Advances in Water Resources 113 (2018) 260–271

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Advances in Water Resources

journal homepage: www.elsevier.com/locate/advwatres

Pore-level influence of micro-fracture parameters on visco-capillary

behavior of two-phase displacements in porous media

M.R. Rokhforouz

a , H.A. Akhlaghi Amiri b , c , ∗

a Department of Chemical and Petroleum Engineering, Sharif University of Technology, Tehran, Iran b Chemical Engineering Department, Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad, Iran c Omid Petro-Energy Khavaran (OPEK) Company, Khorasan Science and Technology Park, Mashhad, Iran

a r t i c l e i n f o

Article history:

Received 29 September 2017

Revised 30 January 2018

Accepted 31 January 2018

Available online 2 February 2018

Keywords:

Fractured porous medium

Phase-field method

Two phase flow

Capillary fingering

Viscous fingering

Wettability

a b s t r a c t

In this work, coupled Cahn–Hilliard phase field and Navier–Stokes equations were solved using finite ele-

ment method to address the effects of micro-fracture and its characterizations on water-oil displacements

in a heterogeneous porous medium. Sensitivity studies at a wide range of viscosity ratios (M) and capil-

lary numbers (Ca), and the resultant log Ca–log M stability phase diagram, revealed that in both media,

with/without fracture, the three regimes of viscous fingering, capillary fingering and stable displacement

similarly occur. However, presence of the fracture caused water channeling phenomenon which resulted

in reduction of the number of active fingers and hence the final oil recovery factor. At high Ca (espe-

cially in the stable regime, with log Ca ≥−2.5 and log M ≥ 0), recovery factor for the fractured medium

was relatively identical with the non-fractured one. At log M ≥ 0, the fracture was fully swept, but flow

instabilities were observed inside the fracture at lower M values, especially for log Ca > −4.6. In the case

of the fractured medium at log Ca = −4.6 and log M = 0 (capillary dominant flow), it is observed that the

primary breakthrough took place by a finger progressed through the matrix, not those channeled through

the fracture. Geometrical properties of the fracture, including length, aperture and orientation, highly af-

fected both displacement profile and efficiency. The fracture length inversely influenced the oil recovery

factor. It was observed that there is a critical fracture width (almost half of the medium average pore di-

ameter) at which the recovery factor of the medium during displacement is minimum, compared to the

media having thinner and thicker fractures. Minor channeling effect in the media with thinner fracture

and larger fracture swept volume as well as high fracture/matrix cross flow in the media with thicker

fracture were detected as the main cause of this non-monotonic behavior. In the models with thick frac-

tures (with the thickness higher than the average pore diameter), considerable trapped oil volumes were

observed inside the fracture at low M values. The fracture orientation had the most impressive effect on

oil recovery compared to the other studied parameters; where the oil recovery factor incremented more

than 20% as the fracture rotated 90 ° from flow direction. Due to the dominant effect of the channeling

phenomenon, the change in the medium wettability from slightly oil-wet to slightly water-wet, did not

considerably affect the displacement profile in the fractured medium. However, oil recovery factor in-

creased as the medium became more water-wet. The fracture area was fully swept by the injected water

in the oil-wet and neutral-wet media. However, flow instabilities were observed inside the fracture of

the water-wet medium due to counter-current imbibition between fracture/matrix. Micro-scale mecha-

nisms of pore doublet effect, interface coalesce, snap-off and reverse movements were captured during

the studied unstable displacements.

© 2018 Published by Elsevier Ltd.

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1. Introduction

Displacement of a fluid in porous media by another misci-

ble/immiscible one is a process which has applications in va-

∗ Corresponding author at: Chemical Engineering Department, Faculty of Engi-

neering, Ferdowsi University of Mashhad, Mashhad, Iran.

E-mail address: ha.akhlaghi@um.ac.ir (H.A. Akhlaghi Amiri).

A

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https://doi.org/10.1016/j.advwatres.2018.01.030

0309-1708/© 2018 Published by Elsevier Ltd.

iety of fields, such as enhanced oil recovery, CO 2 sequestra-

ion and contamination transports ( Juanes et al., 2006; Matthai

t al., 2007; Harrar et al., 2007; Iglauer et al., 2012; Arabloo

t al., 2015; Akhlaghi Amiri et al., 2014; Rokhforouz and Akhlaghi

miri., 2017a; Liu and Wu, 2016 ). During last decades, many re-

earchers have studied the governing mechanisms in two-phase

isplacements through porous media, both experimentally and

umerically, at macro- and micro-scales ( Akhlaghi Amiri and

M.R. Rokhforouz, H.A. Akhlaghi Amiri / Advances in Water Resources 113 (2018) 260–271 261

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Fig. 1. Schematic of the simulated non-fractured model geometry. The inlet and the

outlet are specified with arrows. The dark and the white areas are the pores/throats

and the rock matrix, respectively.

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amouda, 2014; Erfani Gahrooei and Ghazanfari, 2017; Joekar-

iasar and Hassanizadeh, 2012; Karadimitriou and Hassanizadeh,

012; Liu et al., 2016; Mason and Morrow, 2013; Meakin and Tar-

akovsky, 2009; Morrow and Mason, 2001; Raeini et al., 2014;

okhforouz and Akhlaghi Amiri, 2017a, b; Li et al., 2017a, b ).

owever, for deep understanding the flow behavior in porous

edia, characterization of pore-level physics of the phenomena

s crucial ( Raeini et al., 2014 ; Rokhforouz and Akhlaghi Amiri,

017a, b ).

