Capillary Flows in Heterogeneous and Random Porous Media 1

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

Series Editor Gilles Pijaudier-Cabot

Capillary Flows in Heterogeneous and Random Porous Media 1

Rachid Ababou

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 John Wiley & Sons, Inc. 27-37 St George’s Road 111 River Street London SW19 4EU Hoboken, NJ 07030 UK USA www.iste.co.uk www.wiley.com

© ISTE Ltd 2018 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

Contents

List of Symbols ...... xi

Introduction ...... xv

Chapter 1. Fluids, Porous Media and REV: Basic Concepts ...... 1 1.1. Geologic porous media: basic concepts ...... 1 1.1.1. Porous soils ...... 1 1.1.2. Porous rocks ...... 2 1.1.3. Geologic porous media: examples ...... 3 1.2. Porous media: basic concepts, porosity and specific area ...... 6 1.2.1. Fluid phases ...... 7 1.3. Single-phase flow and Darcy’s law: basic concepts ...... 9 1.3.1. Darcy’s flux–gradient law ...... 9 1.4. The Darcy–Buckingham law and the Richards equation: basic concepts of unsaturated flow ...... 10 1.4.1. Remarks on unsaturated water flow ...... 11 1.5. Capillarity and two-phase flow systems at different scales: basic concepts ...... 11 1.5.1. Introduction ...... 11 1.5.2. Capillarity pressure jump at different scales ...... 12 1.5.3. Moving from one scale to another: upscaling ...... 14 1.6. A basic approach to pore scale two-phase flow ...... 14 1.7. A basic approach for continuum scale description of two-phase flow in porous media: the Darcy–Muskat model ...... 15 1.7.1. The Buckey–Leverett model ...... 16 1.8. Other issues: capillarity vs. gravity and viscosity, heterogeneity and upscaling ...... 16 1.8.1. Capillarity plus gravity and viscous dissipation ...... 17 1.8.2. Scales and the representative elementary volume ...... 17 vi Capillary Flows in Heterogeneous and Random Porous Media 1

1.8.3. Objectives at various scales of analysis ...... 17 1.8.4. Upscaling: first and second upscaling problems ...... 18

Chapter 2. Two-Phase Physics: Surface Tension, Interfaces, Capillary Liquid/Vapor Equilibria ...... 21 2.1. Summary and objectives ...... 21 2.2. Physics of capillarity and surface tension at equilibrium ...... 22 2.2.1. Observations and practical applications of surface tension, capillary forces and contact angles ...... 23 2.2.2. Interfacial tension: from molecular scale to fluid scale ...... 31 2.2.3. Laplace-Young pressure jump law (capillary pressure) ...... 34 2.2.4. Solid/liquid contact angle θ at equilibrium (Young) ...... 38 2.2.5. Measurements of interfacial tension ...... 40 2.2.6. Immiscibility versus miscibility at fluid interfaces (examples) ...... 42 2.3. Dimensionless groups (characteristic forces, length scales, timescales) .... 43 2.3.1. Introduction: three forces driving multiphase systems ...... 43 2.3.2. Reynolds and Reynolds-Darcy number, viscous dissipation ...... 44 2.3.3. Capillary forces, surface tension and capillary number Ca ...... 46 2.3.4. Gravitational buoyancy forces and the Bond number Bo ...... 49 2.3.5. Dimensionless contrast ratios (viscosity and density contrasts) ...... 49 2.3.6. Recap of dimensionless groups for a two-phase system ...... 50 2.4. Thermodynamics, Gibbs energy, pressure, suction ...... 51 2.4.1. Interpretation of large suctions, bonding forces and Gibbs energy ..... 51 2.4.2. Thermodynamical systems (isolated or not) ...... 58 2.4.3. Gibbs free energy, heat, work ...... 61 2.5. Kelvin’s liquid/vapor relation (suction vs. air humidity) ...... 67 2.5.1. Introduction to Kelvin’s law (applications in flow modeling) ...... 67 2.5.2. Qualitative discussion of Kelvin’s law (liquid/vapor relations) ...... 68 2.5.3. Thermodynamical variables (pressure, air humidity, etc.) ...... 69 2.5.4. Perfect gases (dry air and water vapor) ...... 70 2.5.5. Kelvin’s law: relative air humidity vs. capillary pressure ...... 71 2.5.6. Extended discussion on liquid/vapor thermodynamics (review) ...... 75

Chapter 3. Capillary Equilibria in Pores, Tubes and Joints ...... 77 3.1. Introduction and summary ...... 77 3.2. Capillary equilibrium in a single tube or planar joint of constant diameter or aperture ...... 78 3.2.1. Introduction: problem formulation and notations ...... 78 3.2.2. Capillary tube: pressure jump (Laplace-Young) ...... 79 3.2.3. Capillary tube: water height (Jurin) ...... 83 3.2.4. Capillary tube: extensions and examples (other fluids, etc.) ...... 84 Contents vii

3.2.5. Example of water/air equilibrium in a capillary tube: calculation of water height for a tube of diameter 100 μm ...... 86 3.2.6. Planar joint: introduction – planar geometry of the meniscus ...... 87 3.2.7. Planar joint: pressure jump across the water/air meniscus ...... 89 3.2.8. Planar joint: equilibrium height of meniscus (capillary rise) ...... 90 3.2.9. Example: parameter values for water and “light oil” in a joint ...... 90 3.3. Capillary equilibria in variable tubes and joints (a(x)) ...... 91 3.3.1. Introduction, description of the problem, and hypotheses ...... 91 3.3.2. Non-existence of two-phase equilibria, depending on initial state ..... 93 3.3.3. Geometric correction for variable tubes/joints: wetting angle θ+ϕ(x) in a fixed frame ...... 94 3.4. Capillary equilibrium in a random set of tubes: calculation of water retention curve θ(ψ) ...... 98 3.4.1. Introduction and summary ...... 98 3.4.2. Capillary water/air equilibrium in a random set of “pores”;

moisture retention curve θ(pC) for uniformly distributed radii ...... 99 3.4.3. Capillary water/air equilibrium and moisture retention curve θ(pc) for Pareto distributed radii with exponent ω = 2 ...... 109 3.4.4. Limitations of the Boolean model of random tubes ...... 115 3.4.5. Soil water retention curves in hydro-agriculture (overview) ...... 119 3.5. Capillary equilibrium of soap films: minimal area surfaces and Euler-Lagrange equations ...... 120 3.5.1. Introduction and summary ...... 120 3.5.2. Soap film surface (preliminary formulation) ...... 121 3.5.3. Euler-Lagrange equations for minimizing integrals ...... 122 3.5.4. Euler-Lagrange equation minimizing the area ...... 125 3.6. Case study of soap film equilibrium between two circular rings: minimal area surface (catenoid) ...... 130 3.6.1. Presentation of the case study: soap film between two rings ...... 130 3.6.2. Formulation: minimal area surface between two coaxial circles ...... 131 3.6.3. Expressing Euler-Lagrange for the generating curve Y(x) ...... 133 3.6.4. Solution of Euler-Lagrange equations: catenoid surface between two coaxial circles of different diameters ...... 135 3.6.5. A special solution of the Euler-Lagrange equations: the catenoid surface between two identical coaxial rings ...... 139 3.6.6. Parametric study and conclusions (existence/unicity of the soap film depending on ring geometry) ...... 142 3.7. Additional topic: the equilibrium depth of a bubble ...... 148

Chapter 4. Pore-Scale Capillary Flows (Tubes, Joints) ...... 151 4.1. Introduction and summary: pore-scale flow in capillary tubes and planar joints (steady and transient) ...... 151 viii Capillary Flows in Heterogeneous and Random Porous Media 1

