Water Resources Research
Supporting Information for
Gas Pressure Gradients in Unsaturated Porous Media and the Assumption of Infinite Gas
Mobility
Lili Hou1, Brent E. Sleep2, and Tohren C. G. Kibbey1
1University
of Oklahoma
of Toronto
2University
Contents of this file:
Text S1. Calibration Details For Figures 3 and 4………………………………… p. 2
Text S2. Versions of Fig. 3 and 4 showing corresponding VG-based
simulations…………………………………………………………………. p. 3
Text S3. Simulation Details for Figures 7 and 8 – Stauffer [1978] Data…….… p. 5
Text S4. Simulation Details for Figure 9 – Illustration of Eq. 3…………………. p. 5
Text S5. Derivation of Eq. 3.………………………………………………………….. p. 5
Text S6. Replicate experiments…………………………………………………….. p. 7
Text S7. Outflow comparisons….………………………………………………….. p. 8
1
Text S1. Calibration Details For Figures 3 and 4:
Table S1 below shows the parameters used in the BC and VG simulations of the data in
Figures 3 and 4 of the paper:
Table S1. Simulation parameters used for the data in Figs 3, 4 (BC) and S3, S4 (VG).
BC k (m2)
L (m)*
0.001
Inlet
0.1524
Sand
0.001
System
0.001
VG k (m2)
No added
resistance
Added inlet
resistance
Added outlet
resistance
No added
resistance
Added inlet
resistance
Added
outlet
resistance
(Fig 3)
(Fig. 4A)
(Fig. 4B)
(Fig. S3)
(Fig. S4A)
(Fig. S4B)
n/a
1.1010
-11
9.0610
-16
1.1010
-11
n/a
1.1010
9.0610
-16
n/a
7.8110
-12
7.8110-12
1.5010-13
n/a
6.0410-14
n/a
1.5010-13
n/a
-11
7.8110
-12
6.0410-14
n/a
1.5010-13
Outlet
1.5010-13
n/a
Pd (m)
0.70
0.66
0.66
α (m-1)
1.31
1.40
1.40

8.5
8.5
8.5
n
16
16
16
2.5010-14
2.5010-14
*
L values correspond to length in the model.
Notes on Table S1:
1. All boundary permeabilities (inlet, system, outlet) correspond to the specific length of
the boundary region in the model. (Because all experience single-phase flow, lengths are
arbitrary.)
2. Pc-S scaling parameters (Pd, ) were observed to vary slightly with packings, and from
run-to-run in re-saturated columns, likely due to changes to the packing density caused by
backflushing during re-saturation. Note that no systematic relationship between Pc-S
parameters and flow rate was observed across all experiments conducted.
3. The experiment with added inlet resistance was the third run of a re-saturated column
(the first was the system with no added resistance in Fig. 3, the second was a run with
lower inlet resistance, shown in Fig. S6 below). As such, the slightly lower system
permeability needed in the calibration of the experiment with added inlet resistance is
consistent with the reduction in membrane permeability typically observed over multiple
runs.
4. Comparison with measured permeabilities: As mentioned in the text, the sand
permeability was measured at 7.8110-12 m2 in falling head measurements conducted for
this work. While it is likely that the true k varied from packing to packing, that measured
value was used in all VG simulations. Regardless of boundary properties, BC
simulations were found to drain more slowly for a given k, so a higher k was used in BC
simulations to duplicate the outflow behavior of the VG runs. Note that subsequent
falling head measurements of equivalent k for the tubing used in the experiments found:
koutlet_permeability = 2.4810-14 m2 **
kinlet_permeability= 1.3210-15 m2 **
**NOTE: both values are equivalent permeabilities based on L=0.001 m (Table S1)
2
The measured outlet value is very close to the value determined by calibration in Table
S1. The measured inlet value is approximately 50% greater than the calibrated value.
Note, however, that the measured value was determined with water at a much slower rate
than the air flow during column drainage; it is probable that minor losses (particularly gas
expansion from the small tubing to the larger nylon tubing) are significant in the inlet
during actual experiments, so the measurement is consistent with the calibrated value.
Text S2. Versions of Fig. 3 and 4 showing corresponding van Genuchten-based simulations:
Note that BC simulations capture average saturation behavior slightly more accurately
than VG simulations, but pressure drop magnitudes are quantitatively similar in both. VG
simulations are shown below.
