Page 31 - MSc_thesis_R A Kil

set-up and samples, and the actual implementation of the experiment. The basic principle is that after
the tube is filled with water, a valve is opened at the top, causing the water level to drop. Time starts
when the water reaches the one metre above the sample level, and at fixed intervals the intermediate
times are recorded. This results in a time versus height plot for each sample.

Calculation of permeability from time-vs.-height plots, obtained by performing falling-head experiments
requires some modification of Darcy’s law. First of all, recapitulation of Darcy’s law for flow trough a
core is

Q = −k ∆p Ac (3.4)
Lc µw

Application of the experimental set-up gives expressions for flow rate

Q(t) = dhAt (3.5)
dt

where At is the cross-sectional area of the tube [m2] and dh/dt the drop velocity of the water level in the
tube, expressed in [m/s].

and pressure drop

∆p = ρg h(t) + Lc − H (3.6)

In here, ρ is the density of the water [kg/m3], h(t) the water level in the tube as function of time [m], Lc
the length of the core [m] and H the water level in the large tank [m], where the size of the tank allows
for the assumption of a constant water level in this tank.

Substitution of equations (3.5) and (3.6) into equation (3.4) leads to

dhAt ρg h(t) + Lc − H Ac (3.7)
= −k
dt Lc µw

Reworking the equation makes it possible to integrate on both sides (3.8)
dh = − kCdt

h + Lc − H

where the constant term is written as C = ρgAc , expressed in [1/m2s].
LcµAt

Now, integration with the necessary start condition for the left-hand term defined as t = 0 : h = h0 = 1 m
leads to

ln |h + Lc − H| − ln |h0 + Lc − H| = −kCt (3.9)

14