4.1.1. Introduction and summary ...... 151 4.1.2. Case of steady flow systems (single phase and two phase) ...... 152 4.1.3. Remark on the quasi-static nature of the water retention curve ...... 152 4.1.4. Case of transient flow problems ...... 152 4.1.5. Numerical experiment (2D visco-capillary invasion) ...... 153 4.2. Single-phase steady flow in tubes: Poiseuille, Darcy, Kozeny-Carman permeability ...... 153 4.2.1. Overview: Stokes, Poiseuille, Specific Area, Darcy, Kozeny permeability ...... 153 4.2.2. Specific area concept ...... 154 4.2.3. Poiseuille flow in a cylindrical tube or a planar joint ...... 156 4.2.4. Kozeny-Carman permeability for single-phase flow (from Poiseuille to Darcy) ...... 164 4.3. Unsaturated and two-phase steady flow in sets of planar joints: equivalent

mesoscale quantities (porosity φ, permeability k, capillary length λcap) ...... 187 4.3.1. Summary and overview ...... 187 4.3.2. Upscaling unsaturated flow through a set of joints (equivalent permeability, porosity, and capillary length) ...... 192 4.3.3. Upscaling two-phase flow in smooth or rough statistical joints:

water retention θ(pc); conductivity curves {KW(pc), KNW(pc)} ...... 194 4.3.4. Unsaturated or two-phase constitutive curves from statistical pore-scale models (discussion, review) ...... 205 4.4. Transient two-phase visco-capillary dynamics: interface motion X(t) in axially uniform or variable tubes/joints ...... 211 4.4.1. Introduction, objectives, and literature review ...... 211 4.4.2. Eulerian/Lagrangian equations for transient two-phase flow: axial interface displacement in tubes and joints ...... 213 4.4.3. Quasi-analytical results on transient dynamics of immiscible fluids: axial displacement in variably constricted tubes and joints ...... 220 4.4.4. Geometrical correction on interface dynamics X(t) in the case of very rough, highly variable tubes or joints (remarks) ...... 230 4.4.5. Interface dynamics X(t) in tubes, pores, joints (prospects) ...... 231 4.5. Two-dimensional two-phase dynamics: transient drainage in a planar joint with randomly variable aperture field a(x,y) ...... 232 4.5.1. Introduction and summary ...... 232 4.5.2. The 2D “rough fracture” and its random aperture field a(x,y) ...... 232 4.5.3. The 2D synthetic drainage experiment (two-phase flow) ...... 233 4.6. Other transient capillary phenomena in fluid dynamics: waves, bubbles, etc. (brief indications) ...... 237 4.6.1. Capillary waves ...... 237 4.6.2. Rayleigh-Plateau instability ...... 238 4.6.3. Bubble dynamics and cavitation ...... 239 4.6.4. Liquid/vapor phase changes, boiling, bubbles in porous media ...... 239 Contents ix

Chapter 5. Darcy-Scale Capillary Flows in Heterogeneous or Statistical Continua (Richards and Muskat) ...... 241 5.1. Introduction, objectives and applications ...... 241 5.1.1. Introduction and summary ...... 241 5.1.2. Flow regimes and potential applications ...... 242 5.1.3. Hierarchy of scales and related issues (discontinuities) ...... 243 5.1.4. Material discontinuities in Darcy-scale flows ...... 244 5.2. Concepts: porous media, Darcy scale and REV (revisited) ...... 252 5.3. Single-phase Darcy-scale continuum flow equations (Navier–Stokes, Poiseuille, Darcy) ...... 255 5.3.1. Introduction and summary ...... 255 5.3.2. Darcy’s law: from Navier–Stokes to Darcy in a nutshell ...... 256 5.3.3. Darcy’s law for isotropic media (scalar permeability, single phase flow) ...... 264 5.3.4. Darcy’s law for anisotropic media with tensorial or directional permeability (single-phase flow) ...... 266 5.3.5. Darcy’s law from single-phase “Poiseuille flow” in fractures ...... 271 5.4. Richards equation for unsaturated water flow with fixed air pressure in the porous medium...... 274 5.4.1. Introduction and summary (unsaturated flow) ...... 274 5.4.2. Darcy–Richards unsaturated flow equations ...... 274 5.4.3. Constitutive relationships θ(h), K(θ), K(h), C(h), D(θ), U(θ) ...... 280 5.4.4. Unsaturated curve models (θ(ψ), K(ψ)): overview ...... 285 5.4.5. Van Genuchten/Mualem (VGM) constitutive model for unsaturated moisture and conductivity curves (θ(ψ), K(ψ)) ...... 286 5.4.6. Gardner’s exponential K(ψ) conductivity curve and extensions ...... 293 5.4.7. Nonlinear relations {K(ψ,x), θ(ψ,x)} for heterogeneous media ...... 301 5.4.8. Matching different nonlinear models for {θ(ψ),K(ψ)}: exponential versus Van Genuchten/Mualem (parameter analyses) ...... 302 5.5. Philip’s theory of infiltration – vertical unsaturated flow ...... 309 5.5.1. Introduction: literature and background on infiltration problems ...... 309 5.5.2. Philip’s θ-based unsaturated flow equation for θ(z,t) ...... 311 5.5.3. Philip’s analytical solution: sorptivity and gravitational term; infiltration rate i(t) and volume I(t); moisture profiles θ(z,t) ...... 312 5.5.4. Philip’s analytical solution versus numerical infiltration experiments (comparisons and identification of soil parameters “A” and “S”) ...... 315 5.5.5. Ponding time under a fixed rainfall rate, from Philip’s quasi-analytical solution i(t) with both gravitational and capillary terms ...... 320 5.5.6. Recapitulation, discussion, conclusions ...... 322 5.6. Darcy–Muskat equations for immiscible two-phase flow ...... 324 5.6.1. Introduction and summary (two-phase flow) ...... 324 5.6.2. Mixed formulation of Darcy–Muskat PDEs governing two-phase flow ...... 326 x Capillary Flows in Heterogeneous and Random Porous Media 1

5.6.3. Nonlinear characteristic curves of porous media for two-phase flow ...... 334 5.6.4. Other two-phase quantities derived from the Darcy–Muskat equations ...... 337

Conclusion to Volume 1 and Outline of Volume 2 ...... 343

References ...... 345

Index ...... 367

List of Symbols

Symbol Physical Description Units 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 ℾ – Curve (closed) in 2- or 3-dimensional space θΘ; [m3/m3] Volumetric water content (wetting fluid content) 3 3 θS [m /m ] Saturated water content or porosity ρ [kg/m3] Density (i.e. volumetric mass density) p q ρ pq [m /m ] Generalized geometric densities (p ≤ q), e.g. ρ33 for porosity; ρ23 for specific area; etc. σ , σ LnK , … [–] Standard deviation of a random parameter such as log-aperture, log-permeability, etc. xii Capillary Flows in Heterogeneous and Random Porous Media 1

σ , σInterf . , σS [N/m] Interfacial tension (also called “surface tension”) 2 3 σSpecif [m /m ] Specific area (of a set of pores, tubes, joints, etc.) 2 σij [N/m ], [Pa] Stress tensor ⅀ – Surface in 3-dimensional space 2 ττij , [N/m ], [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)

aa, 0 [m] Diameter of tubular pore(s)/tube(s); also aperture of planar joints (fractures) “ℂ” – Field of complex numbers (z ∈ℂ) h [m] Aperture or half-aperture of planar joints (depending on context) – not to be confused with pressure head (also denoted “h”) “ⅈ” or i – Pure imaginary number: i1≡−

k [m2] Permeability (Darcy’s intrinsic permeability) K [m/s] Hydraulic conductivity (saturated or unsaturated) KS [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) List of Symbols xiii

