Directory UMM :Data Elmu:jurnal:A:Advances In Water Resources:Vol24.Issue3-4.2001:

Advances in Water Resources 24 (2001) 345±363
www.elsevier.com/locate/advwatres

Network modelling of strong and intermediate wettability on
electrical resistivity and capillary pressure
H.N. Man, X.D. Jing *
Centre for Petroleum Studies, T.H. Huxley School of Environment, Earth Sciences and Engineering, Imperial College of Science,
Technology and Medicine, London SW7 2BP, UK
Received 1 December 1999; received in revised form 14 July 2000; accepted 31 August 2000

Abstract
A network model that investigates electrical resistivity and capillary pressure curves of oil/water/rock systems for a full-¯ooded
cycle (primary drainage, imbibition and secondary drainage) is presented. This model uses a realistic pore geometry in the form of a
grain boundary pore (GBP) shape and pore constrictions. The model also incorporates pore-scale displacement mechanisms and
pore-scale wettability alteration that are physically based. A range of contact angles (from 0 to 180°) has been investigated. A
detailed description of wettability at the pore scale was simulated to allow both water- and oil-wet regions existing within a single
pore. Our numerical simulated results show experimentally observed non-linear trends in double-logarithmic plots of resistivity
index vs water saturation. Furthermore, our results show that contact angle hysteresis, which leads to di€erent pore scale physics
(e.g., snap-o€ vs piston-like displacement), reveals hysteresis observed in both electrical resistivity and capillary pressure
curves. Ó 2001 Elsevier Science Ltd. All rights reserved.
Keywords: Network model; Electrical resistivity; Capillary pressure; Pore geometry; Wettability; Hysteresis

1. Introduction
Initially, all hydrocarbon-bearing reservoirs contained rocks that were fully saturated with water.
Hydrocarbons may migrate into these regions displacing
water and the ¯uids equilibrate over geological periods
of time to occupy the pore space. This is provided that
the pressure di€erential between these two immiscible
¯uids can be overcome. At equilibrium, this pressure
di€erential known as the capillary pressure, Pc , is related
by the Young±Laplace equation


1 1
‡
;
…1†
Pc ˆ r
r1 r2
where r is the interfacial tension between the immiscible
¯uids and r1 and r2 represent the principal radii of

curvature normal to each other. In oil/water/rock systems, the capillary pressure is often de®ned as the oil
pressure minus the water pressure (i.e., Pc ˆ Po ÿ Pw ).
The relationship between capillary pressure and water
*

Corresponding author. Tel.: +44-20-7594-7320; fax: +44-20-75947444.
E-mail address: x.jing@ic.ac.uk (X.D. Jing).

saturation is important in locating zonal regions, where
there is a transition from water to oil of a hydrocarbon
reservoir. The balance of capillary against gravitational
forces determines initial ¯uid distributions across the
transition zone and, together with viscous forces, a€ects
the eciency of oil recovery by water injection.
Capillary pressure controls the distribution of ¯uids.
Electrical resistivity of a ¯uid saturated rock depends on
the distribution of conducting phase [1±4]. One of the
more reliable techniques used to evaluate hydrocarbon
potential, in a petroleum reservoir, namely electrical
logging is based on an empirical relation called the

Archie saturation equation [5]. The equation relates the
electrical resistivity of the rock sample to water saturation such that
Iˆ

Rt
1
ˆ ;
Ro Swn

…2†

where Rt is the resistivity of the sample at a given water
saturation Sw (i.e., partially saturated with water) and Ro
is the resistivity of the sample at 100% water saturation.
n is an empirical parameter called the Archie saturation
exponent. To determine n, the gradient of a double
logarithmic plot of resistivity index against water

0309-1708/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved.
PII: S 0 3 0 9 - 1 7 0 8 ( 0 0 ) 0 0 0 6 1 - 0

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H.N. Man, X.D. Jing / Advances in Water Resources 24 (2001) 345±363

saturation is measured. The ratio Rt =Ro is usually denoted by I and is called the resistivity index.
Eq. (2) also applies to environmental engineering
problems where, for example, electrical measurements
may be applied to monitor the level of soil contamination by non-aqueous phases. An understanding of the
electrical properties and their relation to ¯uid saturation
in soils improves the assessment of contamination and
helps to design remedial engineering processes.
Generally, capillary pressure and electrical resistivity
curves are a function of saturation history, i.e., which
¯uid is displacing and which ¯uid is displaced, and exhibits hysteresis. The degree of hysteresis is found to be
dependent on pore structure [2] and wettability [1±4].
Wettability is a term describing which ¯uid amongst
at least one other immiscible ¯uid has a tendency to
adhere to a surface. It is a very important parameter
because the surface properties of the rock determine the

¯uid distribution in the pore space. Ultimately electrical
properties, oil displacement and hence oil recovery is
a€ected by the wettability of the rock. A direct technique used by many laboratories to quantify wettability
is through the contact angle. In an oil/water/rock system, it is the angle measured through the water phase by
placing a drop of oil or water onto a ¯at solid surface
that should be representative of the reservoir considered.
A strongly water-wet rock means that water has the
greater tendency to spread on the surface. Hence, the
contact angle is roughly zero. For a strongly oil-wet
rock, the roles of the ¯uids are reversed and the contact
angle is near 180°. Intermediate wettability describes a
rock that has no overall dominant preference to either
¯uid. The contact angle is near 90°.
Hirasaki [6] gave a theoretical insight of wettability
from ®rst principals. The stability and thickness of water
®lms determine wetting in oil/water/rock systems. This
in turn depends on a parameter known as the disjoining
pressure. This additional disjoining pressure is due to
the microscopic interactions between atoms that are
signi®cant for residual ®lms. The disjoining pressure

isotherm has a local maximum: when this is exceeded,
the thick water ®lm ruptures to a molecular thin one

