore water suction, according to Kelvin’s law, under standard atmospheric pressure and temperature conditions as given in the text. Top: cartesian graph of H_R(Ψ). Bottom: semi-log graph of H_R(Ψ) (Calculation and plot performed with a MATLAB® program by R. Ababou, January 2010)

3 Capillary Equilibria in Pores, Tubes and Joints

Figure 3.1. Schematics illustrating the forces applied to a curved meniscus in the case of a cylindrical tube of circular section. The tube is filled with a perfectly wetting liquid with almost zero contact angle (e.g. for water on glass in the presence of air). The meniscus here has a hemispherical shape for the case of a zero contact angle (perfect wetting). It would be a spherical cap for an imperfectly wetting liquid with a non-zero contact angle
Figure 3.2. Schematic illustration of forces applied to a water column at equilibrium in a capillary tube under the influence of surface tension, pressure, and gravitational forces. Here, the interface meniscus is hemispherical (more generally it will be a spherical cap in the case of imperfectly wetting fluid with non-zero contact angle). The equilibrium height h is given by Jurin’s law, as demonstrated in the text
Figure 3.3. Forces applied to the semi-cylindrical meniscus between water and air in a vertical planar joint of constant aperture a (imbibition with perfectly wetting water). The interfacial meniscus has the shape of a half-cylinder of circular section and horizontal axis. It is viewed here in perspective from above (the vertical walls of the joint are not shown). The front part of the schematic also shows the meniscus in vertical (x, z) cross-section
Figure 3.4. Wetting phase (w) and non-wetting phase (nw) in a planar joint of variable aperture a(x), or in a capillary tube of variable diameter a(x), shown in axial cross-section
Figure 3.5. Description of the local geometry of a meniscus in a spatially variable tube or a planar joint (one-dimensional two-phase flow dynamics along axis “x”). This cross-sectional view shows only the top half of the tube in a horizontal position: the top bold curve represents the variable solid wall; the bottom dashed line is the central axis of the tube or the mid-plane of the joint. The meniscus is represented as a spherical cap; its axial position is X(t) on the central axis, and Xo(t) on the solid wall, with Xo(t) > X(t). Note: the wetting fluid is on the left, and the non-wetting fluid is on the right
Figure 3.6. Illustration of the capillary pressure jump, expressed with the geometric correction discussed in the text (here for a vertical planar joint). This quasi-static geometric correction should also be used in the equations describing transient two-phase flow and interface displacement in a tube or joint
Figure 3.7. Schematic representation of capillary rise in a tube, showing the final static height h. Water pressure at point A is less than atmospheric pressure, and the capillary jump ΔP = Pair-Pw_A is given by the law of Laplace-Jurin-Young. The water pressure at point B is approximately equal to Pair (neglecting the weight of the column of air ρ_AIRgh), and B is in static equilibrium with A, implying Pw_B = Pw_A+ρ_Wgh. (NB: the tube radius R is magnified compared to height h, for better visibility)
Figure 3.8. Uniformly distributed random variable X∈U[Xmin, Xmax], with Xmin=1, Xmax=3. (a) The theoretical uniform PDF (solid line) and its empirical estimate (histogram bars) for N=100 replicates. (b) The theoretical uniform CDF (solid line) and its empirical estimate by the method of points (Hazen) for N=100 replicates; the abscissae of the 100 points are the data X(i)
Figure 3.9. Schematic display of a statistical bundle of tubular “pores”, represented by a set of vertical capillary tubes immersed in water. This geometric representation is used in the text, assuming a statistical distribution of tube diameters, to obtain the macroscopic moisture retention curve θ(ψ), where θ is water content, and ψ is suction (capillary pressure scaled by ρg)
Figure 3.10. Hyperbolic water retention curve θ(ψ), obtained analytically for the bundle of tubes model with truncated Pareto distribution of pore radii (exponent ω = 2, bounds Rmin = 10 μm and Rmax = 50 μm)
