ren
re
ra
ring,
ised f
1. Introduction
crease recovery in fractured reservoirs. However, the
performance of waterflooding depends crucially on
fractures.
Imbibition can take place by co-current and/or
nd Chaney, 1966;
Journal of Petroleum Science and EnginFractured carbonate reservoirs are important oil
and gas resources. These reservoirs are composed
of two continua: the fracture network and matrix.
The fractures typically have a high permeability but
very low volume compared to the matrix, whose
permeability may be several orders of magnitude
lower, but which contains the majority of the reco-
verable oil.
Waterflooding is frequently implemented to in-
the wettability of the reservoir. If the reservoir is
oil-wet, water will not readily displace oil in the
matrix and only the oil in the fractures will be
displaced, resulting in poor recoveries and early
water breakthrough. In water-wet fractured reservoirs,
imbibition can lead to significant recoveries. Imbibi-
tion is the mechanism of displacement of non-wetting
phase by wetting phase. Strong capillary forces lead
to the imbibition of water as the wetting phase into
the matrix and the discharged oil is displaced into theAbstract
Counter-current imbibition, where water spontaneously enters a water-wet rock while oil escapes by flowing in the opposite
direction is a key recovery mechanism in fractured reservoirs. Fine grid, one- and two-dimensional simulations of counter-current
imbibition are performed and the results are compared with experimental measurements in the literature. The experimental data are
reproduced using two sets of relative permeabilities and capillary pressures. One set is derived from a counter-current experiment
and one set is computed using pore-scale modelling.
Two-dimensional simulations of water flow through a single high permeability fracture in contact with a lower permeability
matrix are run. The results are compared with experimental measurements of fracture flow and matrix imbibition in the literature
and with one-dimensional simulations that account for imbibition from fracture to matrix using an empirical transfer function. It is
shown that with this transfer function the behavior of the two-dimensional displacement can be predicted using a one-dimensional
model.
D 2005 Published by Elsevier B.V.
Keywords: Counter-current imbibition; Dual porosity models; Fracture/matrix transfer function; Network modellingSimulation of counter-cur
fractured
Hassan Sh. Behbahani, Ginev
Department of Earth Science and Enginee
Received 26 November 2003; received in rev0920-4105/$ - see front matter D 2005 Published by Elsevier B.V.
doi:10.1016/j.petrol.2005.08.001
* Correspondi
7444.
E-mail address: [email protected] (M.J. Blunt).t imbibition in water-wet
servoirs
Di Donato, Martin J. Blunt *
Imperial College London, SW7 2AZ, UK
orm 5 August 2005; accepted 9 August 2005
eering 50 (2006) 2139
www.elsevier.com/locate/petrolal, 1986; Bourbiauxcounter-current flow (Parsons a
Iffly et al., 1972; Hamon and Vidng author. Tel.: +44 20 7594 6500; fax: +44 20 7594and Kalaydjian, 1990; Al-Lawati and Saleh, 1996;
eum SPooladi-Darvish and Firoozabadi, 2000). In co-current
flow the water and oil flow in the same direction, and
water pushes oil out of the matrix. In counter-current
flow, the oil and water flow in opposite directions,
and oil escapes by flowing back along the same
direction along which water has imbibed. Co-current
imbibition is faster and can be more efficient than
counter-current imbibition (Bourbiaux and Kalaydjian,
1990; Chimienti et al., 1999; Pooladi-Darvish and
Firoozabadi, 2000) but counter-current imbibition is
often the only possible displacement mechanism for
cases where a region of the matrix is completely
surrounded by water in the fractures (Pooladi-Darvish
and Firoozabadi, 2000; Najurieta et al., 2001; Tang
and Firoozabadi, 2001). Experimentally this process
can be studied by surrounding a core sample with
water and measuring the oil recovery as a function of
time (Iffly et al., 1972; Du Prey, 1978; Hamon and
Vidal, 1986; Bourbiaux and Kalaydjian, 1990; Cuiec
et al., 1994; Zhang et al., 1996; Cil et al., 1998;
Chimienti et al., 1999; Rangel-German and Kovscek,
2002). The imbibition rate is controlled by the per-
meability of the matrix, its porosity, the oil/water
interfacial tension and the flow geometry (Mattax
and Kyte, 1962; Iffly et al., 1972; Hamon and
Vidal, 1986; Babadagli and Ershaghi, 1992; Al-Lawati
and Saleh, 1996; Ma et al., 1997; Cil et al., 1998;
Chimienti et al., 1999) although the ultimate recovery
is generally only governed by the residual oil satura-
tion in strongly water-wet systems.
Correlations have been developed to predict the
recovery from counter-current imbibition as a func-
tion of time for different samples. Mattax and Kyte
(1962) hypothesized that the oil recovery for sys-
tems of different size, shape and fluid properties
was a unique function of a dimensionless time.
Ma et al. (1997) modified an expression derived
by Mattax and Kyte (1962) to include the effect
of the non-wetting phase viscosity. Their experimen-
tal results showed that the imbibition time is in-
versely proportional to the geometric mean of the
water and oil viscosities. They proposed the follow-
ing correlation:
tD tK
/
sr
lwlop 1
L2c1
where t is time, K is permeability, / is porosity, ris interfacial tension, lw and lo are water and oilviscosities and L is the characteristic length that is
H.Sh. Behbahani et al. / Journal of Petrol22c
determined by the size, shape, and boundary condi-tions of the sample and is defined by Zhang et al.
(1996) as:
Fc 1Vma
Xs
Ama
lma2
Lc 1
Fc
r3
where Vma is the bulk volume of the matrix (core
sample), Ama is the area of a surface open to flow in
the flow direction, lma is the distance from the open
surface to the no flow boundary and the summation is
over all open surfaces of the block.