[6,7] (Fig. 1). The general consensus is that wettability
alteration can take place if oil is left in contact with the
rock surface for sucient time. In reality, wettability
alteration at the reservoir scale occurs over geological
periods of time. The critical capillary pressure, associated with wettability reversal, depends on mineralogy
[8], oil and brine properties, such as pH [9], reservoir
pressure and temperature [6]. The actual mechanism of
wettability reversal is still unclear [10].
Kovscek et al. [7] theorised, in their numerical model,
that if regions with a molecular thin water ®lm are
contacted by oil, then they are susceptible to adsorption
by surface-active components in the oil. Therefore, these
regions will be rendered oil-wet (Fig. 1). Kovscek et al.
[7] assumed that the contact angle for these oil-wet
surfaces were 180°, while 0 for water-wet surfaces.
However, Kovscek et al.'s [7] theoretical pore-scale
model is not a network model, but a model formed by a

capillary bundle of tubes, with a cornered cross-section
of di€erent radii, parallel to each other, but of equal
lengths. The radius of each capillary remains constant
along its length. The use of pore shape with crevices
allows mixed-wettability at the pore scale. This draws on
experimental evidence showing that variations of wetting can occur within a pore [11].
Recently, numerical simulators have been generalised
to allow for any contact angle. It is known that systems
rarely exhibit perfectly wetting (water- and oil-wet)
conditions [12,13]. Dixit et al. [14] predicted capillary
pressure and relative permeability curves incorporating
the e€ects of wettability alteration. They randomly distributed contact angles throughout their network to
simulate non-uniform wettability. They found that in a
water-wet system, oil recovery increases as the randomly
distributed contact angles approach a limit of 90°. This
seems counter-intuitive, but the pore ®lling sequence,
determined by the Young±Laplace equation (Eq. (1)),
reveals oil in larger pores is no longer trapped. Experimentally Li and Wardlaw [15] have shown, for waterwet systems, snap-o€ is less favoured as the contact
angle increases. This greatly reduces the amount of oil
trapped in larger pores. When pores are simulated to

Fig. 1. (a) Thick water ®lms coat water-wet surfaces; (b) molecular water thin ®lms allow wettability reversal.

H.N. Man, X.D. Jing / Advances in Water Resources 24 (2001) 345±363

have a contact angle greater than 90°, Dixit et al. [14]
observed oil recovery is dependent on the fraction of
pores that are oil-wet.
Blunt [16,17] explicitly modelled the ``co-operative
pore ®lling process'', which is a function of contact
angle. The process is so-called because the ¯uid displacement at the nodal pore where each tube meets is
dependent on whether neighbouring tubes meeting at
the junction are water- or oil-®lled [18] (Fig. 2). Incorporating Kovscek et al.'s [7] physical development of
mixed-wettability, Blunt [17] was able to reproduce
Dixit et al.'s [14] observations even though the contact
angle was the same everywhere. éren et al. [19], too,
modelled the co-operative pore ®lling process and their
quantitative predictions of capillary pressure and relative permeability agreed closely with experimental
measurements for a reservoir rock.
While numerical simulation of relative permeability

and oil recovery has been a topic of active research,
there is a lack of theoretical work concerning electrical

347

resistivity of porous rocks fully or partially saturated by
brine. Theoretically, Dicker and Bemelmans [20] realised
the importance of the continuity of water to ensure
electrical conductance at low water saturations. Wang
and Sharma [21], Sharma et al. [22] and Suman and
Knight [23] investigated the e€ects of pore structure and
wettability on electrical resistivity.
However, these preceding authors incorporated a
circular pore geometry. They were not able to model
observed experimental electrical resistivity behaviour,
especially at low water saturations for water-wet rocks.
Although they implemented the conductive contribution
of molecular water ®lms, the curves tended to in®nity
below some ®nite water saturation. On the contrary,
many authors [2,4,24±27] experimentally observed that

the resistivity curves tend to some ®nite resistivity index
and curved towards lower values of saturation exponent
demonstrating the so-called ``non-Archie'' behaviour.
The e€ect of a physical development of pore-scale
intermediate wettability on capillary pressure and

Fig. 2. During water imbibition, pore ®lling depends on the number of neighbouring throats ®lled with non-wetting phase. The pore ®lling process is
in order of preference from most likely (b) to least likely (f). (a) depicts piston-like advance (after [16]).

348

H.N. Man, X.D. Jing / Advances in Water Resources 24 (2001) 345±363

Fig. 3. Di€erent viewpoints (from top toÿ bottom:
3-D, circumferential and transverse) of new pore tube with constrictions. The pore constriction


varies according to y ˆ yb ÿ …yb ÿ yt † sin pL x (after [21,22]).

The network model features key geometrical attributes such as pore connectivity, constrictions along
the pore length, pore size distribution and pore shape

(Fig. 3). We chose the shape of four closely packed
uniform rods, which we refer to as a star-like grain
boundary pore (GBP) shape, to represent the crosssection of the conduits (Fig. 3). The constriction, along
the length of the conduit, is represented by a sinusoidal
variation [21,22,30,31].
In this paper, we de®ne the radius of the nodal pore,
yp , as

relative permeability has been studied by Blunt [17,28]
and éren et al. [19]. However, the e€ect of this detailed
wettability scenario has not been investigated for electrical properties. The variation in the saturation exponent due to pore structure and wettability has important
implications for evaluation of water saturation and,
hence, oil in place. Therefore, this paper primarily simulates electrical resistivity and capillary pressure by incorporating variations of wettability, through a general
contact angle, within a single pore. Contact angle hysteresis may occur due to surface roughness and adsorption [13]. In addition, the reservoir pore space is
interconnected, pores do not have uniform cross-sections and the pore space can be resolved into two interconnected regions: pore throats that interconnect
with pore bodies [29].

where ybi is the pore body radius of the ith tube attached
to each nodal pore. k is the number of tubes attached to
the node.

2. Theoretical model

3. Network simulation

Our previous three-dimensional network model
[30,31] incorporates a physical development of porescale wettability on various petrophysical characteristics
(in particular electrical resistivity and capillary pressure)
starting from primary drainage continuing to spontaneous/forced imbibition and secondary drainage. In
this paper, we have extended the capabilities of the
network model to allow contact angles of any value
between 0 and 180°.