Figure 3.11. Synthetic “pore sample” corresponding to the Pareto pore size distribution, shown here on a 1 mm² sample in planar cross-section. This Boolean set of circular “pores” has the following properties: 2D random Poissonian pore centers, and (independently) Pareto distribution of random pore radii with exponent ω = 2 and bounds [Rmin = 10 μm, Rmax = 50 μm]. For a color version of the figure, see www.iste.co.uk/ababou/capillary1.zip
Figure 3.12. Empirical and theoretical Cumulated Distribution Function of pore radii according to the truncated Pareto law with exponent 2. The theoretical CDF, F(r), is represented as the thin solid line. The empirical CDF (Hazen pointwise estimate) is represented by the symbols (small circles). The abscissa values, R(i), were generated based on the inverse CDF method explained in the text. For a color version of the figure, see www.iste.co.uk/ababou/capillary1.zip
Figure 3.13. Analytical water retention curve θ(ψ) for a uniform statistical distribution of pore radii, using the data given in the text. Suction ψ is in meters. (a) θ(ψ) for the “coarse” medium (pore radii in the range 50μm–500μm). (b) θ(ψ) for the “fine” medium (pore radii in the range 25μm-75μm)
Figure 3.14. Sketch of the soap film surface ∑, assumed to be representable as the elevation function z = Z(x,y) for all (x,y) ∈ Ω ⊂ ℝ², where Ω is the domain of definition of Z(x,y)
Figure 3.15. Example of a non-planar minimal surface: the minimal surface of Schwartz, shaped like a horse saddle, constrained by the four sides of a skewed quadrilateral
Figure 3.16. Left: schematic of the minimal surface corresponding to a soap film between two coaxial rings (two circular wires, not necessarily of same diameters). Right: photograph of an actual soap film between two metallic rings. For a color version of the figure, see www.iste.co.uk/ababou/capillary1.zip
Figure 3.17. Schematic representation of the soap film catenoid surface in vertical cross-section. The cross-section is taken vertically through the common axis of the left and right metal rings. The upper curve corresponds to the analytical curve y = Y(x), solution of the Euler-Lagrange equations for the area minimization problem (as explained in the text). For a color version of the figure, see www.iste.co.uk/ababou/capillary1.zip
Figure 3.18. Analysis of existence and unicity of differentiable solutions for the minimal area problem of the soap film surface between two rings: the function g(u) = cosh(u)/u is plotted for the numerical parameters given in the text. The lower dashed line indicates non-existence of the differentiable surface. The intermediate line passing through the minimum point of g(u) indicates the existence of a single minimal surface (single arrow). Finally the upper dashed line indicates the possible existence of two distinct differentiable surfaces (two arrows)
Figure 3.19. Plot of the cross-sectional profile Y(x), solution of the Euler-lagrange equations, for the soap film between two coaxial rings. See the numerical values given in the text. Here the dimensionless ratio is ρ₀ = (ρ₀)_CRIT = 1/0.6627. In this case, a single minimal surface exists: it is the catenoid surface, displayed here in vertical cross-section. For a color version of the figure, see www.iste.co.uk/ababou/capillary1.zip
Figure 3.20. The relation between parameter c₀ and (x₀,y₀), represented as the multivalued curve c₀/y₀ = f(x₀/y₀), where c₀ is the elevation of the lowest point of the minimal surface Y(x), y₀ is the radius of the two rings, and x₀ is the mid-distance between the two rings. As can be seen, there are two branches to the curve, the “stable” one, and the “unstable” one. See explanations in the text
Figure 3.21. Examples of catenoid-like surfaces. Top: stable catenoid; middle: unstable catenoid; bottom: double catenoid with one disc (non-differentiable). Note: only the surface displayed on top is a viable minimal surface between two rings
Figure 3.22. Schematic illustrating the case of a gas bubble (the gas in the bubble is the non-wetting fluid), resting at isothermal hydrostatic equilibrium in an infinite reservoir of liquid water (water is the wetting fluid). The bubble’s equilibrium depth h or conversely the bubble radius r at a given depth h below the free surface of the liquid are both calculated in the text. For a color version of the figure, see www.iste.co.uk/ababou/capillary1.zip

