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98 7. Permeability reduction in porous materials by in situ formed silica geldxz∆neutralaxisLFig. 7.1: Schematic of 3-point beam bending. The deflection ∆ is sustained with a force W .The diameter of the cylindrical beam is d, and the distance between the supports isL.7.2 Theoretical backgroundFor a cylinder with diameter d made of an elastic porous material, the load W , neededto sustain a constant deflection ∆ (see Figure 7.1), is given by [130]∞W (t)W (0) = R(t) = 1 − A + A ∑8β 2 n=1 n( )exp − β2 nt, (7.1)τ Rwhere A is the amount of hydrodynamic relaxation, β n are roots of the Bessel functionsJ 1 (β n ) = 0, and τ R is the hydrodynamic relaxation time. In case of a gel the solid materialcomprising the porous body is usually very compliant such that compressibility effects inthe system can be omitted. The amount of hydrodynamic relaxation A is then simply afunction of the Poisson ratio ν p , that is, A = (1 − 2ν p )/3. The hydrodynamic relaxationtime τ R is given by [128]τ R = (1 − 2ν p)η L d 24DG p, (7.2)where η L is the pore liquid viscosity, D is the permeability and G p is the shear modulusof the (drained) porous system. The latter follows from Young’s modulus E p through therelation E p = 2(1 + ν p )G p . Young’s modulus is derived from the initial load W (0) andthe moment of inertia, I = πd 4 /64, using the following equationE p =where L is the span in the three-point bending experiment.W (0)(1 − A)L3, (7.3)48∆IFor a rectangular isotropic beam (plate) of elastic porous material, with thickness 2aand width 2b, the normalized load as a function of the reduced time θ = t/τ R is given by[131]W (θ)W (0) = R(θ) = 1 − A + AS 1(θ)S 2 (κθ), (7.4)where κ = a 2 /b 2 and the functions S 1 and S 2 are given approximately by [132][ ( ) ( )]6 θ 0.5 − θ 2.5S 1 (θ) ≈ exp − √π , (7.5)1 − θ 0.551