Ma et al. (1997) showed that recovery as a function
of time for a variety of experiments on different water-
wet samples fall on a single universal curve as a func-
tion of the dimensionless time, tD, Eq. (1). In particular,
imbibition experimental data presented by Mattax and
Kyte (1962) for Alundum samples and Weiler sand-
stones, Hamon and Vidals (1986) results for synthetic
materials and Zhang et al.s (1996) results for Berea
sandstones with different boundary conditions all
scaled onto the same curve that was reasonably well
fitted by the following empirical function first proposed
by Aronofsky et al. (1958):
R Rl 1 eatD 4
where R is the recovery, Rl is the ultimate recovery
and a is a constant that best matches the data with avalue of approximately 0.05.
Eq. (1) was proposed for strongly water-wet media
and ignores the effects of wettability. Gupta and
Civan (1994) and Cil et al. (1998) extended the
definition of dimensionless time, Eq. (1), by including
a cosh term, where h is the oil/water contact angle torepresent core wettability. For water-wet systems it is
assumed that cosh =1. Omitting the impact of wetta-bility in Eq. (1) means that the coefficient a in Eq. (4)can also be a function of contact angle distribution.
Furthermore, other authors have presented extensions
to Eq. (4) that account for transfer from matrix to
fracture and transfer from dead-end pores that better
match experimental data (see, for instance, Civan,
1994; Gupta and Civan, 1994; Viksund et al., 1998;
Terez and Firoozabadi, 1999; and Matejka et al.,
2002).
Zhou et al. (2002) correlated the results of counter-
current imbibition experiments on water-wet samples of
cience and Engineering 50 (2006) 2139a diatomite outcrop with very high porosity and low
q. (1)
; Zhan
H.Sh. Behbahani et al. / Journal of Petroleum Science and Engineering 50 (2006) 2139 23permeability for a wide range of mobility ratios with the
scaling relation:
tD tK
/
srL2c
k rw4 k ro4
p 1M4
p 1M4
p5
where kr*=kr* /l is defined as a characteristic mobilityand M*(=krw* /kro* ) is defined as a characteristic mobil-ity ratio. They used end-point relative permeabilities to
Fig. 1. Normalized oil recovery as a function of dimensionless time, E
Kyte, 1962; Hamon and Vidal, 1986; Bourbiaux and Kalaydjian, 1990
empirical fit to the data by Ma et al. (1997), Eq. (4).calculate kr* and M*.Fig. 1 shows experimental imbibition data in the
literature. While the data considered by Ma et al.
(1997) does approximately lie on a single curve
given by Eq. (4), consideration of other results on
water-wet samples from Bourbiaux and Kalaydjian
(1990), Cil et al. (1998) and Zhou et al. (2000) shows
more scatter in particular for some of the experi-
ments the imbibition rate is much slower than predicted
by Eq. (4). The experiments analyzed by Ma et al.
Fig. 2. Grid system for 1D simulations of counter-curren(1997) were performed on either artificial porous
media (that is, not rock samples) or with an initial
water saturation of zero. The other data presented
(Bourbiaux and Kalaydjian, 1990; Cil et al., 1998;
Zhou et al., 2000) are all performed on sandstone
cores with an initial water saturation present, which is
more likely to be representative of reservoir displace-
ments. Only the data for a nominal aging time of zero
from Zhou et al.s (2000) experiments were used to
, for counter-current imbibition on different core samples (Mattax and
g et al., 1996; Cil et al., 1998; Zhou et al., 2000). The solid line is ancompare with other water-wet imbibition experimental
results in Fig. 1, although in this case crude oil is the
oleic phase and the system may not be strongly water-
wet. It is possible that the scatter in this data comes
from ignoring wettability in Eq. (1). However, while
representing wettability in terms of cosh is appealing,the assignment of a single effective average contact
angle is not physically correct for systems where
there is a distribution of contact angles (Jackson et
al., 2003; Behbahani and Blunt, 2004).
t imbibition. A total of 42 grid blocks were used.
Furthermore the behavior for different initial water
saturations is a challenge. The presence of initial water
saturation reduces capillary pressure but increases the
mobility of invading water. The competition between
capillary pressure and relative permeability determines
the recovery rate. Viksund et al. (1998) demonstrated
rom counter-current imbibition experiments by Bour-
iaux and Kalaydjian (1990) were used (Table 2). They
erformed co-current, counter-current and simultaneous
o- and counter-current wateroil imbibition tests on
trongly water-wet sandstone cores. They simulated
ounter-current imbibition using a 1D numerical
odel. They reduced the measured co-current relative
ermeabilities by approximately 30% to match the
able 2
atrix rock and fluid properties used in 1D and 2D counter-current
imulations
arameter Unit Base models* Network
modelling
data**
orosity frac 0.233 0.2
ermeability m2 1241015 3.1481012il density Kg m3 760 835
able 3
racture and matrix dimensions in 1D and 2D counter-current simula-
ons
imension Unit 1D modelling 2D modelling
Fracture Matrix Fracture Matrix
cm 6.1 6.1 9 9
cm 1.2 28 9 9
cm 2.1 2.1 1 1
H.Sh. Behbahani et al. / Journal of Petroleum Science and Engineering 50 (2006) 213924that the imbibition rate for low permeability Berea,
shown in Fig. 1, is around eight times slower than for
most other media. Baldwin and Spinler (2002) moni-
tored saturation profiles during spontaneous counter-
current imbibition using magnetic resonance imaging
(MRI)they showed a transition from a flat frontal
advance to a more gradual water encroachment as the
initial water saturation was increased.
