Experimental and numerical investigation of one-dimensional
waterflood in porous reservoir
N. Hadia a, L. Chaudhari a, A. Aggarwal b, Sushanta K. Mitra
a
a,*
, M. Vinjamur b, R. Singh
c
IITB–ONGC Joint Research Centre, and Department of Mechanical Engineering, Indian Institute of Technology Bombay, Mumbai 400 076, India
IITB–ONGC Joint Research Centre, and Department of Chemical Engineering, Indian Institute of Technology Bombay, Mumbai 400 076, India
c
IITB–ONGC Joint Research Centre, and Institute of Reservoir Studies, ONGC, Ahmedabad, India
b
Abstract
Experimental and numerical investigation of relative permeability and oil recovery from the porous reservoir are described for short
and long core samples. The relative permeability ratios, which are function of water saturation, obtained from laboratory core flooding
experiments have been used for prediction of oil recovery through numerical simulation of non-dimensional Buckley–Leverett equation.
The simulation results for oil recovery compared well with recovery results obtained from core flooding experiments.
Keywords: Relative permeability; Oil recovery; Saturation; Core flooding
1. Introduction
The oil reservoirs are porous in nature. The oil recovery
processes involve the simultaneous flow of two or more fluids in the reservoir. In waterflooding process, water injected
in the reservoir displaces the oil from the pores of reservoir
bed. The property of the porous reservoir that plays an
important role in determining the performance of the
waterflooding process is relative permeability. The prediction of the relative permeability characteristics of oil reservoir is the primary task of any laboratory waterflood
experiments. For this purpose, laboratory experiments
are conducted on a small representative core samples
obtained form the reservoir.
The oil recovery characteristics can be predicted by
using Buckley–Leverett equation [2]. For such prediction,
individual phase relative permeabilities are required. Geffen
et al. [5] presented a comprehensive study on different
methods available for determining relative permeabilities
of reservoir rocks and suggested the direct measurement
of relative permeabilities in the laboratory using representative core sample. There are two basic methods for obtaining relative permeabilities from laboratory experiments:
steady state method, and unsteady state method.
The calculation of relative permeabilities from steady
state experiment requires very long experimentation time.
Unsteady state method needs less time; this method
estimates relative permeabilities by interpreting the data
collected from laboratory experiments in which one fluid
is injected into a core sample that is saturated with another
fluid. The relative permeability can be calculated by application of Welge [16] equation and method developed by
Johnson et al. [7] or by using graphical technique developed by Jones and Roszelle [8]. The basic limitation of this
approach is the necessity to differentiate the experimental
data which may yield large errors in estimation of relative
permeability [15].
Another approach for determining the relative permeability is history matching [1]. In this method, the relative
permeability curves are adjusted until calculated oil recovery curves match those obtained from the laboratory displacement experiments. The interpretation of relative
356
Nomenclature
English
da
F
K
kr
L
q
S
t
x
letters
mass coefficient
source term
absolute permeability
relative permeability
length of core sample
flow rate per unit area
saturation
time
distance
Greek symbols
C
flux vector
l
absolute viscosity
permeability based on history matching technique is found
to be less precise for the injected fluid than the displaced
fluid because the displacement history is weakly dependent
on the injected phase relative permeability [12].
Macary and Walid [10] used an X-plot technique for oil
recovery prediction and for obtaining relative permeability
ratio and fractional flow curves. However, the technique
can only be applied to reservoirs producing with water
fractions higher than 50% and requires linearization of
the relationship between recovery and water cut.
Experimental studies by Richardson [13] indicated that
the relative permeability ratio is independent of the viscosity of fluids and the rate of water advance. Moreover,
Buckley–Leverett theory can be used to predict the waterflood performance neglecting the capillary pressure effects.
Numerical methods have also been developed to predict
the waterflood behavior including capillary effects [4]. Sheldon and Cardwell [14] solved the Buckley–Leverett equation using the method of characteristics and used the
treatment analogous to that used in the theory of supersonic compressible flow. Solution of Buckley–Leverett
equation needs the individual relative permeability values,
which are obtained either from JBN method [7] or from
empirical correlations [3]. However, their solutions are
not validated through experiments. McEven [11] presented
the results of numerical solution to one dimensional waterflooding in which outlet end effects are neglected. Gottfried
et al. [6] presented two numerical schemes for solving the
equations for one dimensional, multiphase flow in porous
media with assumed correlations for relative permeabilities
and negligible capillary pressure. However, the results
obtained numerically have not been validated through laboratory waterflood experiments.