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) 3 Θ1 , Θ2 [kg/m ] Volumetric mass contents of fluids 1 and 2 (per volume of medium) 3 ρ1, ρ2 [kg/m ] Volumetric mass densities of fluids 1, 2 (per volume of fluid) 3 3 θ1 , θ2 [m /m ] Volumetric fluid contents (per volume of medium) 3 3 θ0 [m /m ] Total porosity of the porous medium 2 F1 , F2 [kg/s/m ] Areal density of mass flux (for each fluid) 3 2 f1 , f2 [m /s/m ] Areal density of volumetric flux (for each fluid) 3 S1 , S2 [kg/m /s] Mass source terms: mass fluxes per unit 3 S1 , S2 [kg/s/m ] volume, or equivalently, volumetric flux densities (for each fluid).

p1 , p2 [Pa] Pressures of fluids 1 and 2

pC= p2−p1 = pNW–pW [Pa] Capillary pressure (non-wetting minus wetting pressure) xiv Capillary Flows in Heterogeneous and Random Porous Media 1

3 −1 K1, K2: Kα=kα(pc)/μα [kg/s/m ] Absolute mobilities, or equivalently, hydrodynamic conductivities, of each fluid (functions of capillary pressure) 2 k1(pc) , k2(pc) [m ] Absolute two-phase permeabilities for each fluid (m2) 2 k0 [m ] Intrinsic Darcy permeability of the porous medium (m2)

μ1 , μ2 [kg/m/s] Dynamic viscosities (properties of fluids 1 and 2).

Introduction

This introduction concerns both the present Volume 1 and the forthcoming Volume 2.

I.1. Introduction

This text, in two volumes, focuses on capillary equilibria and capillary flows (visco-capillary hydrodynamics) in the presence of solids and in porous media. The present chapter serves as a general introduction to both volumes. The first develops a statistical analysis for flow through discrete tubes or joints. The second, forthcoming, volume focuses more on geologic porous media on larger scales.

Several types of immiscible two-phase equilibria and flow systems are examined at various scales, in the presence of solid frames (cylindrical tubes and planar joints), and inside the pore space of porous soils, rocks and other materials. In the case of porous media, this leads to studying capillary effects in unsaturated water flow, and immiscible two-phase flow in homogeneous, heterogeneous, random and/or stratified porous media. Several problems of porous media hydrodynamics influenced by capillarity are treated in the last chapters of Volume 1 and then, at more length, in the subsequent Volume 2.

To sum up, in this text we focus on phenomena involving two fluid phases in the presence of a solid object of varying complexities, including a cylindrical wall (tube), a pair of smooth or rough solid surfaces (fracture), a statistical set of tubes and joints and a porous medium made up of many pores. The latter case may concern granular materials like sand, or, more generally, geological media like soils and rocks, where the grains may be cemented by a finer material (e.g. sandstone, cemented by precipitated calcite and other ions). xvi Capillary Flows in Heterogeneous and Random Porous Media 1

Geological porous media are often used as examples in this text, but, for instance, the first few chapters of Volume 1 do not exclude other interests related to industrial man-made materials. Also, various capillary phenomena not necessarily related to porous media are discussed and analyzed (films, bubbles, etc.). For instance, in this volume we present a study of the equilibrium of a soap film bounded by wire frames, and we also briefly examine the equilibrium of a gas bubble in water. The latter two topics are the only ones treated in this book that are not directly related to porous media; they are nonetheless treated for their interest in relation to capillary phenomena.

A few words on terminology. In this text, the term “two phase” refers only to the fluid phases, not the solid phase. We recognize, nevertheless, that solids constitute in fact a third phase. The two fluid phases are assumed immiscible in most of this text. One of the fluids is considered “wetting” and the other one “non-wetting” (in the presence of the solids). Examples of wetting/non-wetting pairs of fluids include water/air, water/oil and oil/gas. In most of this text, the solid phase is assumed immobile and non-deformable, or in some cases, only slightly compressible (elastic).

Another terminological issue concerns scales. We will use a loose terminology, which can vary depending on context. The “microscale” or “pore scale” may refer to individual pores, tubes or joints. “Mesoscale” may refer to the scale of the REV (representative elementary volume), also named the Darcy scale. “Macroscale” usually refers to larger scales at which the porous medium may be heterogeneous (randomly or otherwise).

The notion of a porous medium is introduced first as a “porous continuum” based on the REV concept, even though layered and fractured porous media – with discontinuities – are then considered as well. In that case, the porous medium is a continuum, though only between fractures or layer interfaces.

There are many other terminological variants (or aliases) in the literature. Let us provide just a few examples concerning capillarity. Interfacial tension is also named surface tension (measured in Newtons/meter). Capillary pressure is also named pressure jump (measured in Pascals). The latter is also related to negative Gibbs energy (volumetric density, measured in Joules/m3). And finally, this can also be converted to suction in meters (equivalent to water head).

At equilibrium, two-phase phenomena are dominated by capillarity and gravity (but not by viscosity, at least not directly). For this reason, these phenomena are designated as “capillary equilibria”.

On the other hand, under non-equilibrium conditions, at least one of the two fluid phases moves. Two-phase hydrodynamics depends on the combined effects of Introduction xvii capillarity, gravity, and viscosity, in the presence of solid phase of varying complexities. These effects intervene either at the “pore scale” (that is, the scale of pores, tubes, planar joints or fissures), or at the larger scale of a continuous “porous medium”. We will develop analyses at different scales, first for statistical systems of tubes and planar joints, and then, for randomly heterogeneous porous media such as soils and rocks. The emphasis will be on visco-capillary effects, and on the geometry of the solids. Statistical or stochastic approaches will be used to describe either discrete or continuous structures, depending on the scale of the analysis and on the type of heterogeneity (i.e. on the pore scale, over discrete layers, or the “continuum” scale of heterogeneity).

These topics are present in both Volume 1 and Volume 2. Statistical analyses of flow through discrete sets of tubes or joints are developed in the first volume. The second volume will focus more on geologic porous media on larger scales, with an overview on macroscale hydrodynamic effects due to heterogeneity, and on upscaling issues. We will focus first on unsaturated water flow in soils and rocks, and then on immiscible two-phase flows such as oil/water or gas/oil flow (as may occur in hydrocarbon reservoirs).

Many of the topics explored in this text are related, more or less directly, to the “upscaling problem”. Briefly, the term upscaling refers to the passage from a local scale description to a larger scale description, usually via an equational hydrodynamic model.

– Pore scale. The “most local” hydrodynamic model corresponds to a Navier– Stokes type description of the flow at the pore scale. – Mesoscale. A somewhat “less local” scale is the “mesoscale” of many pores (also named the REV scale or Darcy scale). The mesoscale can lead to a continuous representation of the porous medium and its hydrodynamic equations (Darcy’s flux– gradient law and mass conservation). In fact, Darcy’s law is strongly related to the “porous medium” concept. The REV is the finest spatial resolution at which Darcian flow equations can be defined. – Macroscale. Finally, if the Darcian porous material is heterogeneous, a larger “macroscale model” may be needed in order to describe flow at even larger scales. Indeed, porous media like soils, rocks, and other materials, may be heterogeneous on much larger scales than the pore or REV scale. This macroscale heterogeneity can be modeled as “random”, that is, with porosity and permeability represented as random functions of space (referred to as random fields). The random media approach will be developed in the text, and a more systematic presentation of random media and random fields will be given in the Appendices in Volume 2. xviii Capillary Flows in Heterogeneous and Random Porous Media 1

By definition, in the literature, the “first upscaling” is defined as the passage pore scale → Darcy scale, and the “second upscaling” is defined as the passage Darcy scale → macroscale.

