PETROLEUM SOCIETY
PAPER 2007-156
CANADIAN INSTITUTE OF MINING, METALLURGY & PETROLEUM
Revisiting the Drainage Relative Permeability
Measurement by Centrifuge Method Using a
Forward-backward Modelling Scheme
M. SAEEDI, M. POOLADI-DARVISH
University of Calgary
This paper is to be presented at the Petroleum Society’s 8th Canadian International Petroleum Conference (58th Annual Technical
Meeting), Calgary, Alberta, Canada, June 12 – 14, 2007. Discussion of this paper is invited and may be presented at the meeting if
filed in writing with the technical program chairman prior to the conclusion of the meeting. This paper and any discussion filed will
be considered for publication in Petroleum Society journals. Publication rights are reserved. This is a pre-print and subject to
correction.
EXTENDED ABSTRACT
µ
Abstract
Introduction
Measurement of drainage relative permeability by the
centrifuge method was first introduced by Hagoort1. It has been
shown that capillary end effects can cause errors in the
measurements if a minimum rotational speed is not honored2.
To determine the !min , we propose maintaining the value of
capillary-gravity number, N cg , to be of the order of 10-2 or
smaller, at which the capillary end effect becomes negligible.
The above conclusions were determined by applying a forward
numerical simulator developed for centrifuge experiments and
applying Hagoort’s method as a backward model. These
forward and backward models form a forward-backward loop
which is a powerful tool for error analysis such as determining
the impact of capillary end effects.
Measurement of drainage relative permeability using the
centrifuge method was first introduced by Hagoort[1]. He
viewed the gravity drainage process as a gravity stable BuckleyLeverett displacement with the same set of assumptions. In
addition, he assumed constant centrifugal acceleration along the
core length and infinite mobility for the gas phase. In the full
version of this paper, it will be shown that these assumptions
are reasonable.
Nevertheless, error caused by capillary end effect is one of
the issues in relative permeability measurements. In the
centrifuge method, if the rotational speed, ! , of the experiment
physical properties of the sandstone-brine combination
(specified in the first row of Table 1), the effect of capillary
hold-up on the calculated relative permeability can be neglected
provided that ω ≥ 1500 rpm . The same procedure was
performed for the other combinations and the respective
minimum rotational speeds were determined for each of those
combinations. These results are shown in the same table.
Some researchers[3] have tried to define ωmin by using the
dimensionless bond number, N B = k∆ρω 2 rm σ . Their efforts
have not been quite as successful, because the bond number
indicates the ratio of the centrifugal to capillary forces at the
pore level and not at a macroscopic level. In relative
permeability measurements, one desires to have a capillary
dominated flow regime at the pore level. Therefore, the bond
number can be used only in defining an upper limit for
rotational speed. Hagoort stated that, in a centrifuge drainage
process, we are allowed to go as high as N B = 10-5 and still
maintain capillary dominated flow at the pore level. The last
column of Table 1 shows that, in all cases, the flow regimes are
capillary dominated, while the rotational speed is high enough
to render the capillary hold-up negligible.
Although performing the above mentioned procedure of
forward-backward modeling for each set of core-fluid
combination is comprehensive, it is certainly not practical in
core analysis laboratories. Therefore, a “rule of thumb” is
needed for applying to experimental design. In this study, we
propose such a rule by making use of the capillary-gravity
number, N cg . Capillary-gravity number is the ratio of the
threshold pressure to the pressure differential created by
centrifugation in the core sample. Larger values of capillarygravity number result in more significant capillary end effects in
calculating relative permeabilities; thus, capillary-gravity
number must be minimized. By using the calculation loop
described previously, we found that, to have a negligible
capillary hold-up effect, capillary-gravity number should be on
the order of 10-2 and smaller. This rule of thumb is validated
experimentally for the experimental cases that we studied.
is not maintained above a critical minimum, the calculated
relative permeability values will be underestimated[2]. Although
Hagoort proposed a correction procedure for cases where the
measurements are affected by capillary end effects, it has been
shown that this procedure introduces some noise into the
calculated relative permeability values. Further, it is not easy to
perform Hagoort’s correction procedure in core analysis
laboratories; rather, it is much easier and preferable to avoid the
problem entirely by designing the experiment with a proper
rotational speed from the beginning. On the other hand, since
the target of such measurements is usually in modelling gravity
drainage processes, one would desire that the relative
permeability measurements be obtained under a capillary
dominated flow regime. That is to say, if the measurements are
performed under severe rotational speeds, the de-saturation flow
regime would not be a capillary dominated flow.
The objective of this work is to introduce a criterion for
designing relative permeability experiments so that the
minimum rotational speed that minimizes capillary end effects
is honoured. To find this criterion, a forward-backward
modelling scheme is developed. The forward numerical model
accepts arbitrary capillary pressure and relative permeability
functions as inputs and simulates the centrifugal displacement
process. The simulated centrifuge drainage results is then used
as raw centrifuge data and processed by Hagoort’s method as
the backward model. The resulting relative permeability curve
is then compared with the original input relative permeability to
quantify the capillary end effect. Figure 1 shows the forwardbackward loop schematically. This procedure was performed
for different sets of capillary pressure and relative permeability
curves (based on our experimental results) to obtain ωmin for
each case. Later on, the definition of capillary-gravity number
( N cg = Pcth ∆ρω 2 rm L ) is used to determine the experimental
design criterion. In the full version of this paper, the
experimental cases and the application of the forward-backward
modelling in assessing the experimental results will be
discussed.