In this paper, the experimental investigation of one
dimensional waterflood performance of short and long
Berea core samples are presented. The waterflood performance has been predicted by solving the Buckley–Leverett
equation using FEMLAB solver. In general, it is observed
/
porosity
Subscripts
D
non-dimensional
o
oil phase
or
residual oil
t
total
w
water phase
wi
irreducible water
w2
terminal water
Abbreviations
PV
pore volume
OOIP original oil inplace
that for oil recovery estimation in one-dimensional core
flood, individual phase relative permeabilities are used in
the simulation. In this study, however, instead of using
individual relative permeability values, a novel method is
used where the relative permeability ratios obtained from
the saturation experiments are used. This reduces the computational time significantly. The results obtained from the
numerical simulation are validated by the laboratory one
dimensional waterflood experiments and a good agreement
has been found between them.
2. Experiments
The schematic of the experimental apparatus used for
permeability measurement and waterflooding studies is
shown in Fig. 1. The dual cylinder syringe pump whose
inlet is connected to the brine reservoir maintained the
desired flow rate through the core. The outlet of the pump
is connected to the three way valve which in turn is
connected to brine and oil accumulators at a common junction. Hence, by operating the three way valve appropriately, either oil or brine can be injected into the core.
The effluents from the core outlet are collected in a fraction
collector. A differential pressure transmitter has been connected between the inlet and outlet of the core to measure
the pressure difference across the core length. A digital
pressure gage is provided at the inlet of the core holder
to record the system pressure.
Experiments have been performed at room temperature
of 24 C and atmospheric pressure on cylindrical Berea
sandstone core sample, 3.8 cm in diameter and 7 cm in
length, as well as on rectangular Berea sandstone core sample, 2.5 · 2.4 · 54 cm. The properties of the core samples
are provided in Table 1. Brine (water with 1% KCl by
volume) has been used as a saturating and a displacing
fluid while heavy liquid paraffin oil is the displaced fluid
and its viscosities at room temperature of 24 C are 0.97
and 130 cP, respectively.
357
Brine
accumulator
3 Way
valve
Oil
accumulator
Pressure gage
Core holder
Dual cylinder
syringe pump
Fraction
collector
Differential pressure
transmitter
Brine reservoir
To Vacuum pump
Fig. 1. Schematic of experimental set-up for 1-D waterflood experiments.
Table 1
Properties of core samples
Core sample
Porosity
(%)
Absolute
permeability
(mD)
Swi
%PV
Sor
%PV
Injection
Rate ml/h
Cylindrical Berea
core
29.4
29.4
29.4
38.5
1131
1131
1131
1584
37.5
48.0
45.0
30
23.2
5.1
14.8
34.7
5
8
10
10
Long Berea core
38.5
38.5
1584
1584
32.5
32.2
33.4
31.7
50
100
Swi = irreducible water saturation.
Sor = residual oil saturation.
mD = mili Darcy (1 mD = 1015 m2).
The cylindrical core sample is first coated with resin and
kept in a stainless steel cylindrical core holder of 5.4 cm
internal diameter. The annular space between the core
and the core holder is filled with an alloy called ‘‘Cerro
Metal’’ (42% tin and 58% Bi by weight) to prevent leakage
of fluids from the surface of the core sample.
Fig. 2 shows schematic of apparatus for long core with
rectangular cross-section. For experiments on this apparatus, the core is first coated with a thick layer of resin. After
drying the resin, leakage test is carried out using soap bubble test with nitrogen(up to 2 bar pressure). The leakage
prone areas are then sealed and the core is made leakproof. Pressures have been measured at three different
locations along the length of the core with pressure
transmitters.
At the outset of experiments, air is removed from the
cores by a vacuum pump. When sufficient vacuum level is
achieved, the core sample is disconnected from the vacuum
P1
5
P2
19
RESIN COATING
P3
5
2.4
INLET
25
pump. Pore volume (PV) is found by determining the volume of brine needed to saturate the evacuated core. The
absolute permeability to brine is then determined by measuring the pressure drop across the core length for a known
flow rate. To mimic the reservoir condition of irreducible
water saturation, heavy paraffin oil is injected till no more
brine at outlet is observed.