Lenormand et al. (1988) and recently Zhang et al. (2011) con-

ucted a series of micromodel experiments to assess the influ-

nces of viscous and capillary forces on the flow regimes. They

tudied simultaneous effects of the capillary number (Ca) and the

iscosity ratio between displacing and displaced phases (M) on

ow behavior and mapped the observed displacement regimes on

log M–log Ca stability phase diagram. By tuning Ca and M, three

ypes of flow regimes, including viscous fingering, capillary finger-

ng and stable displacement were recognized ( Lenormand et al.,

983, 1988; Lenormand, 1989; Zhang et al., 2011 ). Later, similar

bservations were made in pore-scale numerical works done by

andara et al. (2013) , using a pair-wise force smoothed particle

ydrodynamics model, and Akhlaghi Amiri and Hamouda (2014) ,

sing phase field method. Ferer et al. (2004) described the flow

egimes, in air–water displacement experiments through glass

icro-model, by two extreme limits of invasion percolation with

rapping and diffusion limited aggregation. A modified local capil-

ary number was introduced by Cottin et al. (2010) , in an experi-

ental work using micro-fluidic chips, to better explain displace-

ent phenomena. The influence of viscous fingering on dynamic

ressure-saturation curves was studied by Løvoll et al. (2011) , us-

ng a porous medium made of glass beads. Liu et al. (2015) applied

attice Boltzmann technique to simulate the drainage process in

omogeneous and heterogeneous pore network. It was found that

he extent and behavior of the preferential flow, such as fingering,

re strongly dependent upon the Ca, M, and media heterogeneity.

hey also showed that the boundaries of regimes may differ for

ifferent pore networks.

In the case of displacements in porous media containing micro-

ractures, which frequently occurs in naturally fractured hydro-

arbon reservoirs, the characterization of the flow regimes is

ore challenging; due to a more complicated medium geome-

ry and difference in visco-capillary properties of the rock matrix

nd fractures ( Abushaikha et al., 2010 ). However, one can rarely

nd pore-scale experimental and numerical studies on this field.

arzaneh et al. (2010) conducted a series of two-phase flow exper-

ments in fractured porous media, using glass micro-models. The

ffect of the geometrical characteristics of fracture, including their

umber, orientation, discontinuity, overlap, distribution and spac-

ng was investigated. They observed that in a pattern with fracture

riented against the main flow direction, displacement efficiency

ncreased in comparison to other cases. Abedi et al. (2012) and

edaghat et al. (2013) experimentally studied the role of the frac-

ure geometrical properties during polymer flooding, using five-

pot glass micromodel saturated with oil. It was revealed that

he fracture orientation has the most effect on oil recovery com-

ared to the other fracture geometrical parameters. In the case

f orthogonal-to-flow fractures, longer fracture led to a higher

isplacement efficiency. Using 2D fractured glass micromodels,

hariatpanahi et al. (2005) showed that immiscible gas injection

erformance is a strong function of the fracture orientation; while

he displacement by water injection is not very sensitive to the

racture orientation. Kianinejad et al. (2015) observed that dis-

lacement efficiency is higher in the cases of longer fracture and/or

racture perpendicular to the flow direction. Monteagudo and

iroozabadi (2004) used control volume method to model two-

hase immiscible flow in two-dimensional discrete-fractured

edia. Capillary pressure was shown to have a determining effect

n the process.

This work numerically addresses the effects of micro-fracture

haracterizations on displacement regimes in porous media, which

re controlled by viscous and capillary forces. Phase field, an in-

erface capturing method is used for numerical simulation of phe-

omena at pore-scale; since it is capable of handling complex ge-

metries of fractured media, without using model approximations,

ompared to the other approaches ( Akhlaghi Amiri and Hamouda,

013, 2014; Maaref et al., 2017; Rokhforouz and Akhlaghi Amiri,

017a ). Governing equations are solved using a robust finite el-

ment solver, COMSOL Multiphysics (2012) . Viscous and capillary

nstabilities are compared in the simulated non-fractured and frac-

ured heterogeneous porous media by tuning Ca and M and un-

er different wettability conditions. The effects of the geometrical

racture parameters, including length, aperture and orientation are

erified.

. Numerical model

This section presents the computational geometry, governing

quations and the numerical scheme used in this study.

.1. Model geometry and boundary conditions

The heterogeneous pattern of the simulated porous medium,

ig. 1 , was taken from a real dolomite rock section, using Corel-

raw Graphics Suite X7 software ( Maaref et al., 2017 ). Geometrical

roperties of the simulated medium are summarized in Table 1 .

he medium was initially saturated with the oil phase, which was

isplaced by water phase, being injected with a constant velocity

rom the left-hand side of the medium, as shown in Fig. 1 . Zero

ressure was assumed at the outlets, on the right-hand side of the

edium. Symmetry boundary conditions were employed on the

ateral sides. The boundary condition of grain surfaces is consid-

red as a wetted wall with specific contact angles. It was supposed

hat the fluids are incompressible, with constant physical proper-

ies, and phase change does not occur. Gravity was neglected, as-

uming 2D horizontal flow.

262 M.R. Rokhforouz, H.A. Akhlaghi Amiri / Advances in Water Resources 113 (2018) 260–271

Table 1

Properties of the simulated micromodel.

Length (cm) Width (cm) Avg. pore diameter (cm) Avg. throat diameter (cm) Medium porosity Medium absolute permeability (Darcy)

6 6 0.15 0.017 0.502 4.5

Fig. 2. A sample distribution of triangular mesh elements in an enlarged section of

the simulated porous medium containing a pore and a throat. The mesh element

size is coarser in wide pores, while finer around corners and inside throats.

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2.2. Governing equations and numerical scheme

The injected fluid velocity, u inj , in the simulated porous me-

dia was in the range of 5 × 10 −3 and 1 cm/s, resulting in Reynolds

number in the range of 0.05 to 10; hence flow in all the simu-

lations was laminar. Cahn–Hilliard phase-field method ( Cahn and

Hilliard, 1958 ) coupled with Navier–Stokes and continuity equa-

tions were employed to solve the interfacial problems. Phase-field

order parameter ( φ) is defined such that the relative concentra-

tion of the two components are 1+ φ2 and

1 −φ2 . In this definition,

φ = ±1 represents the bulks of the two phases and − 1 < φ < 1

represents the interface. All the fluid physical properties are inter-

polated between the two phases using the relative concentrations

of the phases:

ϑ ( φ) =

( 1 + φ)

2

ϑ 1 +

( 1 − φ)

2

ϑ 2 (1)

where ϑ denotes a phase property such as viscosity ( μ) or den-

sity ( ρ). The modified Navier–Stokes and continuity equations are

coupled with the phase-field equation to capture moving interface.

