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: m.blunt@imperial.ac.uk (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.

References

Al-Lawati, S., Saleh, S., 1996. Oil recovery in fractured oil reservoirs

by low IFT imbibition process. SPE 36688, Annual Technical

Conference and Exhibition, Denver, Colorado, USA.

Aronofsky, J.S., Masse, L., Natanson, S.G., 1958. A model for the

mechanism of oil recovery from the porous matrix due to water

invasion in fractured reservoirs. Trans. AIME 213, 1719.

Aziz, K., Settari, A., 1979. Petroleum Reservoir Simulation. Elsevier

Applied Science, London.

Babadagli, T., Ershaghi, I., 1992. Imbibition assisted two-phase flow

in natural fractures. SPE 24044, Western Regional Meeting,

Bakersfield, California, USA.

Barenblatt, G.I., Zheltov, J.P., 1960. On the basic equations of single-

phase flow of fluid through fractured porous media. Dokl. Akad.

Nauk SSSR 132 (3), 545548.

Barenblatt, G.I., Patzek, T.W., Silin, D.B., 2003. The mathematical

model of non-equilibrium effects in wateroil displacement. SPEJ

8 (4), 409420.

Baldwin, B.A., Spinler, B.A., 2002. In situ saturation develop-

ment during spontaneous imbibition. J. Pet. Sci. Eng. 35 (1-2),

2332.

Behbahani, H.Sh., Blunt, M.J., 2004. Analysis of imbibition in mixed-

wet rocks using pore-scale modelling. SPE 90132, Annual Tech-

nical Conference and Exhibition, Houston, Texas, USA.

Blunt, M.J., Jackson, M.D., Piri, M., Valvatne, P.H., 2002. Detailed

physics, predictive capabilities and macroscopic consequences for1720.Civan, F., Rasmussen, M.L., 2001. Asymptotic analytical solu-

tions for imbibition waterfloods in fractured reservoirs. SPEJ

6, 171181.

Civan, F., Rasmussen, M.L., 2003. Modelling and validation of

hindered-matrixfracture transfer for naturally fractured petroleum

reservoirs. SPE 80918, Production and Operations Symposium,

Oklahoma City, Oklahoma, USA.

Cuiec, L., Bourbiaux, B.J., Kalaydjian, F.J., 1994. Oil recovery

by imbibition in low-permeability chalk. SPE Form. Eval. 9,

200208.

Dake, L.T., 1978. Fundamentals of Reservoir Engineering. Elsevier,

Amsterdam.

de Swaan, A., 1978. Theory of waterflooding in fractured reservoirs.

SPEJ, 117122 (April 1978).

Di Donato, G., Blunt, M.J., 2004. Streamline-based dual-porosity

simulation of reactive transport and flow in fractured reservoirs.

Water Resour. Res. 40, W04203. doi:10.1029/2003WR002772.

Di Donato, G., Huang, W., Blunt, M.J., 2003. Streamline-based dual

porosity simulation of fractured reservoirs. SPE 84036, Annual

Technical Conference, Denver, Colorado, USA.

Du Prey, E.L., 1978. Gravity and capillarity effects on imbibition in

porous media. SPEJ, 195206 (June 1978).

Gupta, A., Civan, F., 1994. An improved model for laboratory mea-

surement of matrix to fracture transfer function parameters in

immiscible displacement. SPE 28929, Annual Technical Confer-

ence, New Orleans, Louisiana, 2528 September.

Hamon, G., Vidal, J., 1986. Scaling-up the capillary imbibition pro-

cess from laboratory experiments on homogeneous and heteroge-

neous samples. SPE 15852, European Petroleum Conference.

London, UK.

Huang, W., Di Donato, G., Blunt, M.J., 2004. Comparison of stream-

line-based and grid-based dual porosity simulation. J. Pet. Sci.

Eng. 43, 129137.

Iffly, R., Rousselet, D.C., Vermeulen, J.L., 1972. Fundamental study

of imbibition in fissured oil fields. SPE 4102, Annual Technical

Conference, Dallas, Texas, USA.

Jackson, M.D., Valvatne, P.H., Blunt, M.J., 2003. Prediction of wet-

tability variation and its impact on flow using pore-to reservoir-

scale simulations. J. Pet. Sci. Eng. 39, 231246.

Kazemi, H., Gilman, J.R., Elsharkawy, A.M., 1992. Analytical

and numerical solution of oil recovery from fractured reser-

voirs with empirical transfer functions. SPE Reserv. Eng. 7,

219227.

