Fluid Flow Fluid Flow and Permeability
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1 and Permeability 215
2 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
3 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
4 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 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
5 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
6 H.1 Schematic porosity/permeability relationship in rocks from Bourbié, Coussy, Zinszner, 1987, Acoustics of Porous Media, Gulf Publishing Co. 220
7 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 κ/d 2 (x10e-6) 10 Sintered Glass 280 µm spheres 50 µm spheres 1 H Porosity (%) Demonstration of Kozeny-Carman relation in sintered glass, from Bourbié, Coussy, and Zinszner, 1987, Acoustics of Porous Media, Gulf Publishing Co. 221
8 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
9 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
10 Fused Glass Beads (Winkler, 1993) H.5 224
11 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
12 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
13 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
14 Examples of Diffusion Behavior Sinusoidal pressure disturbance λ Disturbance decays approximately as τ d = λ2 4D 228
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