2. Simulation of 1D-counter-current imbibition
Counter-current imbibition in one dimension (1D)
was modeled using EclipseR-100, which is an industry-standard reservoir simulator. In the 1D model, illustrat-
ed in Fig. 2, a region initially full of water with fracture
properties (Tables 1 and 3) was connected to a thin
matrix slab (Tables 2 and 3). A total of 42 grid blocks
was used in the simulation, with smaller blocks at the
inlet to capture accurately the initial advance of water
into the matrix. Sensitivity studies using 200 grid
blocks demonstrated that sufficient grid blocks were
used to obtain converged results. All other faces not
in contact with the fracture were closed (no flow bound-
aries). The relative permeabilities of the fractures were
assumed to be linear functions of saturation with no
irreducible or residual saturation and the capillary pres-
sure in the fracture was zero. The matrix relative per-
meabilities and capillary pressures are discussed later.
Conservation of water volume assuming incom-
pressible flow and Darcys law in one dimension with
no total velocity can be expressed as follows:
/BSw
Bt BBx
kwkokt
KBPc
BSw
BSw
Bx gx qw qo
0
6where Pc is the capillary pressure, the mobility k =kr /lwhere kr is the relative permeability and kt=kw+ko. q
Table 1
Fracture properties used in 1D and 2D counter-current simulations
Parameter Unit Value
Initial water saturation Fraction 1.0
Porosity Fraction 1.0
Capillary pressure Pa 0.0Permeability m2 501012f
b
p
c
s
c
m
p
T
F
ti
D
X
Y
Zis the density and gx is the component of gravity in the
flow direction. The reservoir simulator solves Eq. (6)
on the grid shown in Fig. 2 using an implicit finite
difference formulation with upstream weighting of the
mobility terms (Aziz and Settari, 1979). It is assumed
that this conventional treatment of the flow equations
with saturation-dependent capillary pressure, Pc, and
relative permeabilities, kr, is sufficient to predict the
results of the experiments. Recently it has been sug-
gested by Barenblatt et al. (2003) that a fundamentally
different formulation of the conservation equations with
rate-dependent coefficients is necessary to explain the
results of imbibition experiments.
2.1. Base models
For the base case, the matrix and fluid properties
Oil viscosity Pa s 1.5103 39.25103***Water density Kg m3 1090 1010Water viscosity Pa s 1.2103 0.967103***Interfacial tension mN m1 35 30Initial water saturation Fraction 0.40 0.25
Residual oil saturation Fraction 0.422 0.75
*: From Bourbiaux and Kalaydjian (1990).
**: From Jackson et al. (2003).
***: From Zhou et al. (2000).T
M
s
P
P
P
OLc cm 28 3.18 28 3.18
counter-current experiment results and concluded that biaux and Kalaydjian (1990) using the same relative
Fig. 3. Matrix imbibition relative permeabilities and capillary pressure used in the base models from Bourbiaux and Kalaydjian (1990).
H.Sh. Behbahani et al. / Journal of Petroleum Science and Engineering 50 (2006) 2139 25co-current relative permeability measurements cannot
be used to simulate counter-current imbibition. Count-
er-current relative permeabilities and imbibition capil-
lary pressure from Bourbiaux and Kalaydjian (1990),
Fig. 3, were used in all base models.
Fig. 4 shows the measured and predicted oil recov-
ery from Bourbiaux and Kalaydjians (1990) experi-
ments. In the experiments, the core was held
vertically, and so gravity effects are included in the
simulation. The simulation results are identical to
experiments and to simulations performed by Bour-Fig. 4. Comparison of experimental and simulated (crosses) recoveries for co
Kalaydjian (1990). Also shown are the simulated results from Bourbiaux anpermeabilities. The relative permeabilities and capillary
pressure used for the simulations were chosen by Bour-
biaux and Kalaydjian (1990) to match their experimen-
tal results and so this is not a genuine prediction.
Fig. 5 shows comparisons of the simulation results
with the experimental data in Fig. 1. In most of these
experiments the cores were held horizontally, and so for
comparison we show a simulation result where gravi-
tational effects have been neglected. This makes little
difference to the results, except at late time and slightly
improves the match to some of the other data.unter-current imbibition. The experimental data is from Bourbiaux and
d Kalaydjian (1990) that agree very well with our results.
sing
the M
H.Sh. Behbahani et al. / Journal of Petroleum Science and Engineering 50 (2006) 2139262.2. Experimental data predicted by network modelling
The pore-scale model used a network derived from
a sample of Berea sandstone and can accurately predict
steady-state waterflood relative permeabilities for
Berea (Blunt et al., 2002; Jackson et al., 2003; Val-
vatne and Blunt, 2004). The void space of the rock was
described by a network of pores connected by throats.
The pores and throats were assigned some idealized
Fig. 5. Comparison of 1D vertical and horizontal simulation results u
Kalaydjian (1990) compared to experimental data in the literature andgeometry and rules were developed to determine the
multiphase fluid configurations and transport in these
elements. The rules were combined in the network to
Fig. 6. Matrix waterflood relative permeabilities and capillary pressure derivecompute effective transport properties on a mesoscopic
scale some tens of pores across. The appropriate pore-
scale physics combined with a geologically represen-
tative description of the pore space gives a model that
can predict average behavior, such as capillary pressure
and relative permeability (Blunt et al., 2002; Jackson et
al., 2003; Valvatne and Blunt, 2004). The imbibition
relative permeabilities and capillary pressure predicted
for Berea by network modelling and used in the simu-
relative permeability and capillary pressure data from Bourbiaux and
a et al. (1997) expression, Eq. (4).lations are shown in Fig. 6.