The numerical solution of the electrical resistivity and
the absolute permeability are found by solving sets of
simultaneous equations that relate the network ¯ow
properties to those of individual pore elements. The
procedure starts with initial estimates of voltage and
hydraulic potential for each junction. No-¯ow or periodic boundary conditions can be imposed in the direction perpendicular to the potential gradient. With
each pore tube, there is an electrical and hydraulic

yp ˆ max…yb1 ; yb2 ; . . . ; ybk †;

…3†

H.N. Man, X.D. Jing / Advances in Water Resources 24 (2001) 345±363

conductance. The voltages of the junction, where the
tubes meet, are calculated using Ohm's and Kircho€'s
laws. The hydraulic potential of the junctions are calculated based on laminar ¯ow in pipes and mass conservation. A ¯ow chart of our algorithm to calculate the
electrical resistivity during primary drainage and absolute permeability can be found in Fig. 4.
The main approximations that have been made in our
network model are:
1. The circumferential radius of curvature of the oil/
water interface near the corner of the crevices is much
less than the transverse radius of curvature (Fig. 5).
Therefore, the transverse radius of curvature has been

349

neglected in the calculations using the Young±Laplace equation [32].
2. The head meniscus has also been neglected. This is
because the radius of the head meniscus is in the
order of microns, whereas the length of the pore tube
has been calculated to be in the order of hundred microns for a range of realistic porosities of granular
sandstones.
3. Although the co-operative pore ®lling process has
been modelled, we do not model explicitly the volumetric and electrical contribution of the pore nodes.
These properties are taken into account by the constricted pore elements.

Fig. 4. A simpli®ed ¯ow chart of our algorithm to calculate absolute permeability, electrical resistivity and capillary pressure curves during primary
drainage.

350

H.N. Man, X.D. Jing / Advances in Water Resources 24 (2001) 345±363

Fig. 5. Schematic picture showing oil moving into a GBP shape constricted tube. Note that the diagram is not to scale, because the length
is about an order of magnitude greater than the radius of the tube.
How far the oil ganglion moves towards the pore throat is given by the
equation shown. For each in®nitesimally thin section, the amount of
water retained in the crevices is calculated. Once the head meniscus
reaches the pore throat, the pore tube will centrally spontaneously ®ll
with oil.

4. Displacement cycles
We will now describe the ¯uid displacements occurring at the pore scale to calculate the full-¯ooded cycle
for electrical resistivity and capillary pressure curves.
The primary drainage cycle (in which, for clarity,
drainage will be consistently de®ned as ``oil displacing
water'' and imbibition as ``water displacing oil'' irrespective of which ¯uid preferentially wets the solid surface) have been discussed mathematically elsewhere [31].
The mathematical analysis describing the pore-scale
events that occur during water injection and oil re-injection has been extended from strongly wetting conditions to any contact angle.
4.1. Primary drainage
At the beginning of each simulation, the network is
fully saturated with water. The entire surface of the
network is designated as water-wet with h ˆ 0. By increasing the capillary pressure, oil displaces water in a
piston-like fashion, mimicking oil migration into a reservoir. Primary drainage continues until a maximum

capillary pressure is reached. Thick water ®lms that reside along the water-wet surfaces may rupture during
the course of primary drainage to a molecular thin ®lm.
The contact angle is no longer zero here [6,7]. We may
account for contact angle hysteresis by assuming that all
regions with a molecular thin ®lm after primary drainage are assigned a new contact angle.
To calculate capillary pressure and electrical resistivity curves, the pore tube is divided into sections of
in®nitesimal constant thickness by making cuts perpendicular to the x-axis. An increase in the capillary
pressure pushes the oil/water interface in discrete steps
to the opposite end of the pore tube. Cross-sectionally,
the oil/water interface moves towards the crevices
(Fig. 5).
Under conditions of capillary equilibrium, the capillary pressure is constant across the oil/water interface.
This in turn determines the meniscus shape. The amount
of water retained in the crevices can be expressed
analytically [33], but because our pore tubes are constricted, the volume of water and electrical resistance of
each individual tube is found by integrating along the
length of the pore tube. The mathematical details can be
found elsewhere [31]. Capillary equilibrium is acceptable
as long as the capillary number is not greater than about
10ÿ6 ±10ÿ7 [16,17]. Most reservoir displacements have
capillary numbers less than this. Oil may only enter a
tube if one of the adjacent tubes has already been
penetrated by oil or it is next to the face where oil is
injected. A ¯ow chart of our algorithm to derive both
the electrical resistivity and capillary pressure curves
during primary drainage can be found in Fig. 4.
The drainage process in the simulations stops at a
desired maximum capillary pressure. To investigate the
e€ect of saturation history, water was imbibed into the
partially oil-saturated model.
4.2. Water imbibition
4.2.1. Pore-scale mechanisms
Water may advance into an oil-®lled tube in a pistonlike fashion dictated by Eq. (4)
r
…4†
Pc ˆ K1 …h† ;
y
where K1 …h† is a constant for a particular contact angle
and pore geometry and y is the grain radius. Table 1 lists
the values of K1 …h† for a GBP shape for a range of
contact angles [33].
Water may also displace oil by snap-o€. As the capillary pressure is decreased, water will spontaneously
imbibe away from the crevices. Further increases in
wetting ¯uid may result in non-wetting ¯uid losing
contact with the pore wall [34] (Fig. 6). The non-wetting
¯uid then becomes disconnected (snapped-o€) retreating
to leave much of the adjoining pores ®lled with non-

H.N. Man, X.D. Jing / Advances in Water Resources 24 (2001) 345±363
Table 1
Piston-like advance factor for particular contact angles

a

Contact angle, h°

Piston-like advance factor, K1 …h†a

0
40
80
100
140
180

4.49
3.80
1.14
)1.14
)3.80
)4.49

After [33].

