Inverse determination of saturated and relative permeability with a bench-scale centrifuge

Figures & data

Fig. 1 Centrifuge holder with sample.

Fig. 2 Schematic of the centrifuge parts.

Table 1 Rotational speed and corresponding total stress at base for a sample with density ${\mathit{\rho }}_{t}1.65$ g/cm${}^3$, $r_0=105$ mm, $L=40$ mm. Corresponding depth in normal gravity, corresponding g-level at the midpoint of the sample (given by $N=R{\mathit{\omega }}^2/g$, with R the distance from the rotation center), and corresponding stress at that same midpoint are also given.

Speed	Total stress base ${\mathit{\sigma }}_b$	Depth D	g-level N at midpoint	Total stress midpoint
rpm	kPa	m	$-$	kPa
100	1.78	0.11	1.4	0.82
200	7.11	0.45	5.6	3.27
300	15.99	1.01	12.6	7.35
600	63.96	4.02	50.3	29.42
900	143.90	9.05	113.2	66.19
1200	255.82	16.10	201.2	117.68
1500	399.72	25.15	314.4	183.87
2400	1023.28	64.39	804.9	470.71

Fig. 3 Schematic representation of drainage into an overflow cup. Left: as the water drains, the height $H_B$ is constant. The interface s between partially saturated and saturated zone moves downward. Right: if the centrifuge slows down, part of the water from the overflow cup is sucked into the sample, and the interface s moves upward.

Fig. 4 Saturated permeability for the first 4 filters used.

Table 2 Permeability of the soil samples and filters.

Name	Falling head	Centrifuge model
	$K_S$ [m/s]	avg. $K_S$ [m/s]
Filter tube 1	$5.53{10}^{-8}$	$9.32{10}^{-8}$
Filter tube 2	$4.95{10}^{-8}$	$7.50{10}^{-8}$
Filter tube 4	$9.89{10}^{-8}$	$7.44{10}^{-8}$
Filter tube 5	$8.17{10}^{-8}$	$6.24{10}^{-8}$
Kaolin tube 1	$1.136{10}^{-8}$	$1.173{10}^{-8}$
Kaolin tube 2	$1.209{10}^{-8}$	$1.139{10}^{-8}$
Kaolin tube 4	$1.331{10}^{-8}$	$1.137{10}^{-8}$
Kaolin tube 5	$1.259{10}^{-8}$	$1.117{10}^{-8}$
Filter tube 16	$1.351{10}^{-5}$	–
Filter tube 17	$1.103{10}^{-5}$	–
Mixture tube 16	–	$2.096{10}^{-6}$

Fig. 5 Measured soil retention curve, and interpolated value for the kaolin sample.

Fig. 6 Drainage of a kaolin sample in the centrifuge at $\mathit{\omega }=1000$ rpm (top) and $\mathit{\omega }=6000$ rpm (bottom) over 60 min starting from saturated.

Table 3 Drainage experiment on 90% sand-10% kaolin mixture. Every line is a new measurement, after the given time in seconds.

Time (s)	rpm	Height soil	Height expelled	Cumulative	Grav. Center
		at start (cm)	water (cm)	height (cm)	(cm)
60	600	5.731	0.03896	0.03896	2.525
120	600	5.701	0.04174	0.08070	2.483
240	600	5.701	0.09461	0.17532	2.387
480	600	5.701	0.06679	0.24211	2.255
960	600	5.701	0.03896	0.28107	2.154
1920	600	5.701	0.03061	0.31168	2.085
3840	600	5.701	0.00278	0.31446	2.060
60	1200	5.701	0.07514	0.38960	1.997
120	1200	5.676	0.05566	0.44525	1.947
240	1200	5.676	0.05566	0.50091	1.886
480	1200	5.676	0.03896	0.53987	1.794
960	1200	5.676	0.05009	0.58996	1.686
1920	1200	5.676	0.01113	0.60109	1.671
60	2400	5.676	0.00557	0.60666	2.412
120	2400	5.456	0.01391	0.62057	1.639
240	2400	5.646	0.02505	0.64562	1.592
480	2400	5.646	0.02226	0.66788	1.485
960	2400	5.646	0.00835	0.67623	1.416
1920	2400	5.646	0	0.67623	1.392

Fig. 7 Experiments performed with the centrifuge. Crosses indicate the measured values, Triangles up the model results for the optimal value taking Gravitational center (GC) and Outflow (MO) into account. Triangles down only using GC, Triangles left using only MO. Full lines indicate the results using the pressure plate fitted soil parameters with two different values of $K_S$.

Fig. 8 Measured and inversely determined soil retention curves. Reference curve (Ref.) is the measurements done with ASTM D2325, with the other lines best measurement fit from the model.

Table 4 Resulting parameters arising in the model. If deviation is given, the parameter was determined via non-linear least-squares, if not given, it was known input.

Description	$K_s$ (m/s)	$\mathit{\gamma }$ (cm${}^{-1}$)	n	${\mathit{\theta }}_s$
Pressure plates (ASTM D2325)	–	$-0.0180\pm 0.006$	$2.18\pm 0.53$	$0.368\pm 0.035$
Based on outflow	$(3.8\pm 1.3){10}^{-7}$	$-0.0234\pm 0.0043$	$1.44\pm 0.05$	$0.346$
Based on GC	$(3.1\pm 0.5){10}^{-7}$	$-0.0154\pm 0.0011$	$2.24\pm 0.07$	$0.346$
Based on outflow & GC	$(2.0\pm 1.5){10}^{-7}$	$-0.0121\pm 0.0022$	$2.17\pm 0.37$	$0.346$
Based on outflow & 2 $\times$ GC	$(2.6\pm 0.9){10}^{-7}$	$-0.0136\pm 0.0014$	$2.26\pm 0.19$	$0.346$
Based on 2 $\times$ outflow & GC	$(1.16\pm 1.3){10}^{-7}$	$-0.0136\pm 0.0044$	$1.96\pm 0.40$	$0.346$
Outflow & GC, fixed $K_s$	$2.096{10}^{-6}$	$-0.0109\pm 0.0023$	$1.69\pm 0.07$	$0.346$
Outflow & GC, fixed $K_s$	$2.096{10}^{-7}$	$-0.0122\pm 0.0022$	$2.15\pm 0.13$	0.346
Outflow & GC, 1200 rpm	$(2.3\pm 0.9){10}^{-7}$	$-0.0107\pm 0.0014$	$2.39\pm 0.29$	0.346

Fig. 9 Saturation curves as obtained by the model during operation of the centrifuge.