The equation system is then as follows:

ρ∂u

∂t + ρ(u · ∇) u = −∇p + ∇ ·

[μ(∇u + ∇ u

T ) ]

+ G ∇φ (2)

∇ . u = 0 (3)

∂φ

∂t + u · ∇φ = ∇ ·

(γ λ

ε 2

)∇ψ (4)

ψ = −∇ · ε 2 ∇φ +

(φ2 − 1

)φ (5)

where p denotes pressure, u is the fluid velocity field and t is

the time. In addition, is an auxiliary parameter to decompose

fourth-order Cahn–Hilliard equation to the two second-order equa-

tions, ε is the interface thickness parameter, γ is the mobility,

λ is the mixing energy density, and G is the chemical potential,

G = λ[ −∇

2 φ + φ( φ2 − 1 ) / ε 2 ] . In this model, mixing energy ( f mix )

is obtained by familiar Ginzburg–Landau formulation as follows

( Yue et al., 2006; Zhou et al., 2010 ):

f mix (φ, ∇φ) =

1

2

λ| ∇φ| 2 +

λ

4 ε 2 ( φ2 − 1) 2 (6)

Surface tension is considered as an intrinsic property corre-

sponding to the excess free energy density of the interfacial region.

As ε → 0, the ratio λ/ ɛ produces the interfacial tension in the clas-

sical sense σ =

2 √

2 λ3 ε ( Yue et al., 2006 ). Other details about phase-

field theory and formulation can be find elsewhere ( Badalassi et al.,

2003; Fichot et al., 2007; Jacqmin, 1999; Yue et al., 2006; Zhou

et al., 2010 )

The governing equations are supplemented by standard bound-

ary conditions (e.g., inlet, outlet, no-slip, wetted wall and symme-

try). On the solid wetted grains, the following boundary conditions

are implemented:

u = 0 (7)

n · ε 2 ∇ φ = ε 2 cos (θ ) | ∇ φ| (8)

g

·(

γ λ

ε 2

) ψ = 0 (9)

here n is the unit normal to the wall and θ is the surface contact

ngle. The details about other boundary equations can be found

lsewhere ( Yue et al., 2006; Zhou et al., 2010 ).

The numerical equations were solved using a robust finite el-

ment solver, COMSOL Multiphysics (2012) . The domain was re-

olved by triangular elements. Finer meshes was implemented

round the sharp corners and inside the thin throats and the small

ores ( Fig. 2 ), and the coarser ones in the larger pore bodies. Con-

idering the average pore diameter in porous medium as the char-

cteristic length ( l c ) and defining Cahn number as Cn = ε/ l c , it was

emonstrated by Akhlaghi Amiri and Hamouda (2013) that using

n = 0.03 and mesh size h = 0.8 ε, the model convergence and mesh

onvergence are satisfied for the phase-field method. By imple-

enting these values in the simulations of this work, the average

umber of mesh elements used in the simulated models, including

ractured and non-fractured, was 282,862 with the average size of

.012 mm. The other critical parameter in the phase field formula-

ion is the mobility γ , which according to Yue et al. (2006) , has to

e large enough to retain a more or less constant interfacial thick-

ess and small enough to keep the convective motion. Sensitivity

tudies done by Akhlaghi Amiri and Hamouda (2013) showed that

imulations with 0.1 < γ < 1 results in less volume shrinkage of

uids in the simulation of two-phase flow problems, so γ = 1 was

sed in the current study.

The established simulation model was previously validated with

he analytical solution of the stratified two-phase Poiseuille flow

hrough the channel ( Akhlaghi Amiri and Hamouda, 2013, 2014 )

nd showed perfect accuracy. It also has been verified with micro-

odel experimental results for different capillary – viscous gov-

rning problems ( Akhlaghi Amiri and Hamouda, 2014 ; Rokhforouz

nd Akhlaghi Amiri, 2017a, b ). The simulation results demonstrated

ood agreements with the reported experimental observations.

M.R. Rokhforouz, H.A. Akhlaghi Amiri / Advances in Water Resources 113 (2018) 260–271 263

Fig. 3. The stabilized fluid distributions for different fluid displacements with log Ca = −4.6, −3.6 and −2.6 and log M = −2, −1 and 0 in non-fractured porous medium.

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. Results and discussion

The effects of micro-fracture geometrical parameters on the

isco-capillary behavior of two-phase displacements, compared

ith non-fractured medium, under different wettability conditions

re addressed here using several sensitivity studies.

.1. Fracture effect on visco-capillary flow instabilities

Capillary number is defined as Ca = μw

u in j /σ . Ca and viscosity

atio ( M = μw

/ μo ) quantify the flow characteristics and determine

he types of flow instabilities in absence of the gravity forces. In

ll the simulated models, at a certain time after breakthrough of

he injected water, the displacement was stabilized, so the profiles

f the phases thereafter remained unchanged. The stabilized fluid

istributions inside the non-fractured and the fractured heteroge-

eous porous media are compared for a range of log Ca and log

, in Figs. 3 and 4 , respectively. Ca and M ranged three orders

f magnitude. The fracture dimensionless length ( l D ) is defined as

he ratio of the fracture length to the average pore diameter; and

he dimensionless aperture ( w D ) is defined as the ratio of the frac-

ure width to the average pore diameter. In the simulated fractured

edium here, l D and w D were set to 28 and 0.65, respectively. The

lue and the light red colors in Figs. 3 and 4 represent water and

il phases, respectively; and the color gradient represents the in-

erface mixing zone. The medium here is considered intermedi-

te wet ( θ = π/ 2 ). It is confirmed that the displacement process

trongly depends on Ca and M for the media with/without frac-

ure. Furthermore, the fluid regimes seem corresponding in both

ases at different values of M and Ca. Except for the flow at the

ighest Ca and M, where the water saturation is high for both non-

ractured and fractured media, the other displacement patterns in-

olve instabilities, called fingering. However, two different types of

ngering can be identified, named viscous and capillary fingerings.