The experiments by Zhou et al. (2000) were per-
formed on Berea sandstone. Hence, of all the results
d from network modelling of Berea sandstone by Jackson et al. (2003).
el de
ne.
H.Sh. Behbahani et al. / Journal of Petroleum Science and Engineering 50 (2006) 2139 27shown in Fig. 1, it is expected that the network
model properties best represent this experiment. For
a more careful comparison the oil and water visco-
sities from Zhou et al. (2000) were used in the
simulations.
Fig. 7 shows the results of 1D counter-current
imbibition simulations using network model derived
data. The simulations accurately predict Zhou et al.s
(2000) experimental results. This is a genuine predic-
Fig. 7. 1D counter-current imbibition simulations using network mod
literature and the experiments of Zhou et al. (2000) on Berea sandstotion of the experimental results, in that the relative
permeabilities and capillary pressure used were inde-
Fig. 8. Grid system used for the 2D counter-current imbibitionpendently computed and no parameters were tuned to
match the results. The predicted imbibition rate is
approximately ten times slower than measured on
artificial porous media or systems with no initial
water saturation present, as shown experimentally by
Viksund et al. (1998).
3. Simulation of 2D-counter-current imbibition
rived data shown in Fig. 6 compared to experimental results in theA series of 2D simulations were run to check the
1D results. The grid system used is shown in Fig. 8.
simulations. A total of 115,236 grid blocks were used.
Fig. 9. Comparison of 2D counter-current imbibition simulations using Bourbiaux and Kalaydjian (1990) (base case, Fig. 3) and network model
properties (Fig. 6) with literature experimental data.
H.Sh. Behbahani et al. / Journal of Petroleum Science and Engineering 50 (2006) 213928A total of 115,235 fine grid blocks were used.
Sensitivity studies using 90,000 and 160,000 grid
blocks demonstrated that sufficient grid blocks
were used to obtain converged results. Fig. 9
shows the results for both sets of relative perme-
abilities. For the 2D simulations, including gravity
had no effect on the results. Again the agreement
with experiments is good, particularly at early times.
The results are similar to those obtained in 1D, Figs.5 and 7.
Fig. 10. Effect of different oil to water viscosity ratios,M, on the results of 1D
as a function of the dimensionless time, tD, proposed by Ma et al. (1997), E
validating the correlation.4. Validity of the correlations, Eqs. (1) and (5)
The dimensionless time, Eq. (1), can be derived
using dimensional analysis. However, the scaling with
viscosity is not obvious. Mattax and Kyte (1962) sug-
gested that tD should be inversely proportional to the
water viscosity, while a recent analytical approach by
Tavassoli et al. (2005) suggests that tD is inversely
proportional to the oil viscosity. The scaling in Eq.(1) involving the geometric average of the oil and
simulations using network model derived data. The results are plotted
q. (1). Notice that all the plots approximately fall onto a single curve,
Fig. 11. Using only the oil viscosity in the definition of dimensionless time leads to a large scatter in the results from 1D simulations with different
oil to water viscosity ratios, M, using network model derived data.
H.Sh. Behbahani et al. / Journal of Petroleum Science and Engineering 50 (2006) 2139 29water viscosities was based on experimental results for
systems with different viscosity ratio by Ma et al.
(1997). Zhou et al.s (2002) scaling equation, Eq. (5),
also accounts for the mobility of wetting and non-wet-
ting phases. To check the validity of Eqs. (1) and (5), a
series of 1D-simulations for different oil to water vis-
cosity ratios were performed using the network model
derived data. The experiments by Zhang et al. (1996)
covered a range of viscosity ratios from 4 to 160 inFig. 12. Using only water viscosity in the definition of dimensionless time le
oil to water viscosity ratios, M, using network model derived data.our simulations a range from 0.01 to 200 was used. The
recovery from the simulations plotted as a function of
the dimensionless time, Eq. (1), is shown in Fig. 10.
While it is evident that all the recoveries do not fall
exactly on the same universal curve, there is little scatter
in the results. In contrast using either the oil viscosity
(Fig. 11) or the water viscosity (Fig. 12) only in the
definition of dimensionless time or Eq. (5) (Fig. 13)
leads to a much larger scatter in the curves, verifyingads to a large scatter in the results from 1D simulations with different
scatte
eum Sthat Eq. (1) is at least approximately correct as the
scaling function. These results confirm previous analy-
tical and experimental studies by Ruth et al. (2000) and
Fig. 13. Using Zhou et al. (2002) dimensionless time, Eq. (5) leads to a
ratios, M, using network model derived data.
H.Sh. Behbahani et al. / Journal of Petrol30Li et al. (2003). Ruth et al. (2003) performed a nume-
rical study of the effect of matrix shape on imbibition
rate and confirmed the accuracy of the expression for
Lc, Eqs. (1)(3).
5. Theory and simulation of fracture flow and
imbibition
5.1. Previous experimental studies
Rangel-German and Kovscek (2002) investigated
the rate of fracture to matrix transfer and the pattern
of wetting fluid imbibition using CT scanning. They
studied an air/water system where water flowed in a
constant aperture fracture along the side of the system,
and imbibed into the lower permeability matrix.
Experiments with different flow rates and fracture
apertures illustrated two different flow regimes.