Upscaling

An additional terminological note is in order concerning the term “upscaling”. This term is sometimes used interchangeably with “averaging” or with “homogenization”. Upscaling can indeed be viewed as a non-trivial type of averaging. It is also a procedure used to replace a heterogeneous medium by an equivalent homogeneous medium. The resulting “upscaled” equational model (and its coefficients) can be labeled variously as upscaled, macroscale, homogenized, effective or equivalent, depending on the mathematical technique employed, or the relevant hypotheses and/or objectives. Some authors insist on distinguishing “mathematical homogenization” from all other upscaling methods; and some authors feel strongly about the distinction between effective coefficients (which are more universal) and equivalent coefficients (less universal, and specific only to a given finite sample).

I.2. Organization of the text

The text is organized in two separate volumes as follows.

Volume 1 contains five chapters and the Introduction.

This chapter presents an introduction and outline of the book, briefly summarizing the contents of both volumes.

Chapter 1 presents basic concepts on fluids and on porous media, including qualitative introductions to porosity, Darcy-type laws, the notion of the representative elementary volume and the capillary jump occurring at the interface between two immiscible fluids.

Chapter 2 develops equilibria relations for two-phase systems, starting with the notion of capillarity and surface tension, solid–fluid relations, capillary rise, wetting angles, and also Kelvin’s equilibrium law relating air humidity to capillary pressure, and related thermodynamic concepts.

Chapter 3 develops calculations of capillary equilibria problems for various systems of cylindrical pores and planar joints. A statistical approach leads to characterizing the water retention curve of a porous medium based on the analogy between the distribution of tube diameters and pore sizes. In addition, this section Introduction xix presents another topic of capillary equilibrium: the equilibrium of a soap film bounded by metal wires, treated as a functional optimization problem.

Chapter 4 builds on the previous chapter by moving on to non-equilibrium problems, involving two-phase flows in systems of pores, or rather, tubes and joints. This chapter is split into several sections as follows. The chapter starts with the case of single-phase steady flow in tubes and joints. Poiseuille-type flow in a single tube/joint leads to Darcy-type flow on average for a system of tubes/joints (through Kozeny-type formulae for permeability). The case of unsaturated steady flow is then treated for a system of smooth or rough planar joints, where water flows in the presence of air in each tube or joint (water-filled or not). A simplified averaging over the set of joints leads to capillary water retention and permeability curves – both of which are functions of capillary pressure or suction. The next section analyzes transient visco-capillary dynamics of two-fluid phases along the axial direction in a tube/joint with an either constant or axially variable diameter (aperture); a system of differential equations is obtained and analysed mathematically. The next section moves on to two-dimensional transient flow, with a numerical simulation study of oil/water flow (drainage) in a rough joint with a random field aperture a(x,y). Finally, the last section furnishes brief indications on other relevant topics concerning capillary dynamics in fluid mechanics.

Chapter 5 develops the equational models and theoretical tools to deal with single-phase flow, unsaturated water flow, and more generally immiscible two-phase flows in porous media. Darcy’s law is used, along with its generalizations (Darcy– Buckingham–Richards, Darcy–Muskat). This approach assumes the existence of a representative elementary volume (REV) to describe the porous medium locally on average over many pores and grains. Otherwise, the porous “continuum” may be heterogeneous on scales larger than the REV scale. Relevant nonlinear models and their nonlinear constitutive relations are discussed (e.g. capillary moisture diffusion, unsaturated conductivity curve, capillary capacity, wetting and non-wetting conductivity curves, etc.). Philip’s theory of infiltration is presented (solving Richards equation for transient downward unsaturated flow).

Volume 2 contains five chapters, plus the Appendices for both volumes, as detailed below.

Chapter 6 develops a case study of soil moisture migration under line source irrigation in the presence of evaporation and root water uptake (an experimental and modelling study).

Chapter 7 presents a theoretical study of capillary-driven infiltration and ponding on a soil with randomly heterogeneous properties in the horizontal plane. A simplified version of ponding time, tp, is obtained from Philip’s infiltration theory. xx Capillary Flows in Heterogeneous and Random Porous Media 1

The soil heterogeneity is modelled using the Miller–Miller scaling theory, leading to a random field capillary scaling factor, λ(x,y) or α(x,y). The resulting ponding time is random too, and this causes the saturated areas to be distributed stochastically in the plane (x,y). The time evolution of these “wet patches” and of the corresponding “excess rainfall” are analyzed explicitly.

Chapter 8 focuses on the effects of soil or rock heterogeneity on unsaturated flow at various scales. It consists of a review, supplemented with models and interpretations, based on various experimental observations and numerical simulation results, both from our research and from the literature. This review focuses on effects such as capillary barriers, and suction-dependent anisotropy in moisture migration phenomena, and the consequences for macroscale unsaturated conductivity.

Chapter 9 presents a literature review and some models for upscaling unsaturated flow and two-phase flow in the presence of random heterogeneity and stratification of the porous medium. This advanced topic cannot be treated exhaustively in a single chapter, as it is a very active topic that produces a large amount of literature, both theoretically (regarding, e.g. multiple scales, volume averaging and perturbation methods), and experimentally or numerically (such as validations of upscaling theories). The main difficulty with unsaturated and two-phase flow is the combination of nonlinearity and heterogeneity. The upscaled equations may not resemble the original local scale equations based on Darcy’s law at the REV scale. This chapter, therefore, will only be a modest survey of selected approaches to the problems of nonlinear flow upscaling in heterogeneous media.

The Conclusion presents a summary of the concepts discussed and a global outlook on the field of capillary hydrodynamics.

The chapters of Volume 2 will be numbered sequentially as a continuation of the chapters in Volume 1. The present volume contains some cross-references to chapters and Appendices that will appear in Volume 2. A list of references and an index for the present volume are provided at the end of the book.

I.3. Objectives, contents, and readership

This book provides a reference on the following topics: (i) basic concepts related to surface tension and capillarity in fluid mechanics; (ii) calculation of capillary equilibria, beginning with the classical problem of water height in a capillary tube; (iii) calculation of capillary fluid displacement in pores and joints; (iv) porous media flows driven by capillary pressure gradients (unsaturated Darcy–Richards water flow, two-phase Darcy–Muskat flow); (v) the role of geometry and heterogeneity Introduction xxi

(including statistical systems of tubes or joints, random variability and layering in geological porous media).

The physical aspects of these fluid mechanics problems will be examined, along with mathematical solutions of equational models in a deterministic or stochastic framework. Scale issues will be examined in relation to observations and experiments, and upscaling techniques for obtaining macroscale equations or coefficients will be presented.

Overall, the contents of both volumes can be summarized as follows: – Basic concepts of porous media, including physical properties and flow models. – Two-phase physics of fluids, e.g. capillarity, interfaces, surface tension and thermodynamics. – Capillary equilibria: two-phase equilibria in tubes and joints, and soap film equilibria. – Capillary–viscous flows in tubes and joints (local scale or pore scale flows). – Darcy-scale capillary flows in heterogeneous porous media; the Richards and Muskat models. – Effects of soil/rock heterogeneity on unsaturated water flow, including observations and analyses. – Advanced topics on heterogeneity and upscaling for unsaturated and two-phase flows.