1.0
250
0.9
200
0.8
150
0.7
Pg applied
Pg obs.
Pw obs.
Pg sim.
Pw sim.
Savg sim.
100
50
0.6
Water saturation, S
Pressure, cm water
30 cm
0.5
Smid sim.
0.4
0
-10
0
10
20
30
40
50
Time, s
Figure S3. Measured (symbols) and simulated (green lines) pressures
during drainage in a system with no added inlet or outlet flow
resistance. Average and local saturations from simulations (red
lines) are shown for reference. VG simulations shown.
3
1.0
250
0.8
150
0.7
Pg applied
Pg obs.
Pw obs.
Pg sim.
Pw sim.
Savg sim.
100
50
Smid
0.6
A. Inlet resistance
added
sim.
Water saturation, S
Pressure, cm water
0.9
100 cm
200
0.5
0.4
0
50
Pressure, cm water
250
100
150
Time, s
1.0
10 cm
0.9
200
0.8
150
0.7
Pg applied
Pg obs.
Pw obs.
Pg sim.
Pw sim.
Savg sim.
100
50
Smid sim.
0.6
B. Outlet resistance
added
Water saturation, S
0
0.5
0.4
0
0
50
100
150
Time, s
Figure S4. Measured (symbols) and simulated (green lines) pressures during
drainage in systems with A. added inlet resistance, and B. added
outlet resistance. Average and local saturations from simulations (red
lines) are shown for reference. VG simulations shown.
4
Text S3. Simulation Details for Figures 7 and 8 – Stauffer [1978] Data:
The simulation of the data from Stauffer [1978] was based on BC parameters and other
soil parameters reported by Stauffer. Note that although Stauffer reported soil parameters for
several levels within his column, all were extremely similar; for this reason the simulation was
conducted assuming a homogeneous column. Because the measurements of interest were
conducted at the top of the column, reported soil parameters from the top portion were used
throughout the simulated column. Parameters used were: Ksat = 2.1710-4 m/s; n=0.332;
Swr=0.12; =5.7; Pd=31.8 cm water. The saturated outlet portion of the column (including
tubing) had a reported equivalent Ksat=6.410-4 m/s. The column had a reported height of 63.8
cm, while the outlet portion had a reported height of 37.7 cm. In the simulation, a block height
of 0.1 cm was used, so 638 grid blocks represented the column portion, and 37.8 blocks
represented the outlet portion. A sufficiently high air entry pressure was used for the outlet
portion for it to serve as a capillary barrier. (The actual value was not specified by Stauffer, but
is not critical, as the front does not reach the outlet portion until later times than those shown in
Fig. 7.)
It is interesting to note that the Ksat values reported by Stauffer were determined from his
unsaturated data using a method based on the Brooks-Corey-Burdine relative permeability
relationship. That is the likely reason that BC simulations using those Ksat values agreed so well
with observed outflow behavior, while the permeability values determined from falling head
measurements in this work underestimated flow in BC simulations (but not VG simulations).
Text S4. Simulation Details for Figure 9 – Illustration of Eq. 3:
The simulation in Figure 9 used to illustrate Eq. 3 was based on a hypothetical medium
sand, slightly finer than the sand used by Stauffer, but coarser than the fine sand used in
experiments for this work. Parameters used were: =6.0; Pd=40.0 cm water, k=1.010-11 m2,
n=0.37, Swr=0.1). The simulated vertical column had 100 0.5 cm grid blocks, for a total height of
50 cm. Two high-conductivity, low-air-entry grid blocks located below z=0 simulated a high
permeability bottom membrane. The simulation involved a 200 cm water gas pressure input.
Text S5. Derivation of Eq. 3:
Equation 3 provides an estimate of the rate at which gas pressure varies in space away from the
moving front. The derivation of the equation is based on two main assumptions: 1. the
assumption that the water pressure head gradient remains approximately constant across the
front, and 2. the assumption that the spatial variation in Pc right at the front will be equal to- or
greater than the variation that would be observed in a static Pc-S relationship in a vertical
column. Both assumptions were found to be consistent with BC simulations in both vertical and
horizontal columns.