After the core is prepared for the recovery experiments,
the brine is injected at a known flow rate through the core
using dual cylinder syringe pump. Three experiments have
been performed on each core at different injection rates. In
actual reservoirs, the flow velocity is about 1 ft/day. The
injection rate in the reservoir is porosity times the average
velocity times cross-sectional area available to the flow.
However, for relative permeability measurement by laboratory experiments, the maximum injection rate can be up to
1 PV/h for which the fluid velocity may exceed 1 ft/day.
Accordingly, injection rates are chosen such that they do
not exceed the injection rates of 1 PV/h. For cylindrical
core, the flooding experiments have been performed at
injection rates of 5, 8, and 10 ml/h, respectively whereas
for long core, injection rates of 10, 50, and 100 ml/h are
used. The injection rates are chosen as per the flood front
stability criteria [17] such that for cylindrical short core,
the front is unstable whereas for long square cross-section
core it is stable.
The experiments are terminated after 2 PV of brine
injection; after 2 PV, the recovery reached a plateau. During the displacement test, the fractions of oil and brine are
collected at the outlet in the fraction collector. The relative
permeability ratios, kkrw
, are then calculated from the mearo
sured oil fractions and known viscosities of oil and brine.
After completion of the experiment at a particular injection rate, the same core is flushed with oil and brought to
the irreducible water saturation and again used for experiments at different injection rates.
OUTLET
2.5
2.1. Errors in Experiments
54
P1 ; P2 ; P3
Pressure Transmitters
All dimensions are in cm
Fig. 2. Schematic of long core for waterflood experiments.
The measurement errors in the oil recovery curve are
attributed to the errors involved in the measurement of volume fractions of oil. The fraction measurements are made
in graduated glass tubes which have a least count of 0.2 ml.
358
The errors involved in the relative permeability ratios,
based on the least count, are ±2%. Based on the least count
and the difference in original oil inplace (OOIP) values,
errors in the recovery curve for short and long Berea core
are ±3% and ±1%, respectively.
3. Numerical simulation
5. Prediction of oil recovery
The average water saturation inside the core is estimated
by integrating water saturation through the core length and
can be expressed as
Z 1
S w dxD :
ð6Þ
ðS w Þav ¼
0
In numerical simulation, the capillary pressures and
gravity effects are neglected. Under such conditions, the
non-dimensional form of Buckley–Leverett equation for
the displacing phase (water) is [9]
oS w ofw ðS w Þ
þ
¼ 0;
oxD
otD
1 þ llwo
k ro
k rw
ð2Þ
;
where lw and lo are the viscosities of displacing phase and
displaced phase (oil), respectively, and krw and kro are the
relative permeabilities of displacing phase and displaced
phase, respectively.
tqt
The non-dimensional time can be expressed as, tD ¼ /L
,
where, qt is the flow flux at the inlet, L is the length of
the core, / is the porosity of the porous medium, and t is
the time for injection. (Here tD represents the cumulative
water injection as a fraction of pore volume (PV).)
Eq. (1) is solved using general PDE model of FEMLAB
given as
da
oS w
þ r C ¼ F;
ot
ð3Þ
where da is mass coefficient, F is the source term, and C is
flux vector. For modeling Eq. (1) using Eq. (3), following
parameters are used
d a ¼ 1;
C ¼ 0;
and F ¼ ðS w Þav S wi
100:
1 S wi
ð7Þ
ofw ðS w Þ
;
oxD
where the source term F can be rewritten as
ofw ðS w Þ oS w
F ¼
:
oS w
oxD
6. Results and discussion
The experimental and simulation results of one dimensional waterflood on two different cores are presented here.
The effects of injection rates on water–oil relative permeability ratios and total oil recovery for short cylindrical
core are shown in Fig. 3 and 4, respectively.
It can be observed from Fig. 3 that as the terminal water
saturation, Sw2, increases, the water to oil relative permeability ratio increases. Also, as the injection rate increases
from 5 to 8 ml/h, the relative permeability ratio decreases
which in turn indicates the increase in the oil relative permeability. However, as injection rate increases from 8 to
10 ml/h, the relative permeability ratio increases which
indicates the decrease in oil relative permeability for the
same terminal water saturation. This concludes that for
the injection rate of 8 ml/h, the recovery should be maximum as the oil relative permeability is maximum for the
same terminal water saturation.
