and Permeability 215
Viscosity describes the shear stresses that develop in a flowing fluid. V z Stationary Fluid Velocity Profile x Shear stress in the fluid is proportional to the fluid velocity gradient. where η is the viscosity. Or in terms of the strain rate: ε xz t = 1 2 σ xz =η V x z V x z σ xz = 2η ε xz t Units: dyne sec 1Poise =1 cm 2 newton sec = 0.1 m 2 Water at 20 o C η.01poise 1centiPoise 216
Darcy found experimentally that fluid diffuses through a porous medium according to the relation l P + P U P Darcy s Law: where Q = κ = η = A = Q = κ η A P l volumetric flow rate permeability of the medium viscosity of the fluid cross sectional area Differential form: where V V = κ η grad( P) is the filtration velocity 217
Units Darcy s law: Q = κ η A P l Permeability κ has dimensions of area, or m 2 in SI units. But the more convenient and traditional unit is the Darcy. 1Darcy 10 12 m 2 In a water saturated rock with permeability of 1 Darcy, a pressure gradient of 1 bar/cm gives a flow velocity of 1 cm/sec. 218
Kozeny-Carman Relation The most common permeability model is to assume that rocks have nice round pipes for pore fluids to flow. The classical solution for laminar flow through a circular pipe gives: strong scale Q = πr 4 P dependence! 8η Compare this with general Darcy s law: Q = κ η A P l Combining the two gives the permeability of a circular pipe: We can rewrite this permeability in terms of familiar rock parameters, giving the Kozeny-Carman equation: where: φ is the porosity S is the specific pore surface area τ is the tortuosity d is a typical grain diameter B is a geometric factor 219 l κ = πr4 8A = πr 2 A R2 κ = Bφ3 κ = Bφ3 d 2 τ 2 S 2 τ 8 2R
H.1 Schematic porosity/permeability relationship in rocks from Bourbié, Coussy, Zinszner, 1987, Acoustics of Porous Media, Gulf Publishing Co. 220
Here we compare the permeability for two synthetic porous materials having very different grain sizes. When normalized by grain-size squared, the data fall on top of each other -- confirming the scale dependence. 1000 100 κ/d 2 (x10e-6) 10 Sintered Glass 280 µm spheres 50 µm spheres 1 H.2 0 1 0 2 0 3 0 4 0 5 0 Porosity (%) Demonstration of Kozeny-Carman relation in sintered glass, from Bourbié, Coussy, and Zinszner, 1987, Acoustics of Porous Media, Gulf Publishing Co. 221
A particularly systematic variation of permeability with porosity for Fontainebleau sandstone. Note that the slope increases at small porosity, indicating an exponent on porosity larger than the power of 3 predicted by the Kozeny-Carman relation. H.3 Porosity/permeability relationship in Fontainebleau sandstone, from Bourbié, Coussy, and Zinszner, 1987, Acoustics of Porous Media, Gulf Publishing Co. 222
Kozeny-Carman Relation with Percolation As porosity decreases from cementation and compaction, it is common to encounter a percolation threshold where the remaining porosity is isolated or disconnected. This porosity obviously does not contribute to permeability. Therefore, we suggest, purely heuristically, replacing φ φ φ P giving κ = B φ φ P Hot-pressed Calcite (Bernabe et al, 1982), showing a good fit to the data using the Kozeny- Carman relation modified by a percolation porosity. 3 d 2 H.4 223
Fused Glass Beads (Winkler, 1993) H.5 224
Here we show the same Fontainebleau sandstone data as before with the Kozeny-Carman relation modified by a percolation porosity of 2.5%. This accounts for the increased slope at low porosities, while retaining the exponent of 3. H.6 Fontainebleau Sandstone (Bourbié et al, 1987) 225
Diffusion The stress-strain law for a fluid (Hooke s law) is which can be written as combining with Darcy s law: gives the classical diffusion equation: where D is the diffusivity ε αα = 1 K P V = 1 K P t V = κ η P 2 P = η P κk t 2 P = 1 P D t 226
Examples of Diffusion Behavior 1-D diffusion from an initial pressure pulse P=P 0 δ x Standard result: Px,t = P 0 4πDt e x2 4Dt = P 0 4πDt e τ t Characteristic time scale τ = x2 4D 227
Examples of Diffusion Behavior Sinusoidal pressure disturbance λ Disturbance decays approximately as τ d = λ2 4D 228