The common thread throughout this book is the effect of capillary forces (interfacial tension) combined with gravitational forces and viscous dissipation for two-phase fluid systems in the presence of solids. Another key link is the geometric complexity of porous media, which are heterogeneous at many scales. This heterogeneity affects the configuration and the flow of pore fluids at various scales (e.g. capillary rise or invasion in variable tubes, the capillary barrier effects in soils, moisture-dependent anisotropy, water ponding at layer interfaces and the soil surface, etc.).

Intended readership

This book is intended for a broad readership, including scientists and engineers interested in an introductory presentation on capillarity, capillary equilibria, and the flow of two immiscible fluids through porous media. More generally, we consider two-phase capillary equilibria and two-phase visco-capillary flows in the presence of xxii Capillary Flows in Heterogeneous and Random Porous Media 1 solids, such as grains, fibers and fracture walls. Multiphase flow phenomena are influenced mostly by capillarity, viscosity, and gravity (the term “visco-capillary flow” is used here).

Several of the topics presented in this book come from lectures given by the author 1 in science and engineering, sometimes at third-year level but mostly at fourth- and fifth-year levels2. Several of the courses taught bear some relation to the topics and materials of this book. Here is a short list3: – Junior level (third year): the course “Introduction to Optimization” (in French), the second part of which was devoted to functional optimization (variational optimization), and in which the author first developed the soap film case study that is presented in this book. – Bachelor level (fourth year): the course “Porous Media Hydraulics” for engineering students in hydraulics, focusing on Darcy’s law but also including a specialized topic on infiltration in unsaturated soils. – Master level (fifth year): “Subsurface Hydrology” for engineers, and “Single-phase and Two-phase Fluid Dynamics in Porous Media” for Master students.

Brief guide to the topics of this book

The first few chapters of this Volume 1 start with capillary equilibria, focusing on static fluid–fluid as well as fluid–solid interactions: see section 1.5, the entirety of Chapter 2, and most of Chapter 3, particularly section 3.2 (single capillary tube), section 3.4 (system of random tubes), and section 3.5 (soap film equilibrium). Similarly, the reader may be interested in simple analytical solutions for some special flow systems dominated by capillarity, such as the visco-capillary flows in individual tubular pores or planar joints, as treated in Chapter 4, for instance in

1 Courses taught mostly at the Department of Hydraulics, ENSEEIHT School of Engineering, Institut National Polytechnique de Toulouse (INPT), and at Paul Sabatier University. Both institutions are part of the University of Toulouse, France. 2 Teaching systems vary worldwide; we use here a simple scale in which the term “third year” means the third year after graduating secondary school. In France, the third year of engineering at a Grande École is actually the fifth year of study, as students first complete two preparatory years. In US universities, the first, second, third and fourth years are called the freshman, sophomore, junior and senior years respectively, and the fifth year usually corresponds to a Master degree (sometimes a sixth year). 3 The material presented in this book has also been influenced by other curricula, such as “Mathematics for Engineers”, and “Stochastic Processes”, which the author taught at the Junior and Bachelor levels respectively, in collaboration with colleague W. Bergez at the Department of Hydraulics at ENSEEIHT-INPT. Introduction xxiii section 4.2.3 (classical Poiseuille flow in a single tube or joint), and in section 4.4, where the transient one-dimensional motion of two-phase flow is studied quasi- analytically in a tube or joint under the competing effects of capillarity and viscosity. The remaining chapters deal more directly with porous media: Chapter 5 in Volume 1, and the chapters in Volume 2.

The Darcy flow PDEs (Partial Differential Equations), which are presented extensively in this text, are usually treated in science and engineering curricula at the Bachelor and Master levels. However, some porous media topics presented in the last chapters of Volume 2 become more advanced (Master to PhD level), as we move progressively to scale issues involving stochastic approaches to heterogeneous porous media.

Overall, the material in this book combines basic two-phase problems (where mathematical solutions can sometimes be obtained explicitly), with more involved topics that will be of interest to scientists and engineers looking for more advanced topics on fluid dynamics in heterogeneous media (e.g., pore systems, planar -fractures, and porous soils and rocks with three-dimensional heterogeneity).

For instance, the simplified case of one-dimensional transient two-phase fluid displacement in a tube (section 4.4) is later revisited numerically in two dimensions for the case of two-phase drainage in a variable aperture planar fracture (section 4.5).

The analysis of steady flow in an individual tube or joint is supplemented by an upscaling calculation over a set of many tubes or joints, which we refer to as the first upscaling problem). This is presented in section 4.3.2 for unsaturated water flow, and in section 4.3.3 for immiscible two-phase flow4.

Ultimately, this book could serve as a basic reference on capillary, but also should be of interest for those who wish to focus on more advanced topics of fluid dynamics in porous media, particularly those involving unsaturated water flow and immiscible two-phase flow (water/air, water/oil, or oil/gas) in geologic media. Furthermore, the topics treated in this book are relevant in a broad range of applications: hydrology, environmental pollution and remediation, mining

4 “First upscaling” refers to the problem of upscaling from a discrete pore/grain system to an equivalent continuum; by extension we use the same term to refer to upscaling (averaging) over a set of many discrete tubes or joints. “Second upscaling” refers to another problem – upscaling from a spatially variable continuum (porous medium) to an equivalent homogeneous medium. The term “homogenization” is also used for the second upscaling problem, although in a more specific mathematical sense (usually based on a multiple scale approach). xxiv Capillary Flows in Heterogeneous and Random Porous Media 1 engineering, petroleum reservoir engineering, underground waste disposal, and in many other areas not necessarily limited to the earth sciences and geologic media.

Finally, we list below a few more practical points about the process of browsing through this book.

The reader will find several thematic literature reviews and historical notes, throughout this text. Some of the historical notes are collected in the Appendices to appear in Volume 2. The history-inclined reader, with an interest in epistemology, is invited to browse through the entire book and the first Appendix of Volume 2, where they will find specific references to modern as well as historical literature. See also the index entry “history”.

Some topics are treated in the Appendices provided at the end of the forthcoming Volume 2. Some of the shorter appendices are mere complements to the text. Others are longer and totally self-contained (for example, the Appendix on random media and random fields).

I.4. Acknowledgments

The author of this text has benefited from advice, collaboration, and contributions from a number of researchers over the years. Specific contributions are mentioned in the relevant sections. Here we wish to acknowledge generally those researchers (advisors, contributors and collaborators), with apologies to any persons whose names might have been inadvertently overlooked.

Concerning unsaturated hydrology (unsaturated flow analyses, parametrization, modeling, coupling, upscaling), the author wishes to acknowledge previous contributions and collaborations on these topics with a number of researchers over several decades through co-authored papers, technical reports, research projects, graduate theses and various internships, including G. Vachaud, M. Vauclin and J.-L. Thony† (LTHE Grenoble); L.W. Gelhar, D. McLaughlin and D. Polmann (MIT Cambridge); A.C. Bagtzoglou, G. Wittmeyer and B. Sagar (SwRI/CNWRA, San Antonio, Texas); A. Poutrel (ANDRA, France), G. Trégarot, Y. Wang, A. Alastal and N. Mansouri (former PhD students); Veena S. Soraganvi, M.S. Mohan Kumar and Muddu Sekhar (Indian Institute of Science, Bangalore, India), D. Astruc and M. Marcoux (colleagues at IMFT Toulouse); and L. Orgogozo (OMP/GET, Toulouse).