Li, Y., Morrow, N.R., Ruth, D., 2003. Similarity solution for linear

counter-current spontaneous imbibition. J. Pet. Sci. Eng. 39 (34),

309326.

Ma, S., Morrow, N.R., Zhang, X., 1997. Generalized scaling of

spontaneous imbibition data for strongly water-wet systems. J.

Pet. Sci. Eng. 18, 165178.

Matejka, M.C., Llanos, E.M., Civan, F., 2002. Experimental determi-

nation of matrix-to-fracture transfer functions for oil recovery by

water imbibition. J. Pet. Sci. Eng. 33 (4), 253264.

Mattax, C.C., Kyte, J.R., 1962. Imbibition oil recovery from fractured

water drive reservoirs. SPEJ, 177184 (June 1962).

Najurieta, H.L., Galacho, N., Chimienti, M.E., Illiano, S.N., 2001.

Effects of temperature and interfacial tension in different produc-

tion mechanisms. SPE 69398, Latin American and Caribbean

Petroleum Engineering Conference, Buenos Aires, Argentina.

Parsons, R.W., Chaney, P.R., 1966. Imbibition model studies on

water-wet carbonate rocks. SPEJ, 2634 (March 1966).

Penuela, G., Hughes, R.G., Civan, F., Wiggins, M.L., 2002. Time-dependent shape factors for secondary recovery in naturally frac-

tured reservoirs. SPE 75234, SPE/DOE Improved Oil Recovery

Symposium, Tulsa, Oklahoma, USA.

Pooladi-Darvish, M., Firoozabadi, A., 2000. Experiments and mod-

elling of water injection in water-wet fractured porous media. J.

Can. Pet. Technol. 39 (3), 3142.

Rangel-German, E.R., Kovscek, A.R., 2002. Experimental and ana-

lytical study of multidimensional imbibition in fractured porous

media. J. Pet. Sci. Eng. 36, 4560.

Rangel-German, E.R., Kovscek, A.R., 2003. Time-dependent ma-

trixfracture shape factors for partially and completely immersed

fractures. SPE 84411, Annual Technical Conference, Denver,

Colorado, USA.

Ruth, D., Morrow, N.R., Yu, L., Buckley, J., 2000. Simulation study

of Spontaneous imbibition. International Society of Core Analysts

Annual Meeting, Abu Dhabi, UAE, October.

Ruth, D., Mason, G., Morrow, N.R., 2003. A numerical study of the

influence of sample shape on spontaneous imbibition. Interna-

tional Society of Core Analysts Annual Meeting, Pau, France,

September.

Tang, G.Q., Firoozabadi, A., 2001. Effect of pressure gradient and

initial water saturation on water injection in water-wet and mixed-

wet fractured porous media. SPE Reserv. Evalu. Eng. 4, 516524.

Tavassoli, Z., Zimmerman, R.W., Blunt, M.J., 2005. Analytic analysis

for oil recovery during counter-current imbibition in strongly

water-wet systems. Porous Media Transp. 58, 173189.

Terez, I.E., Firoozabadi, A., 1999. Water injection in water-wet

fractured porous media: experiments and a new model with

modified BuckleyLeverett Theory. SPEJ 4 (2), 135141.

Valvatne, P.H., Blunt, M.J., 2004. Predictive pore-scale modelling of

two-phase flow in mixed wet media. Water Resour. Res. 40,

W07406. doi10.1029/2003WR002627.

Viksund, B.G., Morrow, N.R., Ma, S., Wang, W., Graue, A., 1998.

Initial water saturation and oil recovery from chalk and sandstone

by spontaneous imbibition. SCA-9814, International Society of

Core Analysts, The Hague, September.

Warren, J.E., Root, P.J., 1963. The behavior of naturally fractured

reservoirs. SPEJ, 245255. Trans. AIME 228 (September 1963).

Zhang, X., Morrow, N.R., Ma, S., 1996. Experimental verification of

a modified scaling group for spontaneous imbibition. SPE Reserv.

Eng. 11, 280285.

Zhou, X., Morrow, N.R., Ma, S., 2000. Interrelationship of wettabil-

ity, initial water saturation, aging time, and oil recovery by

spontaneous imbibition and waterflooding. SPEJ 5 (2), 199207.

Zhou, D., Jia, L., Kamath, J., Kovscek, R.A., 2002. Scaling of

counter-current imbibition processes in low permeability porous

media. J. Pet. Sci. Eng. 33 (1-3), 6174.

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