When the flow in the fractures was slow, there was
sufficient time for the matrix to become saturated with
water and there was a frontal advance in both matrix
and fracturethis was called the filling-fracture re-
gime and the amount of water in the matrix increased
linearly with time. At higher fracture flow rates, water
in the fracture moved ahead of water in the matrix.This was called the instantly filled regime and the
amount of water in the matrix increased with the
square root of time.
r in the results from 1D simulations with different oil to water viscosity
cience and Engineering 50 (2006) 21395.2. Theory for dual porosity systems
At the field scale, flow in fractured reservoirs is
simulated using a dual porosity approach (Barenblatt
and Zheltov, 1960; Warren and Root, 1963) where
conceptually the reservoir is composed of two domains:
a flowing fraction, representing the fractures; and a
relatively stagnant matrix. Transfer of fluid between
fracture and matrix is represented by an transfer func-
tion. For incompressible flow of oil and water in 1D the
conservation equations for water are (Kazemi et al.,
1992; Di Donato et al., 2003):
/fBSwf
Bt mt Bfwf
Bx gx BG
Bx T 7
/m BSwm
Bt T 8
where the subscript f stands for fracture and m for
matrix. vt is the total velocity. fwf is the water fractional
flow in the fractures ignoring gravity:
fwf kwfkwf kof 9
lution integral to compute the matrix saturation. This is
consistent with using:
T b/m Swf 1 Somr Swmi Swm Swmi 15
A similar expression was derived by Kazemi et al
(1992) from de Swaans (1978) convolution integral
eum Science and Engineering 50 (2006) 2139 31and:
G qw qo Kfwfkof 10
T is the transfer function that represents the rate at
which water transfers from fracture to matrix.
de Swaan (1978), Kazemi et al. (1992), Di Donato
et al. (2003), Di Donato and Blunt (2004) and Huang
et al. (2004) developed transfer functions that repro-
duce the exponential functional formEq. (4)that
matches imbibition experiments. If Swm is the average
saturation in the matrix:
R
Rl Swm Swmi
1 Somr Swmi 11
where Swmi is the initial water saturation in the matrix
and Somr is the corresponding residual oil saturation,
then from Eq. (4):
Swm Swmi 1 Somr Swmi 1 eatD 12Then from Eq. (12):
/mBSwm
Bt T a tD
t/m 1 Somr Swm
b/m 1 Somr Swm 13
where the rate constant b is defined by Eq. (13).In the experiments reviewed previously the effective
fracture saturation was held at 1. Di Donato et al.
(2003) assumed that the transfer function is indepen-
dent of the fracture saturation, as long as SwfN0. This isconsistent with assuming that the capillary pressure in
the low permeability matrix is much higher than in the
Table 4
Fracture properties used in 2D fracture/matrix simulations
Parameter Unit Value
Initial water saturation fraction 0.0
Porosity fraction 1.0
Capillary pressure Pa 0.0
Permeability m2 151012Water relative permeability krw=Sw
2
Oil relative permeability kro= (1Sw)2
H.Sh. Behbahani et al. / Journal of Petrolfractures the driving force for imbibition is the matrix
capillary pressure and imbibition continues until the
matrix and fracture capillary pressures are equal
when Swf=0 and Swm=1Sorm. Thus:T b/m 1 Somr Swm SwfN0 0 Swf 0 14
The transfer function is a linear function of the matrix
saturation.
de Swaan (1978) derived a similar transfer function
to Eq. (14) when Swf=1. For Swfb1 he used a convo-This formulation is correct if the large-scale fracture
saturation is viewed as being an average of fully satu-
rated fractures undergoing imbibition and completely
dry fractures. However, for uniformly partially saturat-
ed fractures, it implies that imbibition ceases when the
fracture and matrix saturations are proportional to each
other, implying similar values of the fracture and matrix
capillary pressures. This is rarely correct, and certainly
inconsistent with the basic premise of dual porosity
models that there is a huge disparity in permeability
between flowing and stagnant regions. This transfer
function also is linearly dependent on saturation.
Other authors have derived transfer functions for
imbibition based on a more accurate match to experi-
mental data than Eq. (4) (Civan, 1994; Civan and
Rasmussen, 2001, 2003; Penuela et al., 2002; Terez
and Firoozabadi, 1999). In this work we will use Eq.
(14) for our transfer function since it is based on a
simple one-parameter fit to experiments.
5.3. Grid-based simulation of fracture/matrix flow
The purpose of this section is to determine: (1) if the
behavior observed experimentally by Rangel-German
and Kovscek (2002) can be reproduced by simulation;
and (2) if the same results can be obtained using a 1D
simulation to solve Eqs. (6) and (7) with an appropriate
transfer function. The simulation model consisted of a
horizontal fracture (Tables 4 and 5)) initially filled with
oil (Swf=0) connected from both sides to oil saturated
matrix blocks (Swm=Swc). The fracture was defined
explicitly as a high permeability region with a porosity
of 1, and no transfer function was used (Fig. 14). There
were 122 matrix grid blocks along the fracture. The
matrix grid system in the direction perpendicular to the
fracture was exactly the same as those used to simulate
Table 5
Dimensions of the model in 2D fracture/matrix simulations
Dimension Unit Fracture Matrix
(on either side
of the fracture)
X cm .7 28 (20 .3, 221)Y cm 244 244 (1222)
Z cm 1 1.
.
counter-current imbibition in 1D (Fig. 2) that matched
the experimental results. The system was 244 cm long
with a width of 56.7 cm. The fracture aperture was 0.7
Quadratic relative permeabilities were defined in the
fracture (Table 4)this meant that a shock developed
in the 1D displacement in the fracture and reduced
Fig. 14. Grid system used for the 2D fracture/matrix simulation. The grid perpendicular to the fracture direction is exactly as in Fig. 2the darker
area indicates finer grids close to the fracture.