ased on these observations, three different displacement regimes

ay be classified: stable displacement, viscous fingering and cap-

llary fingering, which is in agreement with reported experimen-

al observations ( Lenormand et al., 1988; Zhang et al., 2011 ). Sta-

le displacement is recognized for both media at log Ca = −2.6 and

og M = 0. In this flow regime, the advancing fluid completely oc-

upies pore bodies before approaching the neighboring pores. In

he capillary fingering pattern, however, instabilities take place in

he form of wide forward and lateral moving fronts of the dis-

lacing phase, with an average width more than 3 pore bodies.

his type is evident at low Ca where M is high enough (e.g., log

a = −4.6 and M = 0 in Figs. 3 and 4 ). In the viscous fingering

egime, on the other hand, several multiple loosely connected or

isconnected flow streams are formed, which progress toward the

utlet, with an average width in the order of 1–3 pore bodies. This

ngering type is evident at lower M values (e.g., log M = −2 in

igs. 3 and 4 ). Transition of the fingering type from viscous to cap-

llary is seen at log Ca = −4.6, as log M increases from −2 to 0.

n the case of the fractured medium ( Fig. 4 ) at log Ca = −4.6 and

og M = 0, it is observed that the primary breakthrough took place

y a finger progressed through the matrix, not those channeled

hrough the fracture. This observation is an evidence for the cap-

llary dominant fluid displacement, in which the advancing phase

referentially flows through the smaller paths (of the matrix) with

he higher capillary forces. The agreement of these results with

hose obtained in the reported numerical ( Akhlaghi Amiri and

amouda, 2014 ) and experimental ( Lenormand et al., 1988; Zhang

t al., 2011 ) works demonstrates that the displacement regimes

re independent of the medium properties, such as porosity and

ermeability and presence of micro-fracture. Therefore, a similar

264 M.R. Rokhforouz, H.A. Akhlaghi Amiri / Advances in Water Resources 113 (2018) 260–271

Fig. 4. The stabilized fluid distributions for different fluid displacements with log Ca = −4.6, −3.6 and −2.6 and log M = −2, −1 and 0 in fractured porous medium. In this

model l D = 28 and w D = 0.65.

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visco-capillary behavior of two-phase flow in different porous sam-

ples may be expected. However, the detailed displacement proper-

ties, such as breakthrough time and recovery factor, are functions

of medium geometry.

General comparison of Figs. 3 and 4 demonstrates lower dis-

placement efficiency of the fractured media, compared to the cor-

responding non-fractured ones. This is due to the channeling of

the injected water through the fracture. However, in the case of

the displacement at the highest Ca (log Ca = −2.6), this difference

is less pronounced. It is also evident that the fracture behaved

as a highway, in which the formed viscous or capillary fingers

were joined together and redirected toward the outlet. For exam-

ple, in the case of log Ca = −4.6 and log M = −2, the non-fractured

medium ( Fig. 3 ) contains several loosely viscous fingers, originated

close to the inlet. While, in the fractured medium ( Fig. 4 ), there

is one major finger through the fracture at the beginning, which is

then split into several thin fingers. In the fractured medium ( Fig. 4 )

at lower M values, the fracture area was not fully swept by water.

This implies that at low M, the domain of instabilities is even ex-

tended to the fracture zone. However, in the cases of stable and

capillary dominant displacement regimes, the flow inside the frac-

ture was piston-like.

It is worth to note that, at high Ca values (log Ca ≥−3.6) when

M is low enough (log M ≤−1), the water-oil interface - in both

fractured and non-fractured media - experiences high diffusion

(detected by gradual color variation). Capillary number increases

by either lowering interfacial tension and/or increasing injection

velocity. For example, by lowering interfacial tension (e.g., surfac-

tant injection), oil/water emulsion may be formed within the front,

in agreement with the experimental observations on surfactant in-

jection ( Arabloo et al., 2015; Kianinejad et al., 2015 ); which may

result in more efficient microscopic displacement.

The simulated normalized pressure and the velocity field for

he non-fractured and the fractured porous media are compared

n Fig. 5 . The comparison is made at water pore volume injected

PVI) of 0.06. The pressure is normalized with capillary pressure,

c = 2 σ / l c , and the velocity is normalized with the water injection

elocity ( u inj ). In these models, log M = −2 and log Ca = −4.6, and

or the fractured medium l D = 28 and w D = 0.65. There is almost a

omogenous pressure gradient in the matrix for both media. How-

ver, it is evident that the medium maximum pressure (which is

appened around the inlet) is higher for the non- fractured case

p / p c = 15 × 10 3 ) compared to the fractured one ( p / p c = 13 × 10 3 ).

o, the presence of the fracture causes the flow resistance to de-

rease in the medium during water-oil displacement. Also, in the

ractured model ( Fig. 5 (b) at the top), the fracture orientation af-

ected the pressure distribution. The velocity profile in the non-

ractured medium ( Fig. 5 (a) at the bottom) demonstrates two main

ater streams at the top and bottom of the medium which later

orm the main advancing fingers (see stabilized fluid distribution

f this case in Fig. 3 ). However, in the fractured medium ( Fig. 5 (b)

t the bottom), water streamlines are concentrated in the fracture

nd a maximum velocity ( u / u inj = 140) observed at the beginning

f the fracture. As can be seen, the velocity of the other water

treams in the matrix is minor, when the fracture is present.

Medium water saturation (S w

) after displacement stabilization

s plotted as a function of log Ca and log M in Fig. 6 , for both the

ractured medium (dashed lines) and the non-fractured one (con-

inuous lines). Fig. 6 (a) shows that at different M values, when

a increases, the water saturation of both media approach to-

ether; as become almost equal at log Ca = −2.6, where viscous

orces are dominant. For both fractured and non-fractured cases,

w

increases by Ca with a semi-linear trend at different M val-

es. This is in agreement with the experimental observations by

M.R. Rokhforouz, H.A. Akhlaghi Amiri / Advances in Water Resources 113 (2018) 260–271 265

Fig. 5. Normalized pressure ( p/p c ) profile at the top and normalized velocity ( u/u inj ) distribution at the bottom when PVI = 0.06 for (a) the non-fractured and (b) the fractured

media. In this model, log M = −2 and log Ca = −4.6 and for the fractured medium l D = 28 and w D = 0.65.

