Capillary Flows in Heterogeneous and Random Porous Media

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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