H.Sh. Behbahani et al. / Journal of Petroleum Science and Engineering 50 (2006) 213932cm (Table 5). Water was injected at a constant rate via
an injection well into the first grid block of the fracture
and liquid (oil and water) was produced at a constant
pressure from a producing well that was completed in
the last fracture grid block.
The matrix porosity, relative permeabilities and ma-
trix connate and residual saturations come from Bour-
biaux and Kalaydjian (1990) (Table 2). In order to
produce a situation similar to low matrix permeability
fractured reservoirs, the matrix absolute permeability
was reduced from 124 to 1 md to ensure a huge
disparity in permeability between fracture and matrix.
The matrix capillary pressure had the same functional
form as used by Bourbiaux and Kalaydjian (1990), but
increased by the square root of the ratio of the exper-
imental permeability to the model matrix permeability.
The viscosities of the oil and water were 1.5103 Pas. Capillary pressure in the fracture was zero. The
fracture permeability was 151012 m2 (15 Darcy).Fig. 15. Comparison of fracture water saturations for 1D simulations with
solution (dotted line) is compared to numerical simulation results using a com
(dashed line). The good agreement shows that simulations can accurately renumerical dispersion in the simulation results. While
simple analytical expressions for the fracture properties
were used, the matrix relative permeabilities and capil-
lary pressure were based on experimental data. Two
water injection rates of 20 and 2 cc/h were used.
Two additional simulations were run for comparison
purposes. The first had no matrix and an injection rate
of 20 cc/h. The results were compared with an analyt-
ical solution based on the BuckleyLeverett approach
and 1D numerical solutions using the streamline code
described by Di Donato et al. (2003). The second was
where the matrix had a capillary pressure of zero. This
test shows the effect of viscous forces on the recovery
process.
5.4. Analytical and 1D numerical solutions
Fig. 15 shows a comparison of numerical solutions
for a simulation with no matrix compared to the ana-no matrix present after 3 and 5 h. The BuckleyLeverett analytical
mercial reservoir simulator (solid line) and a streamline-based model
produce a 1D displacement.
lytic BuckleyLeverett solution (Dake, 1978). The
agreement between the numerical and analytical results
confirms that both Eclipse and the streamline code can
accurately reproduce a 1D displacement.
Fig. 16 shows the fracture saturation when there is
no capillary pressure in the matrix. The results are
compared to the analytical solution where there is no
matrix. The agreement between the two simulations
indicates that in this model viscous forces alone have
little effect on the fracture/matrix transfer.
The results of 2D simulations with a finite matrix
capillary pressure are now compared with results
from 1D dual porosity models. The linear transfer
functions developed by Di Donato et al. (2003) and
Kazemi et al. (1992) were used in a dual porosity
model. In the dual porosity model the following rosity simulation is calculated using the following
olid lines show the fracture saturation for flow with matrix present and no
Table 6
Data used in the 1D dual porosity simulations
Parameter Unit Value
Fracture porosity, /f fraction 0.012Matrix porosity, /m fraction 0.233Fracture permeability, Kf m
2 151012Matrix permeability, Km m
2 1015
Matrix initial water saturation, Swmi fraction 0.40
Matrix residual oil saturation, Somr fraction 0.422
Water density, qw Kg m3 1090
Oil density, qo Kg m3 760
Water viscosity, lw Pa s 1.5103Oil viscosity, lo Pa s 1.5103Fracture/Matrix rate constant, b days1 0.08423Fracture water relative permeability Krwf=Sw
2
Fracture oil relative permeability krof= (1Sw)2
H.Sh. Behbahani et al. / Journal of Petroleum Science and Engineering 50 (2006) 2139 33equations were used to define fracture and matrix
porosities and the effective permeability of the sys-
tem from the geometry of the 2D explicit fracture
model:
/f Wf
Wf Wm 16
/m /mWm
Wf Wm 17
Keff WfKf WmKmWf Wm 18
where W and K are the width and permeability,
respectively. The total velocity in the 1D dual po-
Fig. 16. Numerical simulations of fracture flow after 2 and 4 h. The scapillary pressure. The dotted lines are BuckleyLeverett analytical solutions assuming a purely 1D displacement with no matrix. The agreemen
between the two simulations indicate that viscous forces have little effect on the fracture/matrix transfer.equation:
mt QWfH
19
where Q is the injection rate and Wf is the fracture
aperture. Tables 6 and 7 give the data used in the
dual porosity simulations.
5.5. Analysis of the results
Figs. 17 and 18 show the comparison of the simu-
lated water saturation profiles in the fracture and matrix
for different time-steps for an injection rate of 20 cc/h.
Due to the high injection rate it is expected that the
front moves very fast in the fracture and most imbibi-
tion take places when the average water saturation in
the fracture is nearly one. Since the amount of transfer
is small at early time, the fracture saturation profiles fort
all three simulations are similar. The simulated matrix
saturation profiles are different, however, with the dual
porosity models predicting more transfer into the matrix
than the 2D simulation. This is simply because the
transfer functions reproduce the empirical fit, Eq. (4)
that gives a higher recovery than the equivalent 1D
times for an injection rate of 2 cc/h. In this case water
advances in the fracture and matrix at comparable rates.