4 Pore-Scale Capillary Flows (Tubes, Joints)

Figure 4.1. Schematic of an inclined planar joint made up of two parallel plates, in the case of constant aperture (the two plates are flat and smooth). Top: cross-sectional view. Bottom: perspective view. For a color version of the figure, see www.iste.co.uk/ababou/capillary1.zip
Figure 4.2. Schematic illustration of the parabolic velocity profile in Poiseuille /Couette flow between two parallel plates of constant aperture (planar joint). Note: similarly, the axial velocity profile is also parabolic, in any radial cross-section, for the case of a cylindrical tube of circular section
Figure 4.3. Vertical profiles of axial velocity, Ux(z), for Couette flow with a mobile upper plate moving either to the left (V₀ < 0) or to the right (V₀ > 0). The aperture between the plates is 2 cm (half-aperture h=1 cm); water dynamic viscosity is μ = 1E-3 Pa.s, and kinematic viscosity is v = μ/ρ =1E-6 m²/s. The pressure gradient is negative: dP/dx = -1.0 Pa/m. The upper plate velocity V₀ varies between ±0.25 m/s (±0.90 km/h). The different velocity profiles are shown for 0.05 m/s increments of V₀, with a thicker solid line for the null velocity profile V₀ = 0 (that case is driven solely by the pressure gradient). As can be seen, flow velocities Ux(z) can be positive (to the right) or negative (to the left), depending on upper plate velocity (V₀) and on vertical position (z). Note: in this example, V₀ is at most 0.25 m/s, and the V0-based Reynolds is at most: Re_MAX = V₀.2 h/v ≈ 5000. Several of the profiles shown in this figure have much smaller Re, within the range of validity of laminar flow (Re < 2300 or so), except for the two extreme profiles on the right and left. For a color version of the figure, see www.iste.co.uk/ababou/capillary1.zip
Figure 4.4. Vertical profile of axial velocity, Ux(z), for Couette flow with immobile plates (V₀=0)… also known as “planar Poiseuille flow”. Here, the aperture between the two plates is 2 cm (half-aperture h=1 cm); the dynamic viscosity is μ = 1E-3 Pa.s; the pressure gradient is negative (dP/dx = −1.0 Pa/m). The flow velocities Ux(z) are positive (pointing to the right). The maximal velocity is at the mid-plane position z=0 between the plates: Vmax=0.05 m/s. The corresponding Reynolds number is Re_MAX = V₀.2 h/ν ≈ 1000 (laminar flow). For a color version of the figure, see www.iste.co.uk/ababou/capillary1.zip
Figure 4.5. Schematic representation of a tortuous cylindrical “pore”, in a “bundle of tubes” model of a porous medium. Each tortuous tube may represent a “pore” or a fluid pathway through the system of pores. The average mescoscale tortuosity of the bundle of tubes may be taken into account in the Kozeny-type permeability obtained from the bundle of tubes model. Tortuosity (τ) is defined in the text. Schematic based on Fermigier (2002)
Figure 4.6. Schematic depiction of a parallel system of planar joints. Two cases are considered, the M×N parallel set, and the M×1 parallel set (bottom and top, respectively). The M×N set comprises M parallel joints (m=1,…,M), each joint (m) having variable aperture a_m(x_n) discretized into N aperture sites (n=1,…,N) along the flow direction (x). The M×1 set comprises M parallel joints, but each joint (m) has now a unique aperture a_m that is spatially constant along the flow direction (the apertures vary only from joint to joint)
Figure 4.7. Log-normal aperture distribution for “Network#1”, an M×1 set of smooth 1D joints (equivalent to a planar fracture with apertures varying only in the transverse direction, but cosntant along the flow). The log-normal aperture distribution transversely to the axial flow direction is highly variable in this example. The mean aperture is ho = m_H = 50μm; the log-aperture standard deviation is σ_LnH = 0.43 The resulting coefficient of variation of aperture h is high of the same order as a_LnH about 40%