Fig. 6. Medium stabilized water saturation (S w ) as a function of (a) log Ca and (b) log M for different displacement patterns observed in the non-fractured (continuous lines)

and fractured (dashed lines) media.

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ottin et al. (2010) and Zhang et al. (2011) in non-fractured micro-

odels. However, as Ca decreases, S w

reduces with slightly higher

radient in the fractured media, compared to the non-fractured

nes.

As depicted in Fig. 6 (b), for both fractured and non-fractured

ases, S w

increases by M. The trend of S w

-log M for the fractured

edia is linear for all the Ca values. It can be seen that for the

on-fractured cases with log Ca < −2.6, the gradient of the vari-

tion in S w

is relatively lower for log M < −1, compared to log

> −1. It may indicate that a kind of transition happens from vis-

ous to capillary dominant flow regimes at about log M = −1, as

decreases. It is also evident in Fig. 6 (b), that at log Ca = −2.6,

he trend of S w

versus M for the non-fractured medium is almost

inear and approximately coincides the fractured one.

Based on the observed flow patterns (some of them presented

n Figs. 3 and 4 ) and the reported study on the fluid saturations

Fig. 6 ), three identified displacement regimes are mapped on a

og M–log Ca stability phase diagram for both fractured and non-

ractured media, presented in Fig. 7 . Flow regimes (including sta-

le, capillary dominant and viscous dominant) in corresponding

ractured and non-fractured media simulated in this study were

elatively identical (as also observed in Figs. 3 and 4 ); so, the

oundaries in the stability phase diagram ( Fig. 7 ) are considered

o be matched for both media. However, in the case with more

ough matrix structure, there may be difference in the boundaries

f the flow regimes in fractured medium compared to the non-

racture one. The approximate boundaries of stable displacement

re located at log M ≈ 0 and log Ca ≈ −2.5. Log M ≈ −1 is con-

idered as the boundary for the viscous fingering region, and log

a ≈ −4.5 is considered as the boundary for the capillary finger-

ng region. The shape of the three regions was discussed in detail

y Lenormand et al. (1988) . The region boundaries empirically ob-

ained by Zhang et al. (2011) are in agreement with those obtained

n this work. Each of the three domains on the stability phase di-

gram ( Fig. 7 ) corresponds to a different basic mechanism where

ne kind of force is dominant ( Ferer et al., 2004; Fernández et al.,

991; Lenormand et al., 1988 ). In the stable displacement, the prin-

iple force is due to the larger viscosity of the displacing fluid and

266 M.R. Rokhforouz, H.A. Akhlaghi Amiri / Advances in Water Resources 113 (2018) 260–271

Fig. 7. log Ca–log M stability phase diagram showing three flow regimes and the

locations of the performed numerical experiments (black and white dots, for non-

fractured and fractured media, respectively). The white area is called crossover

zone.

Fig. 8. Matrix oil recovery factor (RF) as a function of pore volume injected (PVI) of

water for the simulated models with different fracture dimensionless lengths ( l D ).

(

l

s

s

c

t

(

t

t

t

b

g

t

j

b

t

b

c

m

a

t

t

t

f

o

t

t

t

p

t

m

s

t

t

t

g

f

a

f

t

(

s

I

w

a

t

t

i

n

1

r

t

c

g

(

o

e

v

S

t

f

t

l

the capillary effect is minor ( Lenormand et al., 1988 ). The pattern

shows almost a flat front with some irregularities at the scale of

a few pores ( Figs. 3 and 4 at log M = 0 and log Ca = −2.6). The

saturation of the displacing phase is high after flow stabilization

for the stable displacement. The viscous fingering occurs when the

viscosity of the displaced fluid is dominant, i.e., larger. The pattern

shows tree-like fingers spread across the network, growing toward

the outlet ( Figs. 3 and 4 at log M ≤ −1 and log Ca > −4.6). Viscous

fingering regime is also referred to as open branch because no vis-

cous fluid encirclement occurs ( Fernández et al., 1991 ). At low cap-

illary numbers (e.g., low injection rates or high interfacial tension),

the principal force is due to the capillarity. In this condition ( Figs. 3

and 4 for log M ≥ 0 and log Ca ≤ −4.6), capillary fingering occurs

which spread across the network but the pattern is different from

the previous case. For very small capillary numbers, flows exhibit

capillary fingering even at M = 0 ( Fernández et al., 1991 ).

3.2. Effect of geometrical characteristics of fracture

Fig. 8 shows the oil recovery factor (RF) as a function of wa-

ter pore volume injected (PVI) for the simulated model, in which

log M = −2 and log Ca = −4.6. The results are compared at differ-

ent dimensionless fracture lengths of l D of 0 (non-fractured), 18,

28 and 38. Fig. 9 shows the fluid profiles after breakthrough of

the injected water, as the displacement is stabilized. In general, in-

creasing l D resulted in lower oil recovery factor and earlier water

breakthrough, which is in agreement with the experimental results

Kianinejad et al., 2015 ). Presence of even a short fracture with

D = 18, resulted in a fall in recovery factor (almost 10%), as ob-

erved in Fig. 8 . It can be seen in Fig. 8 that as fracture dimen-

ionless length increases from 18 to 38 (almost doubled), oil re-

overy factor decreases from 0.28 to 0.14 (halved). It is also worth

o note that the time gap between the primary breakthrough time

the instant at which RF deviated from linearity) and the stabiliza-

ion time, is a function of the fracture length (see Fig. 8 ). In this

ime gap active fingers breakthrough (see Fig. 9 ). In the case of

he non-fractured medium, there are several active fingers which

reakthrough one by one, before stabilization time; so, there is a

radual change in RF at this time period. However, in the case of

he medium with the longest fracture ( l D = 38), there is just a ma-

or water finger, channeled through the fracture. So, as this finger

reakthroughs, the stabilization takes place.