In contrast to the previous case, the simulated fracture
saturation profiles are not the same for the different
models. The simulated matrix saturation profiles are
also different. Fig. 5 reveals that at early time (tD of
less than approximately 5, or a real time of approxi-
mately 70 h) the simulations predict a higher recovery
than Eq. (4), while for intermediate and late times
(greater than around 70 h) the simulations predict
lower recovery. In our 2D simulations this relates to
the recovery (or water saturation) in the matrix. Hence
we expect that at early times the dual porosity models
will give lower recovery in the matrix and higher
recovery at late times. This is evident in Fig. 21. If
Table 7
Equivalent injection data used in 1D dual porosity simulations
Parameter Unit Value
High injection
rate case
Low injection
rate case
2D injection rate, Q2D cc h1 20 2
2D total velocity, vt m s1 7.94105 7.94106
1D injection rate, Q1D m3 day1 4.6676104 4.6676105
H.Sh. Behbahani et al. / Journal of Petroleum Science and Engineering 50 (2006) 213934simulations. The results for 240 h correspond to a
dimensionless time tD of 17. It is clear from a com-
parison of Figs. 1 and 5 that at this time the simula-
tions do indeed predict a lower recovery than the
experimental correlation, Eq. (4), for 1D counter-cur-
rent imbibition. The transfer function of Di Donato et
al. (2003) shows little change in matrix saturation with
distance, similar to the 2D simulations, while the
Kazemi et al. (1992) function predicts a significant
change in saturation with distance. This is because the
Kazemi et al. (1992) model predicts less transfer into
the matrix when the fracture saturation is less than
1that is near the outlet.
Fig. 19 shows the average matrix saturation as a
function of time. The average saturation scales with
the square root of time, except at late time, which was
also observed experimentally for the instantly filled
fracture regime by Rangel-German and Kovscek (2002).
Figs. 20 and 21 compare the simulated water satu-
ration profiles in the fracture and matrix at differentFig. 17. Simulated fracture water saturation for Q =20 cc/h after 3, 5 and 7 h
using a transfer function due to either Di Donato et al. (2003) (dashed linethere is less imbibition into the matrix, more water is
retained in the fracture and the waterfront in the fracture
moves faster. This is confirmed in Fig. 20 where the
water saturation in the fracture is moving faster for the
dual porosity models as a consequence of the lower
matrix recoveries. The water advance is still greater at
70 h, which is just in the late time regime. However,
most of the matrix has been next to a water-filled
portion of the fracture for considerably less than 70 h,
so the behavior is still consistent with the early time
behavior.
Another possible explanation for the discrepancy in
the results is that the 2D simulations allow co-current
flow, whereas the transfer functions are based on count-
er-current flow, which is slower. In the 2D simulations
oil can travel in a direction parallel to water towards the
high permeability fracture and/or un-drained matrix
grid blocks, resulting in more rapid recovery than
from counter-current imbibition alone. However, a
careful quantitative analysis of the results shows that. 2D simulation (solid line) is compared with a 1D dual porosity model
) or Kazemi et al. (1992) (dotted line).
Fig. 18. Simulated matrix water saturation for a flow rate of 20 cc/h after 100 and 240 h. 2D simulation (solid line) is compared with a dual porosity
model using a transfer function due to either Di Donato et al. (2003) (dashed line) or Kazemi et al. (1992) (dotted line). The 2D simulation and the
Di Donato et al. (2003) transfer function show a flat profile similar to that observed experimentally by Rangel-German and Kovscek (2002).
H.Sh. Behbahani et al. / Journal of Petroleum Science and Engineering 50 (2006) 2139 35this does not affect recovery in this case, and that the
differences between the results are explained solely by
the differences in predicted 1D counter-current recov-
ery. For instance, Fig. 21 shows that the water satura-
tion at the inlet after 48 h is 43.5% in the 2D
simulations and 42.8% using the Di Donato et al.
(2003) model. 48 h corresponds to tD of 3.4. Using
Fig. 5 to find the oil recovery assuming counter-current
flow only for the simulations and using Eq. (4) will lead
to exactly the same water saturations. This shows that
co-current imbibition has no effect on the results. This
is likely to be due to the extreme contrast in matrix and
fracture permeabilities.
The Kazemi et al. (1992) transfer function always
gives a transfer rate that is less than or equal to thatFig. 19. The simulated average matrix water saturation in the 2D simulation s
reproduces the behavior of the instantly filled regime in Rangel-German anpredicted by the Di Donato et al. (2003) model. As a
consequence the Kazemi et al. (1992) model predicts
less transfer into the matrix and a more diffuse and
further advanced saturation profile in the fracture. Qual-
itatively the 2D simulations give profiles more similar
to the Di Donato et al. (2003) model, since in both cases
a zero fracture capillary pressure is assumed.
Overall the Di Donato et al. (2003) dual porosity
model gives similar results to direct 2D simulation,
indicating that a dual porosity approach can properly
simulate fracture/matrix flow in a simple system. The
differences in the results can all be explained by con-
sidering the differences in the predictions of counter-
current imbibition with no fracture flow. It should be
possible to adjust the rate constant b to obtain a bettercales linearly with the square root of time for a flow rate of 20 cc/h and
d Kovsceks (2002) experimental results.
Fig. 20. The fracture water saturation for a flow rate of 2 cc/h after 30 and 70 h. 2D simulation (solid line) is compared to dual porosity models using
otted
H.Sh. Behbahani et al. / Journal of Petroleum Science and Engineering 50 (2006) 213936match between the dual porosity and 2D simulation
models, or to use a transfer function with more para-
meters that better reproduce experimental data (Civan
and Rasmussen, 2001).
Fig. 22 shows the average matrix water saturation as
a function of time. The average saturation scales ap-
proximately linearly with time at early times (less than
around 200 h), which also was observed experimentally
for the filling-fracture regime by Rangel-German and
Kovscek (2002).