On other topics, such as immiscible two-phase flow in homogeneous and randomly heterogeneous media, the author acknowledges early collaborations with R. Lenormand (then at Institut Français du Pétrole); A. Fadili (then PhD student at Introduction xxv

IMFT and IFP); M. Fourar and G. Radilla (then at LEMTA, Nancy, France); M. Spiller (then graduate student at IMFT, Toulouse, France and RWTH Aachen, Germany).

The author also acknowledges other fruitful discussions and useful indications on various topics (geo-fluids, geochemistry, capillarity, non-aqueous phase liquids, evaporation, upscaling issues, etc.). The non-exhaustive list includes J.-M. Matray (research engineer at IRSN Fontenay-aux-Roses), and IMFT colleagues P. Duru, M. Prat and M. Quintard, among others at IMFT Toulouse.

The author also wishes to acknowledge previous support from the following institutions, from the 1990s through to 2015, in the framework of research projects on unsaturated hydrogeology, pollution and oil reservoir engineering, including (from the 1990s through 2015) US.NRC, CNWRA (SwRI), GIS HydroBAG (Agence de l’Eau Adour-Garonne), GIP Hydrosystèmes, IRSN/CNRS (GNR TRASSE), ANDRA (Programme UPS4 Simulation mécanique), Institut Français du Pétrole and the French Centre National de la Recherche Scientifique (CNRS) for their funding of several research activities. The Porous Media group at the Institut de Mécanique des Fluides de Toulouse (IMFT) has hosted my research on flow and transport in heterogeneous geologic media for the last two decades; its constant scientific and material support is gratefully acknowledged.

The author is indebted to these researchers and institutions, and apologizes for any possible oversight in the previous lists. This author bears all responsibility for any possible glitches, errors or misinterpretations in the present text.

My wife and three children have had to bear with me throughout the years while I was attempting to assemble this text, and of course, they also had to bear with me the rest of the time too! Thank you all for your patience and understanding.

1

Fluids, Porous Media and REV: Basic Concepts

This chapter presents, in a qualitative fashion, several basic concepts on fluid dynamics in pore systems and porous media, etc., including for instance the concept of the representative elementary volume (REV), and related mesoscale quantities like porosity and pore pressure.

Boxes 1.1 and 1.2, at the end of this chapter, provide overviews on the flow regimes and scales of analyses relevant to this text. Specifically, Box 1.1 presents the various types of flow systems, flow regimes and porous structures to be studied throughout this book, and Box 1.2 seeks to clarify the scales of analyses considered in this text.

1.1. Geologic porous media: basic concepts

Examples of geologic porous media are depicted in Figures 1.1, 1.2 and 1.3, to be found below. Let us first emphasize – without attempting any systematic classification – some useful distinctions between several types of geologic porous media.

1.1.1. Porous soils

Soils are complex porous media that may contain various proportions of sand, silt, clay and organic matter, and several types of pores, grains, aggregates, etc. Topsoils (up to perhaps the first few centimeters or decimeters below the surface) are usually structured quite differently from deeper soils.

Capillary Flows in Heterogeneous and Random Porous Media 1, First Edition. Rachid Ababou. © ISTE Ltd 2018. Published by ISTE Ltd and John Wiley & Sons, Inc. 2 Capillary Flows in Heterogeneous and Random Porous Media 1

Soils are often layered (thick layers), and can also be more finely stratified (possibly within layers). Soil stratification is often imperfect and irregular, and “cross-layering”, with interwoven strata can occur. Similar features can occur also for various consolidated sedimentary formations in the subsurface (e.g. sandstone).

In addition, soils can be vertically fissured, for example in dry conditions, due to the occurrence of dessication cracks near the surface, or, they can be fissured multidimensionally due to plant roots and worm holes, particularly in temperate and wet climates.

1.1.2. Porous rocks

Porous rocks include granite, gneiss, sandstone, claystone and carbonates (carbonaceous rocks). Rock formations can be fractured to varying degrees on various scales, and they can sometimes be highly altered and weathered, both mechanically and biochemically. For example, saproliths may be very altered (saprolites) or only slightly (saprocks), and highly fissured carbonaceous formations containing cavities, conduits and fissures of many sizes and shapes are called karsts.

Regarding their usage, we can also distinguish aquifers (containing water) and hydrocarbon reservoirs (containing oil and gas).

Aquifers Sand and gravel aquifers are permeable and porous water-bearing formations, used for water exploitation by groundwater extraction wells.

We may distinguish different cases in practice: (i) relatively shallow unconfined aquifers, with a free surface (water table) and (ii) deep, confined aquifers, sometimes only a few meters thick and located tens or hundreds of meters below ground.

The hydraulics of pumping in shallow unconfined aquifers are quite complex, because the drawdown of the free water table occurs in the unsaturated air/water zone. Complex phenomena like seepage can occur above the water level, particularly in the pumping well. Such water/air flows occurring under partially saturated–unsaturated conditions are difficult to model.

Porous oil and gas reservoirs If a reservoir is sufficiently permeable and the oil is sufficiently light and not too viscous (for example, oil or gas trapped by a cap rock in a sandy reservoir), then exploitation is done classically by pumping – injecting water and other products, then extracting oil along with the other injected fluids and products. Fluids, Porous Media and REV: Basic Concepts 3

Otherwise, special exploitation methods are used to extract heavy crude oil or bitumen (from bituminous shales, tar sands, etc.). Examples of such “enhanced oil recovery” techniques include ISC (in situ combustion) and SAGD (steam-assisted gravity drainage). Let us briefly describe these two techniques. – ISC generates heat that increases oil temperature and diminishes oil viscosity; the generated gases (carbon dioxide) contribute to displacing oil towards production wells. – SAGD is a form of in situ steam stimulation based on a pair of horizontal wells (a lower well and an upper well just a few meters apart). The high pressure steam is continuously injected into the upper wellbore to heat the oil and reduce its viscosity; the heated oil drains into the lower wellbore, where it is pumped out.

If the reservoir is a naturally fractured porous rock, the fractures may contribute to the flow of oil as high transmissivity “conduits”. However, the volumetric fraction of oil in the fractures may be insufficient compared to that in the porous blocks. On the other hand the porous blocks may be less permeable, implying a longer relaxation time to extract oil from the matrix. Thus the exchange between porous blocks and fractures may be a limiting factor for oil exploitation. For this reason, the fracture–matrix exchange phenomenon has been intensively studied in the literature, usually based on a dual medium approach with a fluid exchange term between the porous matrix and the fractures.

Recently, the technique of “fracking” has been developed for enhancing oil exploitation operations in rock reservoirs (fracking is already used on a fairly large scale in the USA). It is done by generating or enhancing rock fractures by hydraulic pressure and/or by other means. Water is injected under pressure along with other fluid(s), solid products (e.g. fine grained powders) and chemicals (the exact nature of which are usually kept confidential by the industry).

1.1.3. Geologic porous media: examples

Let us now comment upon Figures 1.1, 1.2 and 1.3, which illustrate some of the above described geologic media.

Figure 1.1 shows a sand/gravel fluvio-glacial deposit, illustrating the stratification, cross-bedding, lenticular structures and imperfect layering that can occur in a sedimentary geologic formations. There are many examples of such sedimentary geologic structures in the near subsurface. 4 Capillary Flows in Heterogeneous and Random Porous Media 1

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

For example: Jussel et al. (1994) describe the structure of fluvio-glacial deposits (silts, sands, gravels, bimodal gravels, etc.) from gravel pits near Zurich, Switzerland. They carefully analyzed the geometric structures at various scales, and devised a method for synthetic generation of the medium. Their method combines a Boolean approach with randomly oriented ellipsoidal “lenses”, and a continuum random field approach with spatially correlated permeability within lenses.