As shown in Fig. 9 , as the length of the fracture increases,

he progress of the water finger channeled through the fracture

ecomes more dominant and the progress of other formed vis-

ous fingers in the matrix is more declined. In the non-fractured

edium ( Fig. 9 (a)), there are two main fingering areas, at the top

nd the bottom of the medium. First, the one at the bottom break-

hroughs. For the fractured case with l D = 18 ( Fig. 9 (b)), due to

he water channeling through the fracture, the fingering area at

he bottom initiated, but its rate of progress is less than the non-

ractured one. So, at the breakthrough time, it does not reach the

utlet. As the fracture length increases more ( Fig. 9 (c) and (d)),

he propagation of this bottom finger is more depressed, since

he water channeling effect is more pronounced. For the case of

he longest fracture, the water phase just invaded some adjacent

ore bodies around the inlet and the fracture, and an early break-

hrough happened (at PVI = 0.12). It is also evident that for the

edium with the shortest fracture ( Fig. 9 (b)), water swept area in-

ide the fracture is the least (almost 70%). In other words, the frac-

ure water invaded fraction increases by its length. In the case of

he short fracture, the pressure difference between the two frac-

ure ends is low, which is almost in the same order of pressure

radient in the matrix. Therefore, the water crossflow from the

racture to the matrix is more possible. However, in the case with

long fracture, the considerable lower flow resistance inside the

racture compared to the neighboring matrix, results in preferen-

ial water flow through the fracture.

Three different models with the fracture inclinations of α = 0 °in the flow direction), α = 45 ° and α = 90 ° were simulated to

tudy the effect of fracture orientation on displacement behavior.

n the simulated models, log M = −2, log Ca = −4.6, l D = 28 and

D = 0.65. Fig. 10 compares oil recovery factor of these models

s a function of water pore volume injected (PVI). It is observed

hat increasing α results in higher RF and a later breakthrough

ime, which was predicted. Approximately 30% of the matrix oil

s recovered in the case of the model with α = 90 °, while the fi-

al recovery of the model with α = 45 ° and α = 0 ° are 20% and

0%, respectively; i.e., 10% reduction of RF for each 45 ° fracture

otation toward flow direction. Fig. 11 shows the stabilized dis-

ribution of water and oil in the models with different α. In the

ase of the fracture perpendicular to the flow direction ( Fig. 11 (a)),

eneral displacement pattern is similar to the non-fractured case

Fig. 9 (a)). However, it is observed that presence of a fracture with

ther orientation ( Fig. 11 (b) and (c)) adversely affects displacement

fficiency, in agreement with the micro-model experimental obser-

ations made by Abedi et al. (2012), Farzaneh et al. (2010) and

edaghat et al. (2013) . For example, for the case with the frac-

ure in the flow direction ( Fig. 11 (c)), water just went through the

racture highway and resulted in an extremely early water break-

hrough (at PVI = 0.1).

Fig. 12 shows RF as a function of water PVI for the simu-

ated models with different fracture dimensionless apertures in the

M.R. Rokhforouz, H.A. Akhlaghi Amiri / Advances in Water Resources 113 (2018) 260–271 267

Fig. 9. The stabilized fluid distributions for the models with different fracture dimensionless lengths of (a) l D = 0 (non-fractured), (b) l D = 18, (c) l D = 28 (base case), and (d)

l D = 38. In this model, log M = −2 and log Ca = −4.6 and for fractured ones w D = 0.65.

Fig. 10. Matrix oil recovery factor (RF) as a function of pore volume injected (PVI)

of water for the simulated models with different fracture orientation angles ( α).

r

I

R

w

w

w

m

r

s

w

c

a

t

f

Fig. 12. Matrix oil recovery factor (RF) as a function of pore volume injected (PVI)

of water for the simulated models with different dimensionless fracture apertures

( w D ).

p

f

t

f

t

e

b

m

i

e

F

t

F

l

ange of 0.3 to 2. A non-monotonic behavior is observed in Fig. 12 .

t is interesting to see that the case with w D = 0.3 has the highest

F, while the one with w D = 0.65 has the lowest RF. By increasing

D above 0.65, RF gradually increased but did not reach that of

D = 0.3. This observation indicates that there is a critical fracture

idth ( w D ∼ 0.65 in the current model) in which the medium RF is

inimum. In general, variation of the fracture width in the studied

ange, resulted in less than 10% difference in RF. Fig. 13 demon-

trates the stabilized distribution of water and oil in the models

ith different fracture apertures. Several loosely connected or dis-

onnected flow paths can be seen in the model with w D = 0.3,

s the fracture aperture is in the range of the medium pore and

hroat size (See Fig. 13 (a)). Although water channeling through the

racture is minor in the case with w D = 0.3, it is affecting the dis-

ig. 11. The stabilized fluid distributions for models with different fracture orientation an

og Ca = −4.6, l D = 28 and w D = 0.65.

lacement in those with w D ≥ 0.65. So, the water channeling ef-

ect is the major cause of RF reduction as w D increases from 0.3

o 0.65 ( Fig. 12 ). As can be seen by comparison of Fig. 13 (b)–(d),

or w D ≥ 0.65, the size of the single active finger originated from

he inlet at the bottom of the medium is relatively similar. How-

ver, the size of the fingers originated from the fracture enlarged

y w D . The cross-flow out of the fracture and into the neighboring

atrix pores enhanced by w D . Also, the swept area of the fracture

ncreased by w D . Therefore, RF increased as the fracture was thick-

ned from 0.65 to 2 ( Fig. 12 ).

It is also observed that in the case of the wider fractures (See

ig. 13 (c) and (d)), some amount of oil is trapped inside the frac-

ure, both at the middle and at the bottom. As the fracture width

gles of (a) α = 90 °, (b) α = 45 ° (base case) and (c) α = 0 °. In this model, log M = −2,

268 M.R. Rokhforouz, H.A. Akhlaghi Amiri / Advances in Water Resources 113 (2018) 260–271

Fig. 13. The stabilized fluid distributions for four models with different dimensionless fracture apertures of (a) wD = 0.3, (b) w D = 0.65 (base case), (c) w D = 1.3 and (d)

w D = 2. In this model, log M = −2, log Ca = −4.6 and l D = 28.