351

If how 6 w ‡ 90°, then
Pc ˆ

sin…w ÿ how † max
P ;
sin…w ÿ hpd † c

…6a†

ÿPcmax
;
sin…w ÿ hpd †

…6b†

otherwise, if how P w ‡ 90°, then
Pc ˆ

where w is the angle that describes the movement of the
oil/water meniscus, w ˆ w at Pc ˆ Pcmax , the maximum
imposed capillary pressure during primary drainage
(Fig. 7), and hpd is the contact angle of regions with a
molecular thin ®lm during primary drainage.
Provided how < w , the oil/water meniscus at the
crevices may start to move towards the centre of the
pore space at a positive capillary pressure and may meet
the other oil/water meniscus at a positive capillary
pressure. Subsequently, the oil snaps o€ at a capillary
pressure [31]
r
…7†
Pc ˆ K2 …h† ;
y
is a function of h such that
where Kp2 …h†

K2 …h† ˆ … 2 ‡ 1†…cos h ÿ sin h†.
If we substitute y ˆ yb (pore body radius) into Eq. (4)
and y ˆ yt (pore throat radius) into Eq. (7), then snapo€ will occur before piston-like advance if
r
r
…8†
K2 …h† > K1 …h† :
yt
yb
We note that there is a constraint on the constriction
factor …ˆ yb =yt †

Fig. 6. (a) In water imbibition, as water saturation increases, oil may
form an inscribed circle within a water-wet pore leading to snap-o€;
(b) a further increase in water pressure would result in oil retreating
towards the pore body. How far the oil ganglia retreat is given by the
equation shown.

wetting ¯uid. For a strong water-wet GBP shape
geometry (i.e., h ˆ 0), snap-o€ occurs at a capillary
pressure
p
r
…5†
Pc ˆ … 2 ‡ 1† :
y

For an advancing (water displacing oil) contact angle
greater than zero, the oil/water meniscus at the crevices
is pinned at the position it attained at the end of primary
drainage. The pinned oil/water meniscus may start to
move towards the centre of the pore space once h ˆ how
where how is the contact angle of regions after primary
drainage that were in contact with oil. The capillary
pressure at which the pinned oil/water meniscus starts to
move is stated below and derived in Appendix A.

yb K1 …h†
:
>
yt K2 …h†

…9†

For example, if we use h ˆ 0, and values of K1 …h† (from
Table 1) and K2 …h†, Eq. (9) becomes
yb
ˆ 2 > 1:86:
…10†
yt
Previously, we have shown that a constriction factor of 2
agrees very well with experimental data [30]. Hence, in
this case, snap-o€ always occurs before piston-like
advance for strongly water-wet systems. However, if we
increase the contact angle slightly to h ˆ 20°, Eq. (9)
becomes
yb
> 3:01:
…11†
yt
Therefore, if we use a constriction factor of 2, then using
h ˆ 20°, piston-like advance has precedence over snapo€. Snap-o€ can still occur, though, if piston-like
advance cannot. This situation arises when water cannot
piston-like advance in the tube being considered for
displacement, because it is not adjacent to a water-®lled
node.

352

H.N. Man, X.D. Jing / Advances in Water Resources 24 (2001) 345±363

(a)

(b)
Fig. 7. (a) Schematic picture showing oil moving into an entirely water-wet GBP shape. w describes the movement of the oil/water meniscus with h
constant; (b) if the GBP shape undergoes wettability alteration, then w attains the value w ˆ w for the remaining ¯ooding cycles following primary
drainage. Changes in capillary pressure will vary h instead. If the capillary pressure is low enough, then it is possible for the pinned oil/water meniscus
to move. Oil lens breakage will occur when BC in (a) and (b) are equal.

If how > w , then the pinned oil/water meniscus moves
at a negative capillary pressure. Now, for this case, water
invasion is forced and once h ˆ how , the pore space immediately ®lls with water due to the instability of the oil/
water meniscus [17,19]. However, this mechanism is less
favourable than water piston-like displacing oil.
Now, water may piston-like advance into a node and
has been shown to depend on the number of neighbouring tubes ®lled with oil [18] (Fig. 2). For this reason,

the mechanism is a co-operative process. Blunt [28]
modelled this invasion, where he used a square crosssection to represent the conduits, using a parametric
model given by
n
X
2r cos h
ÿr
Ai x i ;
…12†
Pc ˆ
r
iˆ1

where r is the largest inscribed circle within the pore
shape, n the number of neighbouring tubes ®lled with

H.N. Man, X.D. Jing / Advances in Water Resources 24 (2001) 345±363

oil, xi the random numbers between 0 and 1 and Ai are
matching parameters. Experiments show signi®cant
forced water invasion can occur for samples that exhibit
a contact angle of approximately 60° or greater [35].
Therefore, Blunt [28] chose A1 ˆ A2 ˆ


ˆ A5 ˆ
0:015 lmÿ1 to reproduce this e€ect.
For our nodal pores, Eq. (12) is slightly di€erent
Pc ˆ K1 …h†

n
X
r
ÿr
A i xi :
yp
iˆ1

…13†

For our model, we also found that when A1 ˆ A2 ˆ



ˆ A5 ˆ 0:015 lmÿ1 , this gives the same desired e€ect
as Dullien's [35] observation. A node can be water-®lled
only if (1) the imposed capillary pressure is lower than
the capillary pressure given by Eq. (13), (2) it is adjacent
to a water-®lled tube and (3) oil can escape to the outlet.
Water may also displace oil in the nodal pores
by snap-o€. In this case, Eq. (7) is again used with y
substituted for yp .
4.2.2. Oil layer breakage
During the forced water imbibition of water displacing oil, ``oil layers'' may form between the water in
the crevices and the water in the pore centre (Fig. 8).
According to éren et al. [19], this occurs when
h P 180° ÿ w :

…14†

If the oil/water meniscus on either side of the oil layer
touch, then the oil layer collapses preventing the ¯ow of
oil through the tube (Fig. 9). However, we assume that a
molecular thin oil ®lm still endures after the menisci
touch [7,17]. For a general contact angle how , this occurs
when the distances BC for the inner and outer oil/water
menisci are identical, i.e., BC…inner† ˆ BC…outer†
(Fig. 7). The capillary pressure at which the oil layer
collapses has been, again, deferred to Appendix A.
4.2.3. Cross-sectional area of water with oil layer
If oil layers are present within an in®nitesimal thin
section, then the calculation of the cross-sectional area

