Measurement of permeability of microfluidic porous media with finite-sized colloidal tracers

Abstract

We present two methods how the permeability in porous microstructures can be experimentally obtained from particle tracking velocimetry of finite-sized colloidal particles suspended in a liquid. The first method employs additional unpatterned reference channels where the liquid flow can be calculated theoretically and a relationship between the velocity of the particles and the liquid is obtained. The second method takes advantage of a time-dependent pressure drop that leads to an exponential decrease in the particle velocity inside a porous structure. From the corresponding decay time, the permeability can be calculated independently of the particle size. Both methods lead to results comparable with permeabilities derived from numerical simulations.

Appendices

Appendix 1: Theoretical calculation of mean particle velocity

First we assume that gravity leads to a non-uniform particle distribution described by the Boltzmann distribution

$$ p(z)=e^{-z/\lambda}, $$

(4)

where λ is the gravitational length.

Second we use numerical values from (Staben et al. 2003) to describe the local particle velocity u _d (z). Third we consider the biased sampling, as the inception rate of the particles (coming from an infinite reservoir) is proportional to the particle speed, which results in an effective particle distribution

$$ p_{{\rm eff}_{d}}(z)=u_{d}(z)p(z). $$

(5)

Combining these assumptions, the mean particle velocity is given by

$$ \tilde{u}(d)=\frac{\int\nolimits_{d/2}^{h-d/2}u_d(z)p_{{\rm eff}_d}(z)dz}{\int\nolimits_{d/2}^{h-d/2}p_{{\rm eff}_d}(z)dz}. $$

(6)

Appendix 2: Exponential decrease in mean particle velocity

First we use that $\Updelta P=\rho g \Updelta h$ and the excess volume is given by $V=A_{{\rm res}}\Updelta h$ which yields

$$ V=\frac{A_{{\rm res}} \Updelta P}{\rho g}. $$

(7)

Second we know the time derivative of the excess volume is related to the total flux, so that

$$ \frac{d}{dt}V=2Q, $$

(8)

where the factor 2 is due to the relative increase and decrease in the water column in the two reservoirs. Now we insert Eq. 7 and Darcy’s law into Eq. 8, which gives

$$ \frac{d}{dt} \Updelta P = -\frac{2 A \rho g k}{\eta L A_{{\rm res}}} \Updelta P. $$

(9)

Using $\Updelta P \propto \bar{u}$ we arrive at

$$ \frac{d}{dt}\bar{u}=-\tau^{-1}\bar{u}, $$

(10)

where τ = ηL A _res/(2Aρg k).