Fig. 14. Snapshots of fluid displacements inside the fracture for the case with

w D = 2 at different PVIs of 0.1, 0.14, 0.17 and 0.2, from the left to the right, respec-

tively. In this model, log M = −2, log Ca = −4.6 and l D = 28.

t

r

(

l

i

t

fi

d

t

p

t

t

t

t

m

c

c

f

t

f

i

b

a

t

t

t

t

n

r

o

s

i

t

c

b

p

P

o

c

l

A

w

c

I

m

s

fl

increases, the trapped oil volumes become larger. This effect is due

to the water front instability (fingering) inside the fracture, which

is exaggerated as the fracture becomes wider.

For detail study of the fluid instabilities inside the fracture, fluid

displacement within the fracture for the medium with w D = 2 is

plotted in Fig. 14 at different instants during water invasion. At

the first instant (PVI = 0.1), it is observed that three small drops

of oil have been formed. Water front is advancing inside the frac-

ture; however, the displacement is not stable (piston-like). It is by-

passing some amount of oil, due to low M (log M = −2) and the

large fracture width. The bypassed oil forms more droplets, as can

be seen in Fig. 14 at PVI = 0.14. During displacement, some of the

formed oil droplets join together and form bigger drops (PVI = 0.17

and 0.2). It is very interesting to see the advancing and the re-

ceding contact angles inside the fracture for the bigger drop at

PVI = 0.17 and the single big drop at PVI = 0.2, which are happened

under the influence of water momentum within the fracture. At a

later time after PVI = 0.2, the oil drop slightly grows and its shape

is changed (see Fig. 13 ), which may be due to the counter-current

displacement between the fracture and the neighboring matrix.

Of course, the effect of the fracture geometrical parameters

(including length, width and orientation) on displacement behav-

ior is a function of Ca and M. Combining the results obtained

in Sections 3.1 and 3.2 , may suggest that increasing Ca and M

(approaching stable displacement) will reduce the effect of the

fracture parameters. However, lowering M and Ca may exacerbate

these effects.

3.3. Effect of medium surface wettability

To quantify the degree of wettability, contact angle ( θ ) is usu-

ally studied, which is a boundary condition in determining the in-

terfacial shape ( Brown and Neustadter, 1980; Rabbani et al., 2017 ).

A grain surface is generally considered water wet if θ < π /2 and

oil wet if θ > π /2. The influence of wettability is addressed in this

section for water–oil displacement in the fractured medium as log

Ca = −4.6, log M = −2, l = 28 and w = 0.65. It is important to note

D D

hat when the medium is water wet ( θ < π /2), water injection rep-

esents an imbibition process, while when the medium is oil wet

θ > π /2) it represents drainage process.

Fig. 15 shows the fluid distributions after displacement stabi-

ization for different θ values of 5 π /12, π /2 and 2 π /3, correspond-

ng to slightly water wet, neutral wet and slightly oil-wet, respec-

ively. As shown in Fig. 15 , the general distribution of the water

ngers is quite similar for all the studied wettabilities. However, as

epicted in Fig. 15 (a), in the case of the water-wet medium, wa-

er preferentially tends to invade thinner pores and throats, com-

ared to the oil-wet medium. Another important difference be-

ween these cases, is the displacement stability inside the frac-

ure area. As can be seen in Fig. 15 (c) for the oil-wet medium,

he fracture is fully swept by the invading phase. Capillary force in

he porous matrix is negative for water in the case of the oil-wet

edium; so, water prefers to fully invade the fracture (with lower

apillary forces) before drainage into the matrix. However, in the

ase of the water-wet medium ( Fig. 15 (a)), water flow within the

racture is unstable and hence some oil droplets were trapped in

he fracture. In this case, water imbibition takes place between the

racture and the matrix. In other words, positive capillary forces

n the porous matrix in the water-wet medium resulted in im-

ibition of water from the fracture into the porous matrix. It is

lso observed that the growth of the active finger at the bottom of

he matrix increases as the medium becomes more water-wet. In

he case of strongly water-wet media, it was observed that due to

he considerable water imbibition rate between fracture and ma-

rix, the displacement profile and efficiency approached that of the

on-fractured medium. This is in agreement with the simulation

esults of Monteagudo and Firoozabadi (2004) and experimental

bservations made by Pooladi-Darvish and Firoozabadi (20 0 0) .

The detailed water displacement profiles inside the fracture ver-

us time are depicted in Fig. 16 for the water-wet medium. As the

nterface advances through the fracture (PVI = 0.3), water imbibes

o the matrix; hence oil drops enter the fracture zone, due to the

ounter-current imbibition process (in agreement with the results

y Rokhforouz and Akhlaghi Amiri, 2017a, b ). Small oil drops, ex-

elled from the porous matrix into the fracture, join together (at

VI = 0.44) and the formed larger droplets moves toward the end

f the fracture (PVI = 0.58). This counter-current imbibition pro-

ess goes on during displacement process (PVI = 0.72) until stabi-

ization. However, based on the previous studies ( Rokhforouz and

khlaghi Amiri, 2017a, b ), a different porosity and permeability as

ell as wettability of the matrix would affect the imbibition pro-

ess and the resultant oil recovery factor.

RF increases as the medium becomes more water-wet ( Fig. 17 ).

t is also evident in Fig. 17 that in the case of the neutral wet

edium the time gap between the primary breakthrough and the

tabilization time is wider. In both water-wet and oil-wet media,

ow is stabilized just after primary breakthrough. Theses means

M.R. Rokhforouz, H.A. Akhlaghi Amiri / Advances in Water Resources 113 (2018) 260–271 269

Fig. 15. The stabilized fluid distributions for the simulated model with different grain contact angles of (a) 5 π /12, (b) π /2 (base case) and (c) 2 π /3. In this model, log

M = −2, log Ca = −4.6, l D = 28 and w D = 0.65.