353

of water is very similar to the calculation during primary
drainage, but with the roles of oil and water reversed.
E€ectively, the cross-sectional area of water of a section
with an oil layer has two components. The water imbibing away from the crevices, i.e., the outer oil/water
meniscus that is pinned at the wettability discontinuity,
and the inner oil/water meniscus that is not pinned
(Fig. 8). The calculation of the cross-sectional area of
water with an oil layer has been simpli®ed, for any
contact angle, and is shown in Appendix A.
4.3. Secondary drainage
During oil re-injection (secondary drainage), the
capillary pressure is increased from a high negative value
to a high positive value. Accordingly, pores that exhibit
oil-wet regions spontaneously imbibe oil ®rst.
Competition between the snap-o€ and piston-like
displacement mechanism also occurs during oil reinjection. This is very much analogous to water injection
with the roles of oil and water reversed and, for brevity,
we will not repeat the mathematical derivations. However, the snap-o€ mechanism may only occur if the in®nitesimally thin section contains an oil layer or the
molecular thin oil ®lm that persisted after oil layer
breakage during forced imbibition. In tubes with a
molecular thin oil ®lm, the oil layer will spontaneously
re-form at the same capillary pressure at which it broke
(given by Eq. (A.13) in Appendix A). As the capillary
pressure is increased, the oil layer grows larger. The
outer meniscus of the oil layer moves towards the
crevices and the inner meniscus moves towards the centre
of the pore space. Cross-sectionally, the displacement
process is the reversal of Fig. 9.
The oil layer could swell large enough such that the
oil/water menisci may meet one another analogous to oil
becoming unstable during spontaneous imbibition.
Here, then, water snaps-o€ at a capillary pressure
p
r
…15†
Pc ˆ … 2 ‡ 1† …cos h ‡ sin h†:
y

Fig. 8. For mixed-wet pores water can centrally enter the pore tube, but oil is present in the form of layers. These are layers, because water still
resides in the crevices. The layers are formed, because the capillary pressure is not low enough for the central portion of the water to advance well into
the crevices.

354

H.N. Man, X.D. Jing / Advances in Water Resources 24 (2001) 345±363

Fig. 9. At a low enough capillary pressure, the oil/water menisci on either side of the oil layer meet. We assume a molecular oil ®lm remains after this
event, preventing oil to ¯ow through the pore tube.

Fig. 10. Water may form an inscribed circle within a mixed-wet pore leading to snap-o€. Water in the centre of the pore space is, then, trapped but
the water in the crevices is always connected.

When this happens, the centre of the thin section ®lls
with oil (Fig. 10). Displacement is only possible, if there
is continuity of oil from the inlet to the tube being
considered for displacement either via tubes that are
penetrated by oil or oil layer spanning tubes. Furthermore, if the tube contains an oil layer, then it must be
adjacent to a totally water-®lled section for the central
portion of water to escape. If the tube being considered
for displacement is not adjacent to a totally water-®lled
section, water cannot escape and remains trapped. Since
our chosen pore shape contains crevices, water can only
remain trapped in the centre of the pore space. Accordingly, the contribution to the electrical conductance
is zero. Here, then, only water residing in the crevices
that always remain connected contributes to the electrical conductance.

In terms of pore ®lling, neighbouring tubes again
in¯uence the capillary pressure for ®lling the pore with
oil. But this time, water inhibits the procedure [28]
n
X
r
Ai x i ;
…16†
Pc ˆ K1 …h† ‡ r
yp
iˆ1

where n is the number of neighbouring tubes ®lled with
water.
Snap-o€ of water by oil, in the nodal pores, is given
by Eq. (15) with y substituted for yp .

5. Results and discussion
We will now present simulations investigating the
e€ects of contact angle on electrical resistivity and

Table 2
Input parameters used in the network model for Figs. 11±16a
Figure
Contact angle after primary drainage (°)

11
0

12
40

13
80

14
100

15
140

16
180

a
Porosity ˆ 20%. Modi®ed Rayleigh pore size distribution with minimum, maximum radii and skewness factor ˆ 0, 100 and 1250 lm, respectively.
Constriction factor ˆ 2. Co-ordination number ˆ 6.

H.N. Man, X.D. Jing / Advances in Water Resources 24 (2001) 345±363

capillary pressure curves. Table 2 lists the input parameters used in our simulations. We represented the
pore structure of our network with a modi®ed Rayleigh
distribution. We used the same values of minimum/
maximum radii and skewness factor as those adopted by
McDougall and Sorbie [36]. The same authors argued
that this distribution gave more realistic pore size distributions of sandstone rocks. In the simulations here,
Pcmax was chosen such that a portion of all pore tubes are
not penetrated by oil in order to achieve initial water
saturation.
Table 3 presents the simulated formation factor,
cementation exponent, absolute permeability and saturation exponent for this particular rock type. Lesscemented sands usually have high porosities, and from
Archie [5], low formation factors: the porosity raised to
the power of the cementation exponent equals the reciprocal of the formation factor. The cementation
exponent, amongst other factors, is a function of shape,
sorting, packing of the grains, overburden pressure,
presence of clay and reservoir temperature. The simulated formation factor and cementation exponent presented here are typical of those measured experimentally
Table 3
Simulated petrophysical output data for this particular rock type
Figures

11±16

Formation factor
Cementation exponent
Absolute permeability (mD)
Saturation exponent

25.1
2.0
91
1.5

355

by Moss and Jing [4] of sandstone samples. The saturation exponent during primary drainage was calculated
by considering the line of best ®t between the saturation
range 30±100%.
We ®rst simulated a case that exhibits no contact
angle hysteresis following the primary drainage cycle,
i.e., h ˆ 0 throughout the simulation (Fig. 11). During
spontaneous imbibition, the resistivity indices are
slightly lower than primary drainage (Fig. 11). This is
because during primary drainage, oil migrates into the
system as a front. In spontaneous imbibition, oil is
already in place, but because snap-o€ is increasingly
favourable as the contact angle decreases, the oil tends
to exist as isolated ganglia. Therefore, there are more
large open spaces for the electrical current to ¯ow
through the sample. This type of hysteresis has been
observed experimentally [1,37].
Note that the saturation exponent is at the lower
range of saturation exponents observed experimentally.
Representing all the conduits of the pore space with a
cross-sectional GBP shape will retain appreciable
amounts of capillary-bound water due to the sharp
crevices. In fact, scanning electron microscope (SEM)
studies on quartzose sandstones show that overgrowths
can produce relatively smooth surfaces with no sharp
corners. We matched the saturation exponent, and other
electrical and ¯uid ¯ow experimental data for a North
Sea reservoir core sample, by assigning GBP and some
pores with a circular cross-sectional area [31]. This reduces the amount of capillary-bound water in the system. Detailed petrographic analysis on speci®c rock

Fig. 11. Capillary pressure and electrical resistivity curves when how ˆ 0.