Figure 4.8. Equivalent capillary pressure versus saturation curve pc(S) for “Network#1”, the M×1 set of smooth joints with highly variable log-normal aperture distribution across the set of joints (mean aperture ho=50μm, log-aperture standard deviation σ_Ln(h) = 0.43)
Figure 4.9. Equivalent capillary pressure versus saturation curves, pc(S), for drainage (upper curve) and imbibition (lower curve). Case of an M×N set, or “network”, comprising M=50 parallel joints (m=1,…,50), each joint m having variable apertures a_m(x_n), discretized here into N=10 aperture sites (n=1,…,10) along the flow direction. In this case, the apertures are uniformly distributed. They were independently generated along each rough joint (apertures n=1,…,10) and across the set of joints (joints m=1,…,50)
Figure 4.10. Relative permeability curves versus saturation (case of imbibition), for a rough fracture made up of a statistical parallel set of 1D joints, with M=100 joints, each with N=100 aperture sites (M×N = 100×100). The permeability curves are plotted here for several (four) realizations of the statistical joints in order to assess the effect of statistical fluctuations on the results. Here the aperture distribution is uniform over a broad range of aperture values (100 < h < 200 μm)
Figure 4.11. Capillary pressure curves versus saturation (case of imbibition). Comparisons of various aperture distributions (uniform, exponential, log-normal). The pc(S) curve does not seem sensitive to the probability distribution of apertures in this case (at least for comparable aperture moments). The Quasi-1D Analytical solutions are plotted as solid curves, compared to 1D Numerical Simulations (symbols)
Figure 4.12. Relative permeability curves versus saturation (case of imbibition). Influence of aperture probability law for a rough 100×100 fracture (parallel set of 1D joints with M = 100 joints, each with N=100 aperture sites). Comparison of uniform law with 100 < h < 200 μm, exponential law with mean m_H = 150 μm, and log-normal law with mean m_H = 120 μm and deviation s = σ_LnH = 0.02
Figure 4.13. Relative permeability curves versus saturation (case of imbibition). Influence of the range of variability of aperture h (symbols) for a uniform distribution, in the case of a rough fracture with M×N = 100 × 100. The Quasi-1D Analytical solutions are plotted as solid curves, and compared to 1D Numerical Simulations (symbols)
Figure 4.14. This schematic depicts the detailed configuration of fluid 1, fluid 2, and the fluid/fluid/solid contact angles. The situation depicted here corresponds either to a cylindrical tube (axial symmetry), or to a planar joint (planar symmetry), with variable diameter or aperture a(x) in the axial flow direction (x)
Figure 4.15. Shows the case of variable joint. In this example, the joint aperture is variable at both small and large scales. The walls are “corrugated” surfaces with periodic micro-scale fluctuations. In addition, the mean joint aperture diverges with respect to flow direction
Figure 4.16. Schematic of a variable smooth joint with exponentially varying aperture. (a) case of monotonically “divergent” aperture (if the flow is from left to right); aperture equation: a(X) = a₀ exp(bX), b > 0. (b) case of monotonically “convergent” aperture (if the flow is from left to right): a(X) = a₀ exp(bX), b > 0. For a color version of the figure, see www.iste.co.uk/ababou/capillary1.zip
Figure 4.17. Schematic of a variable joint with sinusoidally varying aperture a(x). Note: if the amplitude of the sinusoidal fluctuations of a(x) is increased, the two walls will touch (zero apertures), and even interpenetrate (negative “apertures”): this can be avoided by changing the model to sinusoidal log-aperture. For short wavelengths of the aperture a(x), the joint becomes “rough” rather than “smooth”. For a color version of the figure, see www.iste.co.uk/ababou/capillary1.zip