Concerning “random fields”, see the relevant Appendix in Volume 2, in which a section is devoted to statistically anisotropic random fields. These may be used for representing “imperfectly stratified” geologic media, such as sedimentary deposits.

Figure 1.2 shows a photograph of a saprolite (here, a highly altered fractured gneiss) lying beneath a layer of red clayey soil, and overtopping a fractured gneiss rock at depth (not visible in the photo). This saprolite is soft enough to yield under the pressure of a finger. This is a case of sharp spatial variation of the subsurface porous medium, occurring over a fraction of a meter vertically.

This photo was taken by the author in 2005 at an instrumented site of the Moole Hole basin (coordinated by the Indian–French Cell for Water Resources and Indian Institute of Sciences of Bangalore). The Moole Hole basin is located in the Kerala/Karnataka region, near the city of Mysore in southern India. Fluids, Porous Media and REV: Basic Concepts 5

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 shows a physical model that reproduces, at laboratory scale, a vertical slab of groundwater flow in a heterogeneous aquifer system. Other useful information to note about this image follows. – The porous medium contains several “layers”: a fine sand layer overtopping a coarse sand layer; a bottom clay layer; and an upper clay “lens” within the fine sand. – On the right, the water table is located above the clay lens, as indicated by the water level in the dye injection tube. – On the left, the water table coincides with the water level in the “river”. – The fine–coarse sand interface is water-saturated, as it is located below the water table. – The fluorescent dye serves as a tracer for the saturated flow system (it was initially injected below the water table). As can be seen, the tracer plume splits into two sub-plumes at the fine/coarse sand interface (see also the discussion in Ababou 2008). 6 Capillary Flows in Heterogeneous and Random Porous Media 1

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

It should be emphasized that material interfaces play an important role in subsurface hydraulics (like soil hydrology or hydrogeology). A material interface corresponds to a sharp contrast of porous medium properties; such interfaces are clearly visible in Figures 1.1, 1.2 and 1.3. In the case of unsaturated porous media, a material interface can create a capillary barrier to water flow due to capillary effects in which a coarser porous layer can block flow at low water content. This will be discussed further in Chapter 2.

1.2. Porous media: basic concepts, porosity and specific area

This section constitutes a brief preliminary introduction to the concept of a porous medium, which will be useful later on when investigating Darcy-type flows in porous media. Recall also that Boxes 1.1 and 1.2 (found at the end of the chapter) provide overviews on flow regimes and on scales of analyses. Box 1.1 presents flow systems, flow regimes, and pore structures to be studied throughout this book, and Box 1.2 seeks to clarify the various scales of analyses considered in this text.

The term porous medium designates, loosely, a solid material (geological or man-made) that is constituted of many pores and “grains” (or other solid components like lamella, fibers, cement, etc.). The usual point of view is that, in a Fluids, Porous Media and REV: Basic Concepts 7 porous medium stricto sensu, the “voids” between the solids are well connected. Classically, a porous medium can be defined by the following two characteristics. 1) A connected network of many pores of various sizes and shapes.

The total volume fraction of pores is called porosity. It should be noted that a fine-grained medium can have large porosity (e.g. over 50% for clayey materials), and conversely, a coarse-grained medium can have small porosity (e.g. 15% for coarse sand and gravel). In addition to porosity, the pore size distribution provides a more detailed characterization of the medium. Grain size distribution can serve as a surrogate for pore size distribution, particularly in the case of granular materials.

2) A large specific area of solids per unit volume.

Specific area can be defined, hydrodynamically, as the total amount of solid/fluid contact areas per unit volume of space (assuming the medium is saturated with a single fluid). This property, the specific area, is also useful for characterizing statistical sets of tubular pores or joints, towards a mesoscale characterization of the resulting porous medium. For more details, see section 4.2.2.

1.2.1. Fluid phases

Concerning the fluid phases: the fluids contained in the pores can be liquid water, oil, air, and/or various vapors and gases other than air. If water entirely fills its pores, the medium is said to be “water saturated” (or simply saturated). If the pores are entirely filled with air at atmospheric pressure, the medium is said to be totally “dry”. More generally, the pores can contain two or more “fluid phases”, for example water and air; water and oil; water and oil and gas, etc.

1.2.1.1. Friction and Reynolds number In a porous medium, flow takes place in the presence of many solids, and therefore flow can be greatly impeded by friction on the numerous solid/fluid interfaces. This internal friction depends on the geometry of the solid/s, and also on the viscosity of the fluid/s. Due to this significant amount of friction, the resulting velocity is usually small in a porous medium. This implies that the Reynolds number Re is usually small (Re < 10 or so).

1.2.1.2. Additional notes on Reynolds–Stokes–Poiseuille, in relation to porous media The Reynolds number Re is a dimensionless number defined in classical fluid mechanics as the ratio of viscous dissipation time scale (tD) over advection time 8 Capillary Flows in Heterogeneous and Random Porous Media 1

scale (tV), both calculated for the same length scale L, which characterizes the flow domain.

See Chapter 4, and in particular section 4.2.3, which is devoted to single-phase Stokes–Poiseuille flow in tubes and joints. The classical Reynolds number characterizes the relative importance of inertial and viscous effects. Usually it is expected that Re << 1,000 is required in order for the symmetries of one-dimensional Poiseuille flow to be maintained.

See section 5.3.2 for the definitions of various quantities like the classical Reynolds number, the concept of Darcy velocity (or flux density q), and a modified version of the Reynolds number adapted to flow in porous media (the Reynolds-Darcy number).

1.2.1.3. Extended definitions of porous media from a hydrodynamic standpoint Let us now briefly consider in a qualitative fashion the “intrinsic” part of friction (that is, the part of friction that depends only on the geometry of solids). A useful intrinsic measure of friction within a porous medium is the specific area of the 2 3 fluid/solid contacts, as defined above. This is denoted by a or σS (m /m ) (not to be confused with surface tension). For example, the total contact area between water and solids in a bathtub filled with silt grains1 of diameter 10 μm can exceed 10 hectares (100,000 m2). A calculation based on a simple cubic arrangement of spherical grains yields a specific area of 32 ha/m3.

Finally, we note that, for practical and methodological reasons, it is useful to include in the field of porous media hydrodynamics, any situation where fluid flow takes place in the presence of many solids, such that the flow is strongly confined or impeded by solids, resulting in relatively low velocities (around Re < 10 or so). This extended notion of porous media hydrodynamics includes the following cases: – a classical, permeable porous medium constituted of pores and grains (the grains may be cemented or not, they may be aggregates of finer grains, etc.); – a fibrous medium, either densely packed like wood, or very loose as may occur in river beds or top soils, where dead leaves, grass, algae, etc., form obstacles to water flow; – a micro-fissured rock or material like concrete, with a number of very tight joints;

1 A silt is a very fine loam, with grain diameters ranging from 2 μm to 20 μm. An example of silt material is loess, which can be transported by winds over long distances. It originates from arid parts of China, through sand abrasion by wind. The Yellow River in China takes its name from the yellow color of loess, which it carries as suspended matter. Fluids, Porous Media and REV: Basic Concepts 9

– a single planar fracture consisting of a pair of rough walls, with variable aperture; – a set of planar fractures of various sizes and apertures (parallel or interconnected set); – a single capillary tube of variable diameter; – a single rough planar joint or fracture with variable aperture in the (x,y) plane; – a set of cylindrical tubes (or indeed a bundle of parallel tubes or pores, or a network of intersecting tubes or pores); and – a set of tubes or planar joints embedded in a permeable and porous matrix.