From Darcy’s Law:
𝜕ℎ
𝑞𝑜𝑢𝑡 = +𝐾𝑠𝑎𝑡
𝜕𝑠
(S1)
where h is piezometric head, qout is Darcy velocity of water leaving the column in the direction
of the movement of the front, and s is a spatial axis the direction toward the inlet of the column.
(The positive sign comes from the opposite conventions for qout and s.) In terms of pressure
head, Eq. S1 becomes:
5
𝑞𝑜𝑢𝑡 = 𝐾𝑠𝑎𝑡
𝜕
(𝑧 + 𝑃𝑤 )
𝜕𝑠
(S2)
where Pw is pressure (in head units), and z is elevation. Recognizing that 𝜕𝑧⁄𝜕𝑠 =sin , where 
is the angle of the column from horizontal:
𝑞𝑜𝑢𝑡
𝜕𝑃𝑤
(S3)
= 𝑠𝑖𝑛𝜃 +
𝐾𝑠𝑎𝑡
𝜕𝑠
In equation form, Assumption 2 is equivalent to:
𝜕𝑃𝑐
= 1.0
𝜕𝑠
(S4)
In a vertical column if no drainage is occurring, Eq. S4 would hold exactly. Simulations suggest
that it is reasonably close right at the front even for moderately rapid drainage, with the 1.0 value
forming a minimum value. Simulations also suggest that Eq. S4 is equally valid in horizontal
drainage. Substituting the definition of Pc gives:
𝜕𝑃𝑐 𝜕𝑃𝑔 𝜕𝑃𝑤
(S5)
=
−
= 1.0
𝜕𝑠
𝜕𝑠
𝜕𝑠
Finally, substituting S3 into S5 and rearranging gives:
𝑑𝑃𝑔 𝑞𝑜𝑢𝑡
=
+ 1.0 − 𝑠𝑖𝑛𝜃
𝑑𝑠
𝐾𝑠𝑎𝑡
(S6)
Or, recognizing that the 1.0 is a minimum value:
𝑑𝑃𝑔 𝑞𝑜𝑢𝑡
≳
+ 1.0 − 𝑠𝑖𝑛𝜃
𝑑𝑠
𝐾𝑠𝑎𝑡
(3)
6
Text S6. Replicate experiments
Experiments a and b shown below correspond to replicates of the experiment shown in Figure 3
(c is the data from Fig. 3). Experiment d corresponds to an experiment conducted with low
added inlet resistance (low enough that it is essentially a replicate of the system without added
resistance).
300
Applied
250
Pressure (cm water)
Gas
200
Water
150
100
a
b
c
d
50
0
-10
0
10
20
30
40
50
time (s)
Figure S6. Experimental data for three systems with no added inlet or outlet
resistance (a, b, c) and one with very low inlet resistance (d).
Note that although the timing differs slightly in the four experiments, the magnitudes of the gas
pressure drops are quantitatively similar for all four. It should be noted that experiments a and b
were recorded at longer, 3 sec. intervals, causing the curves to have an angular appearance where
straight lines connect the more widely-spaced data points. Note also that none of the water
pressure sensor data above are corrected for response rate. Experiments a and b were conducted
in the same packing (re-saturated for expt. b). Experiments c and d were conducted in another
packing (re-saturated for expt. d). The packing used for expt. d was also subsequently resaturated for the experiment with added inlet resistance (Fig. 4A).
7
Text S7. Comparison between measured and simulated outflow
Figure S7 shows a comparison between the measured and simulated outflow for the BC
simulations shown in Figures 3 and 4.
25
No added resistance
Simulation (BC)
Outflow (g)
20
14
12
10
15
Outflow (g)
8
6
10
4
2
t (sec)
0
5
0
50
100 150 200 250
0
0
2000
4000
6000
8000
10000
t (sec)
25
Inlet resistance
Simulation (BC)
Outflow (g)
20
14
12
10
15
Outflow (g)
8
6
10
4
2
t (sec)
0
5
0
50
100 150 200 250
0
0
2000
4000
6000
8000
10000
t (sec)
25
Outlet resistance
Simulation (BC)
Outflow (g)
20
14
12
10
15
Outflow (g)
8
6
10
4
2
t (sec)
0
5
0
50
100 150 200 250
0
0
2000
4000
6000
8000
10000
t (sec)
Figure S7. Comparison between experimental and simulated outflow data for
the experiments in Figure 3 and 4.
8