Figure 4.18. Schematic of a variable joint with non-harmonic periodic variation of aperture a(x), shaped like the exponential of a sinusoid. Note: in this case, even for a large amplitude of variation of a(x), the two plates will never touch each other (the aperture field remains everywhere non-zero). For short wavelengths of the log-aperture field, Ln(a(x)), the joint becomes “rough” rather than “smooth”. For a color version of the figure, see www.iste.co.uk/ababou/capillary1.zip
Figure 4.19. Top: schematic of an experimental set up for a horizontal left-to-right imbibition experiment in a tube/joint with variable diameter/aperture. Two immiscible liquids are involved: “Liquid 1” (at left) is wetting; “Liquid 2” (at right) is non-wetting. Initially, the tube/joint is filled with the non-wetting Liquid 2. The open reservoirs control the left/right pressures. Bottom: schematic of a vertical liquid/gas imbibition experiment. The entry pressure of the liquid at bottom can be adjusted using the level Δh in the reservoir, e.g. for studying capillary rise in the vertical tube/joint
Figure 4.20. Above: analytical versus numerical solutions of the dynamic visco-capillary equations for axial flow in horizontal joint with exponentially varying aperture, either decreasing (left) or increasing (right). The numerical results X(t) (crosses +) fit perfectly well the analytical solutions X(t) (solid lines). Below: comparison of interface displacement X(t) for convergent and divergent aperture, respectively
Figure 4.21. Visco-capillary imbibition dynamics of water/air in a horizontal joint with periodic aperture a(x) = ao cos(wx). Top: aperture field a(x); the scale of the ordinate (±half-aperture) is about ±1 × 10^–4 m (±one tenth of a millimeter); the length of the joint is 5 m. Middle: time evolution of air/water interface position X(t), up to t = 50,000 s ≈ 14 h. Bottom: axial profile of Lagrangian interface velocity U(x) represented versus mobile positions x = X(t); scale of the U-axis: 3 × 1E-3 m/s (3 mm/s)
Figure 4.22. Visco-capillary imbibition dynamics for water/air in a vertical joint with periodic aperture a(x) = ao.cos(wx). Right: aperture field a(x); the scale of the aperture axis is about ±1E-4 m (± one tenth of a millimeter), and the vertical length of the joint is 5 m. Top left: time evolution of air/water interface X(t), up to t = 5,000 s ≈1 h 20 mn (the final static position is X ≈ 77 cm, U ≈ 0). Bottom left: axial profile of Lagrangian interface velocity U(x) plotted here versus the mobile position x = X(t); scale of U-axis: 1.2×1E-3 m/s (1.2 mm/s)
Figure 4.23. Schematics illustrating the 2D algorithm for calculating two-phase visco-capillary invasion (e.g. drainage) in a planar fracture with variable apertures, discretized as aperture cells (2D grid)
Figure 4.24. Plane view of the random aperture field a(x,y), with a color scale from red to blue for large to small apertures. High apertures are red, moderate ones are light blue and yellow, low ones are dark blue. For a color version of the figure, see www.iste.co.uk/ababou/capillary1.zip
Figure 4.25. Landscape view of the random aperture field a(x,y), with the same color scale as before. The high altitude peaks in this “landscape” correspond to large apertures, and the troughs correspond to small but non-zero apertures (it is known that a(x,y) > 0 everywhere, as explained in the text). For a color version of the figure, see www.iste.co.uk/ababou/capillary1.zip
Figure 4.26. Configuration of the two fluids in the (x,y) plane of the randomly heterogeneous planar joint during drainage at times times t1 (upper left), t2 (upper right), t3 (lower left), and t4 (lower right). For a color version of the figure, see www.iste.co.uk/ababou/capillary1.zip
Figure 4.27. Spatial configuration of the two fluids during drainage at times t5 (upper left), t6 (upper right), t7 (lower left), and at “final” time t8 (just a few time steps before final equilibrium time). The t8 configuration is stable, except for the last few vertical columns of cells on the right, where displacements will continue to occur for a few more steps. For a color version of the figure, see www.iste.co.uk/ababou/capillary1.zip