Having defined a porous medium, we introduce in the next section the basic concepts associated with Darcy’s law for the case of single phase flow in a porous medium. However, the reader is invited to browse the more advanced Chapter 5, particularly sections 5.2 and 5.3.2, for more details on the concepts of porous media, scales of analyses, the REV (representative elementary volume), and Darcy’s law (including a definition of the Reynolds number adapted for porous media). Furthermore, sections 5.3 and 5.6 will present extensions of Darcy’s porous media law for unsaturated water flows, and for more general two-phase flows, respectively.

1.3. Single-phase flow and Darcy’s law: basic concepts

At relatively coarse scales, called Darcy scales, the porous medium can be viewed as a continuum such that individual pores and grains are no longer distinguished. Each point in space (x,y,z) represents the centroid of a “representative elementary volume”, or REV. The REV should ideally be as small as possible but large enough to contain many pores and grains. The geometry of the pores does not need to be explicitly known. Accordingly, flow through the pore system is not described in detail at the pore scale, but only at the larger, “mesoscopic” scale of the REV. The flow parameters and variables, defined at the REV scale, include porosity, permeability, pressure, flux density (often improperly named Darcy velocity), volumetric fluid content, and others.

1.3.1. Darcy’s flux–gradient law

In this framework, let us first consider the flow of a single fluid phase (like water) through a porous medium. Darcy’s law (Darcy 1856) is generally used to describe such a flow, without direct representation of the pores. Briefly, Darcy’s law is a linear flux–gradient relation, of the form q = −(k/μ) grad(P), where k is the intrinsic permeability of the porous medium (or porous matrix), and μ is the fluid’s 10 Capillary Flows in Heterogeneous and Random Porous Media 1 dynamic viscosity. Thus, Darcy’s law states that the flux density of the fluid (q) is proportional and opposite to the gradient of total pressure (grad P). Darcy’s law is analogous to: – Ohms’ law for electrical conduction, – Fourier’s law for heat conduction / diffusion, – Fick’s law for solute diffusion in a solvent, and – Onsager’s general thermodynamic flux-potential relations.

Darcy’s law will be presented in more detail later in this text, in section 5.3.2. We will explain there, briefly, how Darcy’s law emerges from the Navier–Stokes equations of fluid mechanics. For direct references on Navier and Stokes, see Stokes (1845) and Barré de Saint-Venant and Navier (1864). See also Hosch (2015) for a brief historical account on the emergence of the Navier–Stokes equations, and Moffatt (2005) for a historical essay on the work of Stokes (and Kelvin).

Finally, the reader is also referred to the classical reference textbook of Bear (1972), where saturated groundwater flow in aquifers, unsaturated flow in soils, two-phase flow in oil reservoirs, and many other topics pertaining to hydrology and hydrogeology (e.g. variable density flows, or freshwater/saltwater flow in coastal aquifers), are treated from an equational point of view2.

1.4. The Darcy–Buckingham law and the Richards equation: basic concepts of unsaturated flow

Water flow actually takes place in the presence of air in many practical applications: – in soils (watershed hydrology, subsurface hydrology); – in underground excavations and galleries (mining engineering); and – in civil engineering works (seepage through earth dams or other man-made structures).

In many cases, however, it is reasonable to simplify the “two-phase flow” of water/air by neglecting air dynamics, while still taking into account the presence of air via the variable water content (and the variable air content) of the porous medium. This leads to the “unsaturated flow” model of Richards (1931), a mass conservation equation for water flowing in a porous medium in the presence of air.

2 We will return to more specific hydrologic and hydrogeologic topics in later sections. Fluids, Porous Media and REV: Basic Concepts 11

The Richards equation is based on Darcy’s law as modified by Buckingham (1907) to account for variable water content in the presence of air (as occurs in soils).

1.4.1. Remarks on unsaturated water flow

In the unsaturated flow model based on the Richards equation, air is assumed to be constantly in contact with the atmosphere. It is assumed that, during water motion, the pressure of air equilibrates rapidly to atmospheric pressure because of its low viscosity. Therefore, the gas phase is essentially “passive” in this simplified representation of water/air flow.

On the other hand, even though air pressure remains fixed, it should be noted that the coefficients of the Richards equation depend on the (variable) water content θw (x,t). Since air content is porosity minus water content, we see that air content will influence water flow in Richards’ unsaturated flow model.

Since the total water plus air content must always be equal to porosity, only one variable is needed, either water content θw or air content. Usually, water content θw is the variable used, and it is often denoted θ.

The unsaturated flow model is very useful in hydrology (soils, watersheds) and in other areas (hydrogeology, civil engineering). Several chapters in this text will be devoted to unsaturated flow models, with particular attention to capillary effects, heterogeneity, and upscaling (see section 5.4, and also Chapter 7 in Volume 2).

In summary, the unsaturated flow equation of Richards is a simplified two-phase model describing the flow of water in the presence of air in a porous medium. To account for the effect of water content on “unsaturated” flow, Darcy’s law is extended to the Darcy–Buckingham law.

An even more general law (the Darcy–Muskat law) describing two-phase flow for water/air or water/oil is introduced in section 1.5.

1.5. Capillarity and two-phase flow systems at different scales: basic concepts

1.5.1. Introduction

This section introduces the basic concepts for multiphase systems of immiscible fluids in the presence of many solids (such as grains or solid surfaces). At the microscale of pores, tubes or joints, the key concepts for multiphase fluid systems are: 12 Capillary Flows in Heterogeneous and Random Porous Media 1

– the hypothesis of immiscibility between two fluids, and – capillary forces, and the capillary “pressure jump” between the two fluids.

Moving now from the microscale to the continuum mesoscale (or REV scale), the Darcy–Muskat model emerges, which governs two-phase flow in a porous continuum containing many pores and grains. This model is expressed in terms of mesoscopic variables like fluid velocities, fluid pressures and capillary pressure.

In fact, we are particularly interested here in two-phase flow problems, where there are only two “immiscible” fluid phases present in the pore system. We distinguish a wetting phase (water) and a non-wetting phase (air or oil). Each fluid phase has its own viscosity and density3.

As observed earlier, a special instance of two-phase flow is the case of “unsaturated flow”, in which water flows in the presence of air in the porous medium, while the air phase is fixed at atmospheric pressure. This has many applications in soil hydrology, watershed hydrology and hydrogeology.

Let us continue with our introductory description of the main concepts needed to understand capillary effects in two-phase flows. By the same token, we will also encounter for the first time the problem of passing from pore scale flow (tubes, joints) to continuum scale flow (Darcy–Muskat).

1.5.2. Capillarity pressure jump at different scales

1.5.2.1. Capillary pressure at the pore scale At the pore scale, the two immiscible fluids are separated by a geometrically complex set of interfaces, bounded by solids (grains, rock walls, etc.). The movement of these interfaces is governed by local fluid pressures, and by the pressure jump that occurs across the interface(s). This pressure jump is also called capillary pressure, denoted pC.

pC(x)=pN(x) − pW(x) [1.1] where pN is pressure in the non-wetting fluid, pW is pressure in the wetting” fluid, and x is a point located on any of the wetting–non-wetting interface/s Γ(t).

This pressure jump is due to the interfacial tension σ, also called the surface tension. The pressure jump depends on σ and on the geometry and curvature of the

3 “Density” here usually means the volumetric mass density (kg/m3), unless stated otherwise.