356

H.N. Man, X.D. Jing / Advances in Water Resources 24 (2001) 345±363

types will provide information on the degree of cementation and alteration of sharp crevices.
Realistically though, experiments show that advancing and receding contact angles are generally not
identical [12]. We introduce contact angle hysteresis by
increasing the contact angle to di€erent values for a
number of simulations. We started by inputting h ˆ 40°
following the primary drainage cycle.
From the capillary pressure curve (Fig. 12), we notice
that during imbibition more water imbibes into the
system compared to the strongly water-wet case
(Fig. 11). This is because snap-o€ is almost suppressed
for this contact angle (snap-o€ is totally suppressed
when h ˆ 45°) allowing the piston-like advance displacement mechanism to dominate at the pore scale.
Morrow et al. [38] experimentally observed that weakly
water-wet cores gave improved oil recoveries compared
to strongly water-wet cores. As commented in the preceding paragraph, snap-o€ leaves oil existing as isolated
ganglia and, therefore, remains trapped during spontaneous imbibition. Conversely, the suppression of snapo€ allows the water to migrate as a front that in turn
displaces the well-connected oil.
In the resistivity curve, the cycles following primary
drainage shifts upward (Fig. 12). Note that for the
strongly water-wet case, the resistivity indices during
secondary drainage are lower than that during primary
drainage. In Fig. 12, there does not appear to be much
hysteresis between the three cycles compared to the
strongly water-wet case. The increase in the resistivity
indices, again, indicates oil being well connected and not

existing as isolated ganglia that allowed water to conduct more e€ectively for the strongly water-wet case.
In Fig. 13, we simulated a weakly water-wet system
with a contact angle of h ˆ 80°. Here, snap-o€ does not
occur at all. We note, in the resistivity plot, the water
imbibition cycle is even higher than the primary drainage cycle compared to Fig. 12. The trend appears because of the dominance of water piston-like displacing
oil. However, we now see a reversal of trend during
secondary drainage: the resistivity indices during this
cycle are now lower than the water imbibition cycle.
This could be explained by the fact that the suppression
of snap-o€ allows piston-like displacement to leave the
system almost water-®lled at the end of spontaneous
imbibition. Therefore, re-injecting with oil is very similar
to the displacement process occurring during primary
drainage. Note that there is some saturation change
during negative capillary pressures even though the
contact angle is less than 90°. This is a consequence of
the co-operative pore ®lling process and agrees with
Dullien's [35] experimental observation. Water may
only enter some nodes at negative capillary pressures
(cf. Eq. (13)).
Fig. 14 presents a weakly oil-wet system with a contact angle h ˆ 100°. During water imbibition, signi®cant
saturation changes occur only for negative capillary
pressures. For oil re-injection, there is some saturation
change during positive capillary pressures. Again,
analogous to the weakly water-wet case, this is due to
the co-operative pore ®lling process. Comparing
previous resistivity curves, we note that now at the

Fig. 12. Capillary pressure and electrical resistivity curves when how ˆ 40°.

H.N. Man, X.D. Jing / Advances in Water Resources 24 (2001) 345±363

357

Fig. 13. Capillary pressure and electrical resistivity curves when how ˆ 80°.

Fig. 14. Capillary pressure and electrical resistivity curves when how ˆ 100°.

beginning of water imbibition, the slope is much less.
This is because water is mainly entering large pores, but
not penetrating them. Once the tubes are penetrated by
water, we then see a signi®cant drop in the resistivity
indices. For water-wet samples, we do not observe this
two-tier gradient regime. During spontaneous imbi-

bition, once water enters the pore body of a constricted
tube, water spontaneously penetrates the whole tube.
Furthermore, the hysteresis between the water imbibition and secondary drainage resistivity indices appear to be less than for a weakly water-wet system
(Fig. 13). For a weakly oil-wet system, water invades the

358

H.N. Man, X.D. Jing / Advances in Water Resources 24 (2001) 345±363

Fig. 15. Capillary pressure and electrical resistivity curves when how ˆ 140°.

Fig. 16. Capillary pressure and electrical resistivity curves when how ˆ 180°.

largest pores ®rst and advances dendritically [17]. Note
that oil layers do not form at all (cf. Eq. (14)). This leads
to more trapping of oil in the smallest pore tubes. Oil
being trapped in the smallest pore tubes lead to higher
resistivity indices during secondary drainage and a
higher residual oil saturation compared to the weakly
water-wet case.

In Fig. 15, we increased the contact angle even further
to h ˆ 140°. Oil layers can form now. Oil layers can
provide continuity for the oil to escape. Observe that the
residual oil saturation at the end of water injection is
lower than for the weakly oil-wet system (Fig. 14). Since
oil layers may be present, water may snap-o€. Note that
the resistivity indices during secondary drainage are

H.N. Man, X.D. Jing / Advances in Water Resources 24 (2001) 345±363

visibly lower. The hysteresis between the resistivity indices during water imbibition and secondary drainage is
greater compared to the weakly oil-wet case. This is due
to the formation of oil layers that allow less trapping of
oil in smaller pore tubes.
For a strongly oil-wet case (Fig. 16), h ˆ 180°, the
residual oil saturation is even less. This is because a very
large negative capillary pressure is needed in order to
break the oil layers. Analogous to a strongly water-wet
sample, i.e., h ˆ 0, snap-o€ has precedence over pistonlike advance during secondary drainage. This implies oil
does not invade the system in the form of piston-like
advance, but spontaneously displaces water throughout
the sample. Snap-o€ occurs, analogous to spontaneous
imbibition, for the smallest pore tubes ®rst. The dramatic increase in the resistivity indices during secondary
drainage is a result of the oil occupying the smallest pore
tubes (Fig. 16). In fact, there appears to be virtually no
hysteresis between the resistivity indices during water
imbibition and secondary drainage (Fig. 16).