5 Darcy-Scale Capillary Flows in Heterogeneous or Statistical Continua (Richards and Muskat)

Figure 5.1. Schematic illustration of the refraction of streamlines for incompressible single-phase flow in a Darcian porous medium containing a single inclined planar interface of material discontinuity (Γ). This interface separates two semi-infinite porous domains Ω₁ and Ω_2. The porous medium properties (permeability, porosity) are continuous within each sub-domain, but discontinuous at the interface. This schematic illustrates the case K1 < K2 or k1 < k2, i.e. the upper domain has lower permeability than the lower domain. Top: the upper schematic shows two streamlines forming a streamtube crossing the interface. Bottom: the lower schematic shows a zoomed in version the coordinate systems (x,z) and (s,n)
Figure 5.2. Schematic illustration of the REV concept of a poly-disperse packing of disk-shaped “grains” in the plane (cylindrical “grains” in 3D-space), with three grain sizes. Macroscopic quantities like porosity can be computed for different sub-domains using moving windows or telescopic sets of embedded windows, etc. The “REV” is the smallest sub-domain such that the property (porosity) does not change much by increasing its size further. For example, the square shown in the lower left-hand corner, which contains about 100 grains, is roughly an REV. (Schematic courtesy of L.W. Gelhar, Ralph Parsons Laboratory, MIT, Cambridge, 1986)
Figure 5.3. This slide shows, in a nutshell, the conceptual path from the Navier–Stokes equations to Darcy’s law, for single-phase flow in a porous medium (this is the conceptual path, not the historical path to Darcy’s law). Top: the Navier–Stokes equations (to be completed with no-slip boundary conditions on fluid/solid interfaces). Bottom: a theoretical simplification that leads from the Navier–Stokes conservation of momentum to Darcy’s law
Figure 5.4. Summary of Darcy’s law expressed in terms of hydraulic conductivity K (m/s) and total hydraulic head H (m), for a porous medium saturated with incompressible or weakly compressible liquid water. This conductivity law is the classical formulation of Darcy’s law for hydrogeologic applications. The relation between conductivity K and permeability k is shown for water under standard conditions: K = 0.873 × 1E+7 × k, with K in (m/s) and k in (m²). Conductivity and permeability are scalars because the porous medium is assumed to be isotropic
Figure 5.5. Photograph of a layered column of “flysch”, isolated by erosion, in the coastal area of Saint-Jean de Luz, Pyrénées Atlantiques, France. (Photo credit: R. Ababou, 2007). For a color version of the figure, see www.iste.co.uk/ababou/capillary1.zip
Figure 5.6. A numerically generated sample of a perfectly layered medium, with periodic alternation of layers. For a color version of the figure, see www.iste.co.uk/ababou/capillary1.zip
Figure 5.7. Photographs of a columnar basalt at Devil’s Postpile, California, USA. Top: top view showing hexagonal column structures. Bottom: front view. (Photo credit: Cian B. Dawson, USGS). For a color version of the figure, see www.iste.co.uk/ababou/capillary1.zip