From Fig. 4, it can be observed that as injection rate
increases from 5 to 8 ml/h, the total recovery increases.
However, for an injection rate of 10 ml/h, the total recovery decreases which indicates that there is an optimum
injection rate for which the recovery is maximum. This is
ð4Þ
ð5Þ
It is to be noted that the experimental relative permeability
ratio curves for short cylindrical and long square cross-section Berea core have been used for the numerical
simulations.
4. Initial and boundary conditions
Initially, the core will be at the irreducible water saturation, Swi which is determined from the laboratory
experiments.
At the inlet (xD = 0), water saturation, Sw is 1.
Relative permeability ratio (K rw /K ro )
1
%OOIP ¼
ð1Þ
where Sw is the water saturation, fw is the fractional flow of
water, tD is the non-dimensional time, and xD ¼ Lx is the
non-dimensional distance from the inlet. The fractional
flow of water neglecting the capillary pressure, fw is given as
fw ¼
The percent original oil in place (OOIP) recovered is calculated from
101
10
0
1
10
5 ml/hr
8 ml/hr
10 ml/hr
10 2
0.4
0.5
0.6
0.7
0.8
0.9
1
Terminal water saturation (S w2 ), % PV
Fig. 3. Experimental water to oil relative permeability ratio curve for
short Berea core sample for different injection rates. PV = 23.3 ml,
K = 1131 mD. Lines joining the points show the trend of curves.
359
60
50
80
Recovery, % OOIP
Recovery, % OOIP
100
60
40
5 ml/hr
8 ml/hr
20
0
40
30
20
10 ml/hr
0
0
0.5
1
1.5
Injection, PV
2
10 ml/hr
50 ml/hr
100 ml/hr
10
2.5
0
0.5
1
1.5
2
2.5
Injection, PV
Fig. 6. Effect of injection rate on recovery of long Berea core.
PV = 126 ml, K = 1584 mD. Lines joining the points show the trend of
curves.
also supported by the relative permeability ratio curves
(Fig. 3). At an injection rates higher than the optimum,
the recovery reduces, which can be attributed to the phenomena of viscous fingering [18]. Moreover, the flood front
in cylindrical core is unstable and for such flood fronts, the
recovery depends on injection rates. Hence, there is an optimum injection rate for such unstable fronts and above this
injection rate, due to viscous fingering, recovery decreases.
The relative permeability ratios obtained from the experimental data have been directly implemented in the numerical simulation.
The experimental relative permeability ratio curves for
long Berea core at different injection rates are shown in
Fig. 5. It can be observed from Fig. 5 that the relative permeability ratio curves exhibit an exponential relationship
with water saturation. Moreover, there is marginal effects
of injection rates on relative permeability ratios for the
injection rates considered in flooding experiments.
The effect of injection rates on recovery performance of
long Berea core is shown in Fig. 6. It can be observed from
Fig. 6 that as injection rate increases from 10 to 100 ml/h,
the recovery increases. Unlike the recovery performance of
short core, optimum injection rate is not observed for the
injection rates considered here. Moreover, as the injection
rate is increased from 50 to 100 ml/h, only 3% increase in
the ultimate recovery is observed after 2 PV of injection.
This can be attributed to the marginal decrease in the
water–oil relative permeability ratios as can be observed
20
P
1
Pressure (psi)
15
P3
0.25
0.5
0.75
Injection, PV
1.0
Fig. 7. Measured pressure variations with PV injection for pressure
transmitters P1, P2, and P3 in long Berea core. 1 psi = 6987.9 N/m2.
1
0
10
101
10 ml/hr
50 ml/hr
100 ml/hr
35
40
45
50
55
0.8
Oil
0.6
60
Terminal water saturation (S w2 ), % PV
Fig. 5. Effect of injection rate on water/oil relative permeability ratio for
long Berea core. PV = 126 ml, K = 1584 mD. Lines joining the points
show the trend of curves.
Brine
0.4
0.2
2
30
10
0
101
10
P2
5
Water saturation (S w )
Relative permeability ratio (K rw /K ro )
Fig. 4. Effect of injection rate on recovery of short Berea core sample.