Fig. 16. Snapshots of fluid displacements inside the fracture for the water-wet case

with θ = 5 π /12 at different PVIs of 0.3, 0.44, 0.58 and 0.72, from the left to the

right, respectively. In this model, log M = −2, log Ca = −4.6, l D = 28 and w D = 0.65.

Fig. 17. Matrix oil recovery factor (RF) as a function of pore volume injected (PVI)

of water for the simulated model with different grain contact angles ( θ ).

t

m

d

f

s

s

s

c

Fig. 18. Three enlarged sections of the simulated model with θ = 2 π/ 3 at two suc-

cessive instants during the displacement process. Different events are specified with

circles and arrows.

t

e

i

p

l

fi

i

c

i

s

a

d

F

p

t

hat the number of active fingers in the case of neutral wet

edium is larger, compared to the other verified wettability con-

itions.

Fig. 18 demonstrates two instants during water invasion process

or three enlarged sections of the oil-wet medium ( θ = 2 π/ 3 ). The

imulated model was able to successfully capture different pore-

cale mechanisms during the displacement process. Some pore-

cale displacement mechanisms including pore doublet, interface

oalesce, snap-off, and reverse displacement were captured. Due to

he capillary pressure effect, the non-wetting phase (water) pref-

rentially invades the larger pores and the oil phase is bypassed

n the smaller pores (shown in Fig. 18 (a) with black circle). Water

hase bridges between the adjacent pores and the interface coa-

esce happens (specified with green circle in Fig. 18 (a)). As speci-

ed in Fig. 18 (b) with black circle, the water phase forms collars

n the pore throat, which snaps off the oil droplet. The presence of

oncave, convex and flat interfaces in the oil wet porous medium

s also detected (specified with black, purple, and red arrows, re-

pectively). Complex interplay between the contact angle and pore

ngularity causes the variation of the interface curvature, in accor-

ance with the theoretical results of Mason and Morrow (1994) .

ig. 18 (c) shows a pore-level reverse displacement as a result of

ressure gradient decline across the throat. In other words, when

he interface’s curvature changes from convex to concave, the in-

270 M.R. Rokhforouz, H.A. Akhlaghi Amiri / Advances in Water Resources 113 (2018) 260–271

p

a

s

A

a

R

A

A

A

A

A

B

C

C

C

E

F

F

F

H

J

J

K

K

L

L

L

L

terface is forced back into the pore throat against the direction of

the main stream flow.

4. Conclusion

This work evaluates the effects of micro-fracture geometri-

cal parameters on visco-capillary behavior of two-phase displace-

ments, compared with non-fractured medium, under different wet-

tability conditions. A heterogeneous pattern, taken from a real

dolomite rock section, was considered as the computational do-

main. To perform the simulations, the coupled Cahn–Hilliard phase

field and Navier–Stokes equations were solved by a finite ele-

ment method, using COMSOL Multiphysics solver. The values of

mesh sizes and interface thickness parameter were set in order to

achieve mesh convergence and model convergence.

First, numerical experiments were performed on the simulated

medium with/without fracture to study the effects of viscosity ra-

tios (M) and capillary numbers (Ca). For both media, the bound-

aries of stable displacement were identified at log M ≈ 0 and log

Ca ≈ −2.5, and the boundaries for viscous fingering and capillary

fingering regions were located at Log M ≈ −1 and log Ca ≈ −4.5,

respectively. Presence of the fracture in the fractured medium re-

sulted in water channeling effect which caused reduction of the

number of active fingers and hence reduction of the final oil re-

covery factor. At log M ≥ 0, the fracture was fully swept by the in-

jected water, but flow instabilities were observed inside the frac-

ture at lower M values, especially when log Ca > −4.6. At high Ca

(especially in the stable regime, with log Ca ≥−2.5 and log M ≥ 0),

recovery factor for the fractured medium was relatively identical

with the non-fractured one.

The effects of geometrical characteristics of fracture including

length, aperture and orientation, were evaluated in the simulated

models. The presence of even a short fracture affected oil recovery

factor. It was observed that increasing the fracture length resulted

in lower oil recovery factor due to earlier water breakthrough time

caused by stronger channeling effect.

It was observed that there is a critical fracture width (almost

half of the medium average pore diameter) at which the recovery

factor of the medium during displacement is minimum, compared

to the media having thinner and thicker fractures. Minor channel-

ing effect in the media with thinner fracture and larger fracture

swept volume as well as high fracture/matrix cross flow in the me-

dia with thicker fracture were detected as the main cause of this

non-monotonic behavior. In the models with thick fractures (with

the thickness higher than the average pore diameter), considerable

trapped oil volumes were observed inside the fracture at lower M.

The fracture orientation showed the most impressive effect on

oil recovery, compared to the other studied fracture geometrical

properties. The oil recovery factor increased ( > 20%) when the frac-

ture orientation changed from flow direction to the direction per-

pendicular to the flow. The medium with perpendicular to the flow

fracture behaved almost the same as the non-fractured medium.

The effect of medium wettability was also studied. The change

of the medium wettability from oil-wet to neutral or slightly

water-wet, did not highly affect the displacement profile in the

fractured medium. However, oil recovery factor was increased as

the medium became more water wet. Although the fracture was

fully invaded by water in the oil-wet and neutral-wet media, flow

instabilities caused by counter-current imbibition were observed

inside the fracture for water-wet medium. Different pore-scale

events were captured for different simulated models. For exam-

ple, pore doublet, interface coalesce, snap-off, and reverse move-

ments were observed during the studied displacement in oil-wet

medium. The presence of concave, convex and flat interfaces were

also detected.

The results of this numerical study confirmed the ability of the

hase field model to realistically predict different pore-level mech-

nisms during two-phase displacements processes through highly

ophisticated media, containing fractures.

cknowledgment

This research did not receive any specific grant from funding

gencies in the public, commercial, or not-for-profit sectors.

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