Conclusion
Results and Discussion
Capillary hold-up affects the relative permeability
calculation significantly unless the test is run with a high
enough rotational speed. On the other hand, the flow regime
should be maintained in a capillary dominated regime so that it
appropriately represents a gravity drainage process. This puts an
upper bound to the rotational speed. We introduced the forwardbackward loop as a powerful tool to determine the proper
rotational speed for relative permeability experiments. The
results of this study show that the capillary-gravity number
criterion can be used with confidence in designing relative
permeability experiments. Specifically, if capillary-gravity
number is in the range of 10-2 or smaller, the capillary end effect
becomes negligible. If the capillary-gravity-number criterion is
not honoured, the Hagoort correction procedure is an alternative
way to improve the quality of the relative permeability data.
However, that procedure may introduce a significant noise into
the data, so smoothing the corrected data becomes necessary.
In this work, five core-fluid combinations were used; these
are listed in Table 1. As described above, by knowing the
capillary pressure data of each core sample and using the
1 → 2 → 3 → 4 → 1 route in the loop depicted in Figure 1,
we can determine the minimum rotational speed needed for
correct relative permeability measurements, as follows. Each
combination involves several core samples. For each
combination, an average capillary pressure curve was found
experimentally (using the capillary pressure curves obtained for
each core sample in that combination). The resulting capillary
pressure curves for each combination with an arbitrary relative
4
) were used as inputs to the
permeability function ( k rw = S wD
forward model to simulate single speed step experiments. The
simulated recovery curves were processed by Hagoort’s method
to obtain a relative permeability curve. This curve was
compared to the input relative permeability function. If the
calculated and input relative permeability curves did not match,
the same procedure would be repeated at a higher rotational
speed.
Figure 2 shows the recovery curves generated by the forward
model at different rotational speeds for the sandstone-brine
combination. The data sets were then processed by Hagoort’s
method to give the relative permeability curves shown in Figure
3. This figure shows that, for this case, if ω ≥ 1500 rpm , the
calculated relative permeability values would be very close to
the input relative permeability function. As a result, for the
Acknowledgements
The inspiration for this work was Dr. Ali M. Saidi. We
thank him for this and his support. The authors would like to
thank NSERC for funding.
2
REFERENCES
NOMENCLATURE
Hagoort, J.: “Oil Recovery by Gravity Drainage” SPEJ
(June 1980), 139-150.
Saeedi, M.: “Modeling and Experiments of Drainage
Relative Permeability and Capillary Pressure Functions
Using a Centrifuge”, M. Sc. dissertation, University of
Calgary, Canada (2007).
Skauge, A., Haaskjold, G., Thosrsen, T., and Aara, M.:
“Accuracy of Gas Oil Relative Permeability from Two
Phase Flow Experiments”, Soc. Core Analyst – 9707.
1.
k
k rw
L
NB
N cg
=
Absolute permeability [L ]
=
Relative permeability of the wetting phase
=
Length of the core sample [L]
=
Bond number
=
Capillary-gravity number
Pcth
=
Threshold capillary pressure [MLT-2]
rm
=
Mean rotational radius [L]
S wD
∆ρ
=
Wetting saturation, Normalized
=
Density difference [ML-3]
ω
σ
=
=
Rotational speed (rpm)
Surface Tension [MT-2]
2
2.
3.
Figures and Tables:
Input
kr
function
1
Input
Pc
function
Forward model
2
Single speed test
3
kr
4
from Hagoort
Multi speed test
Pc
from BW methods
Figure 1 - The forward-backward loop
Table 1 - The maximum value of the
Core Type
Sandstone
High perm
sintered metal
High perm
sintered metal
Low perm
sintered metal
Low perm
sintered metal
N cg for correct measurement of relative permeability
Fluid k (mD) ∆ρ (kg/m3) rm (cm)
L (cm)
σ
(mN/m)
Pcth (KPa) ω min (rpm) N cg (max)
NB
-2
3.1 × 10 7.7 × 10-6
7
1500
14
2000
3.8 × 10-2 5.6 × 10-6
Brine
IPA+
Water
150
1008
15
6
72.5
50
926
15
6
54.5
IPA
50
785
15
6
21
14
2200
3.7 × 10-2 1.5 × 10-5
IPA+
Water
5
926
15
6
54.5
35
3000
4.3 × 10-2 1.3 × 10-6
IPA
5
785
15
6
21
35
3300
4.4 × 10-2 3.4 × 10-6
3
1
Cumulative Production
0.9
0.8
0.7
0.6
0.5
0.4
w = 350 rpm
0.3
w =500 rpm
0.2
w =715 rpm
w =1000 rpm
0.1
w =1500 rpm
0
0.1
1
10
Dim ensionless tim e (sec)
100
Figure 2 - Simulated recovery data at different rotational speeds for the sandstone-brine combination
Relative permeability
1.E+00
Input krw
krw , 350 rpm
krw , 500 rpm
krw , 715 rpm
krw , 1000 rpm
krw , 1500 rpm
1.E-01
1.E-02
1.E-03
1.E-04
0.0
Figure 3 - Comparison of the calculated
0.2
0.4
0.6
Norm alized Saturation
0.8
1.0
k rw (S w ) for the sandstone-brine combination at different rotational
speeds with the input relative permeability function
4