Penuela et al. (2002) developed a transfer function
for dual porosity simulation that matched fine grid
simulations of imbibition. They used a transfer function
with the same definition of shape factor used conven-
tionally by Kazemi et al. (1992), but allowed the shape
factor to be time-dependent. In comparison our formu-
lation is mathematically simpler, and physically more
the Di Donato et al. (2003) (dashed line) and Kazemi et al. (1992) (dappealing, since the governing partial differential equa-
Fig. 21. The matrix water saturation for a flow rate of 2 cc/h after 48 and
porosity models using the Di Donato et al. (2003) (dashed line) and Kazemtions are written as a function of saturation. However
incompressible two-phase flow is assumed in this
work, whereas Penuela et al. (2002) also considered
compressibility.
Rangel-German and Kovscek (2003) also proposed
a time-dependent shape factor to match their experi-
mental results. They found a formulation that matched
the overall recovery for different flow rates; however, in
essence they upscaled a 2D system to a zero-dimen-
sional average property. The dual porosity model pro-
posed here upscales a 2D or 3D system into a 1D
problem along the flow direction. Their formulation
except for the representation of a time rather than a
saturation dependent shape factoris broadly equiva-
lent to Kazemi et al.s (1992) transfer function. When
placed in a dual porosity model, this transfer function
underestimates the matrix recovery and overestimates
line) transfer functions.the water advance rate in the fractures. The formulation
240 h. Results from 2D simulation (solid line) are compared to dual
i et al. (1992) (dotted line) transfer functions.
ation
ture r
eum S(2) The validity of the Ma et al. (1997) scaling
function, Eq. (1), was tested by performing simu-
lations for different oil/water viscosity ratios
using network model derived data (Jackson et
al., 2003). The results indicated that recoveryin this paper is identical to theirs for a constant shape
factor.
6. Conclusions
(1) Simulation of one-dimensional and two-dimen-
sional counter-current imbibition matched the
results of core experiments. This confirms that a
conventional Darcy treatment of multiphase flow
is adequate to describe counter-current imbibition.
Fig. 22. The simulated average matrix water saturation in the 2D simul
for a flow rate of 2 cc/h and reproduces the behavior of the filling-frac
H.Sh. Behbahani et al. / Journal of Petrolplots for different viscosity ratios could be scaled
onto the same curve using a dimensionless time
that is inversely proportional to the geometric
mean of the water and oil viscosities, as estab-
lished experimentally by Ma et al. (1997).
(3) The results of 2D simulations of flow in a long
fracture connected to a horizontal water-wet ma-
trix showed the same flow regimes as observed
experimentally by Rangel-German and Kovscek
(2002).
(4) A 1D dual porosity model with a transfer function
that matched core-scale imbibition experiments
was able to reproduce the behavior observed
using explicit 2D simulation of fracture/matrix
flow.
Nomenclature
Ama area of a surface open to flow in the flow
direction, m2Fc shape factor, m2
fj fractional flow
H vertical thickness, m
K absolute permeability, m2 or Am2
kr relative permeability
L, Lc characteristic length, m
lma distance from the open surface to the no flow
boundary, m
M viscosity ratio
M* mobility ratio
P pressure, Pa
Q flow rate, m3 s1
Rl final oil recovery, m3
R recovery, m3
S saturation, fraction
scales approximately linearly with time at early times (less than 200 h)
egime in Rangel-German and Kovsceks (2002) experimental results.
cience and Engineering 50 (2006) 2139 37T transfer function, s1
t time, s
tD dimensionless time
Vma bulk volume of matrix (core sample), m3
Vt total fluid velocity, m s1
W width, m
Greek
b rate constant, s1
/ porosity, fractionk mobility, Pa1 s1
lo oil viscosity, cp, Pa slw water viscosity, cp, Pa sq density, kg m3
r interfacial tension, Nm1
Subscripts
f fracture
i initial
pore-network models of multiphase flow. Adv. Water Resour. 25,
H.Sh. Behbahani et al. / Journal of Petroleum Science and Engineering 50 (2006) 21393810691089.
Bourbiaux, B.J., Kalaydjian, F.J., 1990. Experimental study of cocur-
rent and countercurrent flows in natural porous media. SPE
Reserv. Eng. 5, 361368.
Chimienti, M.E., Illiano, S.N., Najurieta, H.L., 1999. Influence of
temperature and interfacial tension on spontaneous imbibition
process. SPE 53668, Latin American and Caribbean Petroleum
Engineering Conference, Caracas, Venezuela.
Cil, M., Reis, J.C., Miller, M.A., Misra, D., 1998. An examination of
countercurrent capillary imbibition recovery from single matrix-
blocks and recovery predictions by analytical matrix/fracture
transfer functions. SPE 49005, Annual Technical Conference,
New Orleans, Louisiana, USA.
Civan, F., 1994. A theoretically derived transfer function for oil recov-
ery from fractured reservoirs by waterflooding. SPE 27745, Ninth
Symposium on Improved Oil Recovery, Tulsa, Oklahoma, Aprilm matrix
o oil
r residual
w water
Acknowledgments
We would like to thank the sponsors of the ITF
project on dImproved Simulation of Flow in Fracturedand Faulted Reservoirs for funding this research. Has-
san Behbahani would like to thank NIOC for providing
financial support.
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H.Sh. Behbahani et al. / Journal of Petroleum Science and Engineering 50 (2006) 2139 39
Simulation of counter-current imbibition in water-wet fractured reservoirsIntroductionSimulation of 1D-counter-current imbibitionBase modelsExperimental data predicted by network modelling
Simulation of 2D-counter-current imbibitionValidity of the correlations, Eqs. (1) and (5)Theory and simulation of fracture flow and imbibitionPrevious experimental studiesTheory for dual porosity systemsGrid-based simulation of fracture/matrix flowAnalytical and 1D numerical solutionsAnalysis of the results
ConclusionsAcknowledgmentsReferences