359

through these oil layers. Oil entrapment, in this case, is
less than for a weakly oil-wet system. This explains why
the hysteresis between the resistivity indices during
water imbibition and secondary drainage is actually
greater than for a weakly oil-wet case. When snap-o€ of
water dominates (i.e., h ˆ 180°) there, again, appears to
be no hysteresis between water imbibition and secondary drainage similar to the weakly oil-wet case. However, the saturation exponents during water imbibition
and secondary drainage are much higher than that
during primary drainage. This is in close agreement with
experimental observations. Our results for a variety of
contact angles indicate that the saturation exponent
during secondary drainage does not monotonically increase from h ˆ 0° to h ˆ 180°.
Acknowledgements
Hing Man would like to thank Martin Blunt for
discussion on the co-operative pore ®lling process. We
thank EPSRC for ®nancial support.

6. Conclusions
We have presented a network model based on a
physical development of pore-scale wettability with the
main intention to simulate electrical resistivity and
capillary pressure of oil/water/rock systems, in a full
drainage and imbibition cycle, for any contact angle
from 0° to 180°.
For a strongly water-wet system, snap-o€ is the
dominant event occurring at the pore scale, and oil tends
to exist as isolated ganglia during spontaneous imbibition. Therefore, strongly water-wet samples are more
conductive during spontaneous imbibition than primary
drainage.
During water imbibition, for a weakly water-wet
system, the resistivity indices are high, because as the
contact angle increases, snap-o€ is suppressed. Oil no
longer exists as isolated ganglia that allow the sample as
conductive as during primary drainage. The suppression
of snap-o€ also results in the weakly water-wet system to
be almost water-®lled at the end of water imbibition.
Therefore, re-injecting with oil almost mimics the primary drainage process. The resistivity indices during
secondary drainage maybe even as low as those attained
during primary drainage.
Increasing the contact angle further into the weakly
oil-wet regime, during water imbibition, water displaces
oil from the largest pores ®rst. Oil layer formation is not
possible and leads to trapping of oil in the smallest
pores. The resistivity indices during secondary drainage
can be quite high and as high as those attained during
water imbibition.
When formation of oil layers is possible (strongly oilwet), oil in smaller pores may escape to the outlet

Appendix A
A.1. Pore-scale mechanisms
The pinned oil/water meniscus may move during
water imbibition (Fig. 7) once h ˆ how . That is,


Pc

†
ˆ how :
…A:1†
sin…w
ÿ
h
w ÿ sinÿ1
pd
Pcmax
If ÿ90° 6 w ÿ how 6 0, then
Pc ˆ

sin…w ÿ how † max
P ;
sin…w ÿ hpd † c

…A:2a†

otherwise, if w ÿ how 6 ÿ 90°, then
Pc ˆ

ÿPcmax
:
sin…w ÿ hpd †

…A:2b†

We note that this is in agreement with éren et al. [19].
They used triangles to represent the cross-section of
their conduits (Fig. 17). They labelled bl as the largest
half-angle of their triangle. Accordingly, our w is related to bl via (Fig. 17)
bl ˆ 90° ÿ w :

…A:3†

Substituting this bl into their derived expression for
when the pinned oil/water meniscus moves [19], yields
our result.
Referring to Eqs. (A.2a) and (A.2b), the pinned oil/
water meniscus may move. However, in general, once

360

H.N. Man, X.D. Jing / Advances in Water Resources 24 (2001) 345±363

Finally, w is given by


f

:
w2 ˆ w1 ÿ

imp
of =ow

…A:5†

 p i
sin…w ÿ hpd † Pcmax h
ÿ
y
†
sin
ˆ
y
x ;
ÿ
…y
b
b
t
r
…1 ÿ sin w †
L

…A:6†

Pc ˆPc

Now

hence, the furthest position of the oil/water interface
moving towards the pore throat is,
xmove ˆ



L
yb ÿ …r=Pcmax †…sin…w ÿ hpd ††=…1 ÿ sin w †
sinÿ1
:
p
yb ÿ yt
…A:7†

A.2. Oil layer breakage
The capillary pressure at which the oil layer collapses
occurs when BC…inner† ˆ BC…outer† (Fig. 7).
Now
r
BC…inner† ˆ y cos w ‡ f1 ÿ cos‰w ÿ …p ÿ how †Šg;
Pc
…A:8†
BC…outer† ˆ y cos w ‡

r
f1 ÿ cos…w ÿ h†g;
Pc

…A:9†

where

Fig. 17. (a) éren et al. [19] used triangles to model their pore shape
where bl is the largest half-angle of the triangle; (b) we can relate w
with bl by imagining a triangle enclosing the arc meniscus as in (c).

the oil/water contact moves, the imposed capillary
pressure, Pcimp , is low enough for the oil/water contact
to move further towards the throat of the tube. The
furthest position the oil/water contact moves is denoted
by x ˆ xmove . For non-zero values of hpd ; w cannot be
explicitly solved in terms of Pc . Therefore, a function
f …Pc ; w †  0 was de®ned in order to solve for w via the
Newton±Raphson bisection method [39]. Hence, if
ÿ90° 6 w ÿ how 6 0, then