PV = 23.3 ml, K = 1131 mD. Lines joining the points show the trend of
curves.
0
0.2
0.4
0.6
0.8
Non dimensional distance (x D )
Swi
1
Fig. 8. Numerical simulation flood front position in a short Berea core
sample after 0.1 PV of injection at an injection rate of 5 ml/h.
PV = 23.3 ml, K = 1131 mD, Swi = 37.5% PV, Sor = 23.2% PV. The black
region is thick because of more data points.
360
from Fig. 5. Moreover, the flood front in long core is stabilized and hence oil recovery is independent of injection
rates.
The pressure variations with respect to the volume of
brine injection, along the length of the long core, is shown
in Fig. 7 for three pressure transmitters P1, P2, and P3,
respectively. It can be observed from the pressure plot that
as PV injection increases, the pressure inside the core
decreases. Eventually, the pressure at each location
becomes constant when no further oil is recovered.
The numerical simulation of the flood front position in a
short cylindrical core for an injection of 0.1 PV at an injection rate of 5 ml/h is shown in Fig. 8. The area below the
curve represents the total oil displaced by the brine above
the irreducible water saturation, Swi, whereas area above
the curve represents the amount of oil left in the core after
0.1 PV of brine injection.
The comparison of numerical simulation results with the
experimental results at different injection rates for short
cylindrical core are shown in Fig. 9. In the numerical simulation, experimentally obtained relative permeability
ratios are used instead of individual phase relative permeability curves. It can be observed from Fig. 9a that for
injection rate of 5 ml/h, the simulation and experimental
results are in good agreement. The initial slope, however,
of the experimental recovery curve is not well predicted
by the simulation. For injection rates of 8 and 10 ml/h
(Fig. 9b and c, respectively), it can be observed that the
simulation results are underpredicting the experimental
60
Recovery, % OOIP
Recovery, % OOIP
60
40
Experiment
Simulation
20
0
0
50
40
30
20
Experiment
Simulation
10
0.5
1
1.5
Injection, PV
2
2.5
0
0
0.5
1
1.5
Injection, PV
2
2.5
100
Recovery, % OOIP
Recovery, % OOIP
60
80
60
40
Experiment
Simulation
20
0
0
50
40
30
20
Experiment
Simulation
10
0.5
1
1.5
Injection, PV
2
2.5
0
0
0.5
1
1.5
Injection, PV
2
2.5
80
Recovery, % OOIP
Recovery, % OOIP
60
60
40
Experiment
Simulation
20
0
0
0.5
1
1.5
Injection, PV
2
2.5
Fig. 9. Comparison of numerical and experimental recovery performance
of 1-D waterflood of short Berea core sample for injection rates of (a)
5 ml/h (Swi = 37.5% PV, OOIP = 14.5 ml, Sor = 23.2% PV) (b) 8 ml/h
(Swi = 48% PV, OOIP=12.1 ml, Sor = 5.1% PV), and (c) 10 ml/h
(Swi = 45% PV, OOIP=12.8 ml, Sor = 14.8% PV). PV = 23.3 ml,
K = 1131 mD for all the cases.
40
20
0
Experiment
Simulation
0
0.5
1
1.5
Injection, PV
2
2.5
Fig. 10. Comparison of numerical and experimental recovery performance of 1-D waterflood of long Berea core sample for injection rates of
(a) 10 ml/h (Swi = 30% PV, OOIP = 88.8 ml, Sor = 34.7% PV) (b) 50 ml/h
(Swi = 32.5% PV, OOIP = 85 ml, Sor = 33.4% PV), and (c) 100 ml/h
(Swi = 32.2% PV, OOIP = 85.4 ml, Sor = 31.7% PV) PV = 126 ml for all
the cases.
361
recoveries. The careful observation from Fig. 9 shows that
the recovery is maximum for the injection rate of 8 ml/h
which is also supported by the experimental recovery
curves (Fig. 4). The error bars in the experimental recovery
curve corresponds to the error in the volume fraction measurement, as discussed earlier.
