9781848215283

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Capillary phenomena occur in both natural and human-made systems, from equilibria in the presence of solids (grains, walls, metal wires) to multiphase flows in heterogeneous and fractured porous media. This book, composed of two volumes, develops fluid mechanics approaches for two immiscible fluids (water/air or water/oil) in the presence of solids (tubes, joints, grains, porous media). Their hydrodynamics are typically dominated by capillarity and viscous dissipation.

This first volume presents the basic concepts and investigates two-phase equilibria, before analyzing two-phase hydrodynamics in discrete and/or statistical systems (tubular pores, planar joints). It then studies flows in heterogeneous and stratified porous media, such as soils and rocks, based on Darcy’s law. This analysis includes unsaturated flow (Richards equation) and two-phase flow (Muskat equations).

Overall, the two volumes contain basic physical concepts, theoretical analyses, field investigations and statistical and numerical approaches to capillary-driven equilibria and flows in heterogeneous systems

Rachid Ababou is Professor at the University of Toulouse – Institut National Polytechnique, France. He holds a Dr.-Ing. in Fluid Mechanics (Grenoble) and a PhD in Civil Engineering (MIT). He researches flow and transport in random porous media at the IMFT laboratory.

1 Introduction and outline

2 Pore scale capillary air/water systems at equilibrium, steady flow, and dynamic flow regimes in tubes and joints

2.1 Quasi-static equilibrium and the macroscopic ??(??) moisture retention curve: a simple analytical example (random bundle of tubes, uniformly distributed radii)

2.1.1 Static equilibrium in a single vertical tube ("pore")

2.1.2 Static equilibrium in a statistical system of "pores" represented by a bundle of vertical tubes: construction of macroscopic moisture retention curve ??(pC) or ??(??)

2.2 Steady-state Poiseuille flow in a bundle of tubes filled with water (no air): simplified analyzis, leading to Darcy's law and Kozeny-Carman permeability

2.2.1 Specific area (general concept + application to bundle of tubes)

2.2.2 Poiseuille flow

2.2.3 From Poiseuille flow to Kozeny-Carman

(the various forms of K-C + discussion of dimensionless constant)

2.3 Steady-state Poiseuille / capillary flow of water in an unsaturated set of planar joints (air/water system): scale analyses leading to relations between porosity (??), permeability (k), and capillary length scale (??c)

2.4 Steady water flow in statistical sets of tubes/joints with variable apertures or constrictions: macroscopic ??(pc) and K(pc) relations for parallel or series systems

2.5 Pore scale dynamics in 1D: immiscible 2 phase “visco-capillary” flows in tubes and joints with uniform or variable radii/apertures (geometrically simplified “quasi-1D” approach)

2.6 Pore scale dynamics in a planar joint with randomly variable 2D aperture field a(x,y): transient drainage under the action of capillary forces & viscous dissipation (2 phase wetting/non wetting flow system)

3 Darcy scale and macroscale capillary flows with Richards/Muskat models (in heterogeneous media and statistical continua)

3.1 Introduction and summary (scales, REV concept, etc…)

3.2 Continuum equations for unsaturated “Darcy-Richards” flow and 2 phase “Darcy-Muskat” flow in spatially variable porous media

3.2.1 Darcy’s law for single-phase flows (including a slide "from N-S to Darcy")

3.2.2 Generalized Darcy’s law for two-phase flows: Darcy-Muskat

3.2.3 From Darcy-Muskat to Darcy-Richards(-Buckingham)

3.3 Observations on capillary effects in heterogeneous unsaturated media (statistical continua) and macro-scale flow behavior: review+analyses

3.3.1 Introduction and overview (…)

3.3.2 Applied context of the study, literature review (and acknowledgments)

3.3.3 Review of macro-scale behavior of unsaturated flow (moisture migration) in randomly heterogeneous/stratified geologic media (rocks, soils): theoretical findings (macro-permeability Kii(??)) and experimental evidence on nonlinear anisotropy

3.3.4 Macro-scale analyzis of unsaturated flow in randomly heterogeneous/stratified geologic media: parametrization of heterogeneity relation between saturated hydraulic conductivity Ks(x) and scale factor ??(x) of local permeability curves Kii(pC,x)

3.3.4.1 Spatial distribution of conductivity-suction curves, cross-correlations, and consequences on flow paths at low vs. high suction (capillary barrier effect at high suction)

3.3.4.2 Anisotropy of macro-scale flow: anisotropy of the nonlinear macro-permeability Kii(??) in randomly heterogeneous and/or randomly stratified soils ?? this to be moved in the next section

3.4 Upscaling unsaturated flow equations in randomly heterogeneous or stratified media: effective macroscale relations (??(pC), K(pC)), and emergence of a capillary number for statistical continua

3.4.1 Review of some upscaling models for (??(pC), K(pC)), and their applications to unsaturated flow in randomly heterogeneous/stratified geologic media

3.4.2 The Power Average Model of Ababou et al. (1993), and the resulting nonlinear anisotropic macro-scale Kii(pC) or Kii(??) curve for randomly heterogeneous/stratified media

3.4.3 Behavior of upscaled permeability Kii(??): the role of capillary length scale, geometric fluctuation scale, and dimensionless "capillary number”

3.5 Capillarity, sorptivity, and localized ponding over a heterogeneous soil surface during infiltration: a simplified stochastic analysis of the genesis of ponding with randomly variable scaling parameter

3.5.1 Introduction and summary (case of stochastic infiltration on a randomly permeable soil surface)

3.5.2 Internal ponding at a material interface (between two layers)

3.5.3 Localized surface ponding under point or line source flux

3.5.4 A simplified model of ponding time under fixed rainfall rate

3.5.5 A simplified scaling model of soil heterogeneity in (x,y)

3.5.6 Stochastic analysis of space-time distributed ponding on a heterogeneous soil surface (analytical results on the evolution of excess rainfall, wet area, and wet patches)

3.6 Immiscible 2-phase capillary flows in stratified and/or randomly heterogeneous media: Darcy-Muskat upscaling / statistical continuum approach

3.6.1 Introduction, summary, and brief literature review…

3.6.2 Steady state 2 phase flow in statistically heterogeneous 1D and 3D porous media under zero-mean capillary gradient: effective macro-scale ??(pc) & K(pc) curves

3.6.3 Transient 2 phase flow in statistically heterogeneous 1D (stratified) porous medium: macro-scale hysteresis and kinetic effects on ??(pc) and K(pc) curves due to heterogeneity

4 Recapitulation, conclusions, and outlook

5 Appendices

5.1 Appendix A1: Summary of governing flow equations (PDE's) at various scales (pore systems, planar joints, continuous porous media)

5.2 Appendix A2: Random numbers, spatial point processes (Poisson point processes), and spatially correlated random fields (covariance functions, spectral densities, Wiener-Khinchine theory)

6 Reference

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Capillary Flows in Heterogeneous and Random Porous Media

Capillary Flows in Heterogeneous and Random Porous Media

1848215282

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