Figure 5.8. This is a simple example illustrating the cubic aperture law (a³) for Poiseuille flow in a planar fracture set. The equivalent hydraulic conductivity of the parallel set of fractures is obtained upon averaging over the set. Answer to the question: in this example, the equivalent conductivity of the fracture set is K = 10⁻⁶ m/s (1E-6 m/s), which is in the same order as the conductivity of a very fine sand or a sandy loam. For a color version of the figure, see www.iste.co.uk/ababou/capillary1.zip
Figure 5.9. Back view of a 3D moisture plume in a randomly heterogeneous, imperfectly stratified soil, shown here after 10 days of strip source infiltration. A part of the domain, on the right, has been excavated to allow for better visualization. The colors draped onto surfaces indicate moisture: brown and yellow for dry zones that have not yet been wetted, light blue for marginally wet zones (spreading horizontally), and dark purple and dark blue for the wettest zones. (R. Ababou/courtesy of SwRI, San Antonio). For a color version of the figure, see www.iste.co.uk/ababou/capillary1.zip
Figure 5.10. Schematic plot of the VGM unsaturated curves. Top: unsaturated conductivity curves versus pressure head K(h). Bottom: moisture content versus pressure head θ(h). For a color version of this figure, see
Figure 5.11. Transient vertical infiltration test in relatively dry Grenoble sand under constant flux q = Ks, showing that the assumption of constant capillary moisture diffusivity D(θ) = Do, or constant gravitational moisture velocity k(θ)=ko, is not adequate. Top: comparison of numerical versus analytical pressure head profiles h(z) (cm) at t = 10 mn, over a depth of 40 cm, respectively with nonlinear (k(θ)) and constant moisture velocity (values k = 4 cm/h and k = 40 cm/h). Bottom: the “true” nonlinear curve k(θ) = dK/dθ (cm/h) of the sand. Note: when conductivity K(h) is exponential, then k(θ) = ko ⇔ D(θ) = Do, as explained in the text
Figure 5.12. Horizontal infiltration in the Guelph Loam soil (no gravity). Top: simulated infiltration volume I(t) [m]. Bottom: simulated curve I(t), plotted as a function of √t, and linear regression fit: the slope yields the estimated sorptivity S. For a color version of the figure, see www.iste.co.uk/ababou/capillary1.zip
Figure 5.13. Vertical infiltration in the Guelph Loam soil (with gravity). Top: simulated infiltration volume I(t) (m). Bottom: simulated curve I(t), plotted as a function of √t, and linear regression fit: the slope yields the gravitational A-parameter, and the intercept yields the sorptivity S. For a color version of the figure, see www.iste.co.uk/ababou/capillary1.zip
Figure 5.14. Vertical moisture profiles θ(z,t) during vertical infiltration in the Guelph Loam soil: comparison of numerical simulation results with the BIGFLOW code (solid line) and Philip’s quasi-analytical solution (circles). Collaboration with K. Alastal (2012). For a color version of the figure, see www.iste.co.uk/ababou/capillary1.zip
Figure 5.15. Vertical distribution of water saturation (Sw) at quasi-equilibrium, in a homogeneous oil reservoir. This schematic shows transition zones defined by two saturation bounds (0 < SWir < SWro < 1), with SWir = irreducible water saturation and SWro = residual or immobile oil saturation. The arrows indicate (slow) drainage or (slow) wetting. (Schematic from Archer and Wall (1986))