The comparison of numerical simulation results with the
experimental results for long Berea core at different injection rates are shown in Fig. 10. It is found that for all injection rates, the initial slope of the recovery curve is well
predicted by the numerical simulation. It is also observed
that the simulation result greatly underestimated the experimental result for an injection rate of 10 ml/h (Fig. 10a),
slightly underestimated for an injection rate of 50 ml/h
(Fig. 10b) and overestimated for an injection rate of
100 ml/h (Fig. 10c). This is because for an injection rate
of 10 ml/h, the flood front is just stable and water/oil relative permeability ratio is higher compared to higher injection rates of 50 and 100 ml/h for which the flood front is
fully stabilized. However, with increase in injection rate
from 50 to 100 ml/h, due to marginal decrease in the relative permeability ratios, the oil recovery increases. However, the maximum oil recovery predicted by the
simulation differs from the experimental recovery by 3%.
7. Conclusions
In this paper, the experimental and numerical studies
have been carried out to predict the one dimensional waterflood performance of short and long Berea cores at different injection rates. The relative permeability ratios
obtained from experiments are used in the numerical simulation. Experimental studies on short core concludes that
there is an optimum injection rate at which recovery is
maximum. Comparison of experimental and numerical
simulation results show reasonable agreement for short
and long cores. It is also observed that there is a good
match of the initial slopes of the recovery curves.
References
[1] J.S. Archer, S.W. Wong, Use of reservoir simulator to interpret
laboratory waterflood data, SPEJ (1973) 343–347.
[2] S.F. Buckley, M.C. Leverett, Mechanism of fluid displacement in
sands, Petroleum Transactions AIME 146 (1942) 107–116.
[3] G.L. Chierici, Novel relations for drainage and imbibition relative
permeabilities, SPEJ (1984) 275–276.
[4] J. Douglas Jr., P.M. Blair, R.J. Wagner, Calculation of linear
waterflood behavior including the effects of capillary pressure,
Petroleum Transaction AIME 213 (1958) 96–102.
[5] T.M. Geffen, W.W. Owens, D.R. Parrish, R.A. Morse, Experimental
investigation of factors affecting laboratory relative permeability
measurements, Petroleum Transaction AIME 192 (1951) 99–110.
[6] B.S. Gottfried, W.H. Guilinger, R.W. Snyder, Numerical solution of
the equations for one dimensional multi phase flow in porous media,
SPEJ (1966) 62–72.
[7] E.F. Johnson, D.P. Bossler, V.O. Naumann, Calculation of relative
permeability from displacement experiment, Petroleum Transaction
AIME 216 (1959) 370–372.
[8] S.C. Jones, W.O. Roszelle, Graphical techniques for determining
relative permeability from displacement experiments, Journal of
Petroleum Technology 30 (1978) 807–817.
[9] L.W. Lake, Enhanced Oil Recovery, Prentice Hall, Englewood Cliffs,
USA, 1989.
[10] S. Macary, A.A. Walid, Creation of fractional flow curve from purely
production data. SPE Paper 56830, in: 1999 SPE Annual Technical
Conference and Exhibition held in Houston, Texas, 1999 October 3–
6.
[11] C.R. McEven, A numerical solution of the linear displacement
equation with capillary pressure, Transactions of AIME 216 (1959)
412–415.
[12] V.S. Mitlin, B.D. Lawton, J.D. McLennan, L.B. Owen, Improved
estimation of relative permeability from displacement experiments,
SPE Paper 39830, in: 1998 SPE International Petroleum Conference
and exhibition of Mexico held in Villahermosa, Mexico, 1998 March
3–5.
[13] J.G. Richardson, The calculation of waterflood recovery from steady
state relative permeability data, Transactions of AIME 210 (1957)
373–375.
[14] J.W. Sheldon, W.T. Cardwell, One dimensional, incompressible, non
capillary flow, two phase fluid flow in a porous medium, Transactions
of AIME 216 (1959) 290–296.
[15] T.M. Tao, A.T. Watson, Accuracy of JBN estimates of relative
permeability: Part 1 – Error analysis, SPEJ (1984) 209–214.
[16] H.J. Welge, A simplified method for computing oil recovery by gas or
water drive, Transactions of AIME 195 (1952) 91–98.
[17] G.P. Willhite, Waterflooding, SPE Textbook Series, vol. 3, Richardson, TX, USA, 2004, p. 89.
[18] Z.F. Zhang, J.E. Smith, Visualization of DNAPL fingering processes
and mechanisms in water saturated porous media, Transport in
Porous Media 48 (2002) 41–59.