Pc
sin…w ÿ how † 
;
…A:4a†
f …Pc ; w † ˆ max ÿ
sin…w ÿ hpd † Pc ˆPcimp
Pc


otherwise, if w ÿ how 6 ÿ 90° then


Pc
1


:
f …Pc ; w † ˆ max ‡

Pc
sin…w ÿ hpd † Pc ˆPcimp

…A:4b†

sin‰w ÿ …p ÿ how †Š
y
ˆ ÿPc ;
1 ÿ sin w
r

…A:10a†

sin…w ÿ h†
y
 ˆ Pc ;
1 ÿ sin w
r

…A:10b†

sin…w ÿ hpd †
y
ˆ Pcmax
1 ÿ sin w
r

…A:10c†

and Pc < 0, because water is being forced into the system.
We now solve for the capillary pressure, Pc , at which
the oil/water menisci touch by equating Eq. (A.8) with
Eq. (A.9). However, we cannot express Eq. (A.8) in
terms of w or Eq. (A.9) in terms of w. But we may
eliminate Pc to obtain,
BC…inner† ˆ

r sin…w ÿ hpd †
cos w
Pcmax …1 ÿ sin w †



r
1 ÿ sin w
sin…w ÿ hpd †
ÿ max
Pc
1 ÿ sin w
ÿ sin…w ‡ how †
 ‰1 ‡ cos…w ‡ how †Š;

BC…outer† ˆ

…A:11†

r sin…w ÿ hpd †
cos w
Pcmax …1 ÿ sin w †



r
1 ÿ sin w
sin…w ÿ hpd †
ÿ max
Pc
1 ÿ sin w
ÿ sin…w ‡ how †

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H.N. Man, X.D. Jing / Advances in Water Resources 24 (2001) 345±363


sin…w ÿ hpd †
Pcmax
ÿ sin…w ‡ how †
)1=2
(

2
1 ÿ sin w
2
;
sin …w ‡ how †
 1ÿ
1 ÿ sin w
‡

r



1 ÿ sin w
1 ÿ sin w



…A:12†

guess wnew is fed into Eq. (A.14) and once again w is
determined by the Newton±Raphson bisection method.
Then, a new capillary pressure under equilibrium conditions, Pcnew , is calculated via Eq. (A.13) such that



1 ÿ sin wnew
ÿ sin…w ‡ how †
P max :
Pcnew ˆ ÿ
1 ÿ sin w
sin…wnew ÿ hpd † c
…A:16†

where



1 ÿ sin w
ÿ sin…w ‡ how †
Pcmax :
Pc ˆ ÿ
sin…w ÿ hpd †
1 ÿ sin w

…A:13†

To solve for Pc , we observe for a given pore tube, the
oil/water menisci will touch at y ˆ yb ®rst. From
Eq. (A.10c), w is known and so we can solve for w by
de®ning
f …w; w †  BC…inner† ÿ BC…outer†  0:

…A:14†

Again, the hybrid root-solver known as the ``Newton±
Raphson bisection method'' [39] was used to determine
w,

f 
:
…A:15†
w2 ˆ w 1 ÿ
of =ow w ˆw …xˆ0†

Hence, Pc is calculated by using Eq. (A.13).
However, in general, once the oil/water menisci touch
at y ˆ yb , the imposed capillary pressure, Pcimp , is low
enough for the oil lens to break further towards the
throat of the pore tube (Fig. 18). Eq. (A.14) can be used
again to determine the furthest position of oil lens
breakage denoted by x ˆ xbreak in Fig. 18. An initial

If
 imp

P ÿ P new  < e;
c
c

…A:17†

where e is the tolerance then we accept wnew and xbreak is
given by
L
xbreak ˆ sinÿ1
p


yb ÿ …r=Pcmax †…sin…wnew ÿ hpd †=1 ÿ sin wnew †
:
yb ÿ yt

…A:18†

A.3. Cross-sectional area of water with oil layer
Fig. 9 illustrates the cross-sectional area of water with
an oil layer. The outer oil/water meniscus that is pinned
at the wettability discontinuity contributes an area of
water of (Fig. 9)
(
p
2
2
Awater1 ˆ 4y ÿ ‡ cot w ‡ cot…w ÿ h†…1 ÿ sin w †
2
ÿ cot w …1 ÿ sin w †2 ‡ w

2 )
1 ÿ sin w

ÿ …w ÿ h†
:
sin…w ÿ h†

…A:19†

The inner oil/water meniscus, which is not pinned,
contributes a cross-sectional area of water of (Fig. 9)
(
p
2
2
Awater2 ˆ y …4 ÿ p† ÿ 4y ÿ ‡ cot w
2
‡ cot‰w ÿ …p ÿ how †Š…1 ÿ sin w†

2

ÿ cot w…1 ÿ sin w†2 ‡ w ÿ ‰w ÿ …p ÿ how †Š

2 )
1 ÿ sin w

;
…A:20†
sin‰w ÿ …p ÿ how †Š
where y is given by Eq. (A.10a), Pc < 0 and
Awater ˆ Awater1 ‡ Awater2 . Note that we cannot explicitly
express Awater in terms of one independent variable unless how ˆ 180°. Equating Eq. (A.10a) with Eq. (A.10c),
we may numerically solve
ÿ
Fig. 18. The oil/water menisci on either side of the oil layer ®rst meet
at the pore body. Further increases in the water pressure could result in
oil lens breakage occurring until the position x ˆ xbreak .

sin…w ‡ how †
Pc sin…w ÿ hpd †
;
ˆ ÿ max
Pc
1 ÿ sin w
…1 ÿ sin w †

…A:21†

for w and substitute into Awater2 . However, to simplify
our calculations, we henceforth approximate that the
cross-sectional area of water is dominated by water

362

H.N. Man, X.D. Jing / Advances in Water Resources 24 (2001) 345±363

entering the centre of the pore space, i.e., we neglect
Awater1 . Hence, the volume of water in the constricted
pore tube is
Z
dx dy
dw;
…A:22†
Vwater ˆ Awater2
dy dw
where
dx
L
1
ˆÿ
dy
p ‰…yb ÿ yt †2 ÿ …yb ÿ y†2 Š1=2

…A:23a†

and
dy
r
1
ˆÿ
‰cos‰w ÿ …p ÿ how †Š ÿ sin…p ÿ how †Š:
dw
Pc …1 ÿ sin w†2
…A:23b†

The electrical resistance of the constricted pore tube can
be found in a similar fashion
Z
1 dx dy
dw;
…A:24†
Rˆ
Awater2 dy dw
where we have given that the resistivity of water is unity,
as it cancels out in the resistivity index calculations.

References
[1] Wei JZ, Lile OB. In¯uence of