I spilled a glass of water and went to mop it up with some paper towels.

“They don’t have very good capillarity”, Olivier said.

“Huh?” I replied, continuing to dab at the puddle.

“Their capillarity isn’t very good.”

“What are you talking about? That’s not even a word.”

Olivier said nothing. A few days later, I noticed a piece of paper lying in the printer tray. It was a page from the Merriam-Webster online dictionary:

Capillarity noun ka-pə-ˈler-ə-të, -‘la-r ə -.

1: the property or state of being capillary

2: the action by which the surface of the liquid where it is in contact with a solid (as in a capillary tube) is elevated or depressed depending on the relative attraction of the molecules of the liquid for each other and for those of the solid.

Ink to a nib, my heart surged.

Lauren Collins, “Love in Translation” (p. 52)

The New Yorker, pp. 52–61, August 8 and 15, 2016

First published 2018 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

ISTE Ltd

27-37 St George’s Road

London SW19 4EU

www.iste.co.uk

John Wiley & Sons, Inc.

111 River Street

Hoboken, NJ 07030

USA

www.wiley.com

The rights of Rachid Ababou to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Control Number: 2018958575

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

ISBN 978-1-84821-528-3

List of Symbols

Symbol	Physical Units	Description
α	[1/m]	Scale factor in Gardner’s exponential model K(ψ) or scale factor in Van Genuchten moisture curve θ(ψ)
β	[1/m]	Scale factor in Gardner’s exponential model K(ψ)
λ	[m]	Spatial correlation scale random field properties or fracture inter-spacing (also λ = spatial density of statistical tubes model)
μ	[Pa.s]	Dynamic viscosity
ν	[m²/s]	Kinematic viscosity
π	[Pa]	Disjunction pressure
Φ	[m³/m³]	Porosity
	–	Curve (closed) in 2- or 3-dimensional space
θ; Θ	[m³/m³]	Volumetric water content (wetting fluid content)
θ_S	[m³/m³]	Saturated water content or porosity
ρ	[kg/m³]	Density (i.e. volumetric mass density)
ρ_pq	[m^p/m^q]	Generalized geometric densities (p ≤ q), e.g. ρ33 for porosity; ρ23 for specific area; etc.
σ, σLnK, …	[–]	Standard deviation of a random parameter such as log-aperture, log-permeability, etc.
σ, σ_Interf, σ_S	[N/m]	Interfacial tension (also called “surface tension”)
σ_Specif	[m²/m³]	Specific area (of a set of pores, tubes, joints, etc.)
σ_ij	[N/m²], [Pa]	Stress tensor
	–	Surface in 3-dimensional space
τ_ij, τ	[N/m²], [Pa]	Shear stress
ψ, Ψ	[m]	Suction (local suction, mean suction).
ω	[–]	Exponent in the Pareto probability law or exponent in the Power Average Model of upscaled Kii(Ψ)
Ω	–	Region or domain in 3-dimensional space (Ω⊂ℝ³)
“℧” or “V”	[m³]	Volume of a region of space; volume of fluid region
ξ	[m]	Separation distance (in spatial covariance functions)

a, a₀	[m]	Diameter of tubular pore(s)/tube(s); also aperture of planar joints (fractures)
	–	Field of complex numbers (z ∈ )
h	[m]	Aperture or half-aperture of planar joints (depending on context) – not to be confused with pressure head (also denoted “h”)
“” or i	–	Pure imaginary number:
k	[m²]	Permeability (Darcy’s intrinsic permeability)
K	[m/s]	Hydraulic conductivity (saturated or unsaturated)
K_S	[m/s]	Saturated hydraulic conductivity
“ℓ”	[m]	Length scale (its meaning depends on context)
“nw”	–	“Non-wetting” (non-wetting fluid)
ℕ	–	Natural numbers (non-negative integers n ∊ ℕ
pC, pCap	[Pa]	Capillary pressure
“ℝ”	–	Field of real numbers (x ∊ ℝ)
R	[m]	Radius of pore(s), tube(s), etc.
“w”	–	“Wetting” (wetting fluid)
ℤ	–	Relative numbers (positive & negative integers p ∊ ℤ

Bo	[–]	Dimensionless Bond number
Ca	[–]	Dimensionless Capillary number
Da	[–]	Dimensionless Darcy number
PDE		Partial Differential Equation
Re	[–]	Dimensionless Reynolds number

Additional list of symbols for two-phase flow

Symbol	Units	Description
α=1 or 2	–	Fluid phases: α = 1 or w (wetting), α = 2 or nw (non-wetting)
Θ₁, Θ₂	[kg/m³]	Volumetric mass contents of fluids 1 and 2 (per volume of medium)
ρ₁, ρ₂	[kg/m³]	Volumetric mass densities of fluids 1, 2 (per volume of fluid)
θ₁, θ₂	[m³/m³]	Volumetric fluid contents (per volume of medium)
θ₀	[m³/m³]	Total porosity of the porous medium
F₁, F₂	[kg/s/m²]	Areal density of mass flux (for each fluid)
f₁, f₂	[m³/s/m²]	Areal density of volumetric flux (for each fluid)
S₁, S₂	[kg/m³/s]	Mass source terms: mass fluxes per unit
S₁, S₂	[kg/s/m³]	volume, or equivalently, volumetric flux densities (for each fluid).
p₁, p₂	[Pa]	Pressures of fluids 1 and 2
p_C= p₂–p₁ = p_NW–p_W	[Pa]	Capillary pressure (non-wetting minus wetting pressure)
K₁, K₂: K_α = K_α(p_c)/μ_α	[kg/s/m³]⁻¹	Absolute mobilities, or equivalently, hydrodynamic conductivities, of each fluid (functions of capillary pressure)
k₁(p_c), k₂(p_c)	[m²]	Absolute two-phase permeabilities for each fluid (m²)
k₀	[m²]	Intrinsic Darcy permeability of the porous medium (m²)
μ₁, μ₂	[kg/m/s]	Dynamic viscosities (properties of fluids 1 and 2).