74-04-05
ENHANCED RECOVERY
Role of Capillary Forces in Determining
Microscopic Displacement Efficiency for
Oil Recovery by Waterflooding
J. C. Melrose and C. F. Brandner,
Mobil Research and Development Corporation,
Dallas, Texas
Abstract
The factors controlling the distribution of two immiscible
fluids, SUJch as oil and wa,ter, within the interstices of a
porous solid are reviewed. It is shown that an improved
understanding of these factors forms the basis of much
Tecent work aimed at the development of new methods fOT
reducing capillary forces and, consequently, eliminating
residual oil saturations.
Introduction
has for many years devoted
much research effort to the development of new processes for achieving improved oil recovery efficiencies. The current status of the most promising of these
new processes has been reviewed by Elkins(l) and by
ArnoldC'). For the most part, the research leading to
these processes has developed along lines pointed out
by Muskat(3) in a review published about 20 years ago.
In addition to thermal processes, which are used
primarily to overcome adverse crude oil viscosity
characteristiCs, a variety of processes are designed
to eliminate or reduce the so-called "surface forces"
within the crude-oil-displacing fluid-reservoir rock
system.
THE PETROLEUM INDUSTRY
James C. Melrose is a senior research
associate with the gxploration and
Production Research Division of Mobil
Research and Development Corporation in Dallas, Texas. He holds an
S.B. degree from Harvard University
and a Ph.D. degree from Stanford
University in physical chemistry. He
joined Mobil in 1947 and has worked
in the areas of drilling fluid rheology,
clay chemistry, reservoir fluid properties and the application of surface
chemistry to improved oil recovery processes.
Carl F. Brandner is a research associate with the Production Research
Division of Mobil Research and Development Corporation in Dallas, Texas.
He holds a B.S. from Iowa State University and a Masters degree from
University of Minnesota in physics.
He joined the Field Research Laboratory in 1949 and has worked in the
rheology of drilling fluids, formation
evaluation, core analysis, reservoir
description, and the physics and chemistry of fluid flow in porous media, especially as applied
to improved oil recovery processes.
54
When displacement is carried out by flooding with
an immiscible fluid such as water or gas, the surface
forces (or, more accurately, the capillary forces) are
responsible for trapping a large portion of the oil
phase within the interstices or pores of the rock. For
typical, homogeneous sandstone rocks, under usual
waterflooding conditions, trapping will occur within
an appreciable fraction of the total number of pores
constituting the pore space of the rock. The fraction
of the pore volume which is involved may range from
10 to 50 per cent. Thus, capillary forces playa major
role in limiting the recovery efficiency of such displacement processes.
Capillary forces will clearly be eliminated if the
drive fluid is preceded by a slug or bank of fluid which
is miscible both with the reservoir oil and with the
displacing phase(4l. However, the costs of the solvent
materials required in processes based on this principle
may be prohibitive. Alternatively, water-soluble surfactants, such as petroleum sulphonates, can be employed to eliminate the tendency of the oil phase to
remain trapped in the pores of the rock. Details of
several processes based on the use of such surfactants
have been recently reported<5-9J.
In this paper, the nature of the surface or capillary
forces encountered in petroleum reservoirs, together
with the microscopic features of the associated trapping mechanism, will first be reviewed. Some features of a new recovery process based on the use of
surfactants, i.e., the low-tension waterflooding
process(7-9J, will then be described. This discussion will
emphasize one of the principal microscopic mechanisms on which the process is based. An analysis of this
mechanism indicates that ultra-low values of the oilwater interfacial tension are required in order to
achieve improved recovery.
Hydrostatics, Capillarity and
Residual Saturations
HYDROSTATIC PRINCIPLES IN
MULTIPHASE FLUID SYSTEMS
Two or more fluid phases confined within a porous
solid phase of very small average pore size will generally be microscopically commingled. The fluid-fluid
interfacial areas associated with fluid distributions
of this type are large and characterized by high curvatures. The precise configurations of the individual
segments of the total fluid-fluid interfacial area in
such a system are described by two equations derivable
from the fundamental principle of fluid hydrostatics<lOl. For simplicity, in what follows, only the
case of two immiscible and relatively incompressible
The Journal of Canadian Petroleum
fluids, e.g., oil and water, will be explicitly discussed.
The extension to cases involving three fluids presents
some complexities(ll), but, in principle, can be carried
out without recourse to any additional fundamental
considerations.
The first of the two hydrostatic equations is the
Laplace equation;
Po - P w =
Yow
Jow. . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
Yos = Yws
+ Yow cos 60ws ·
..
..
.....
(2)
Here, (Jows represents the angle of contact, as measured
through the water phase, which the oil-water interface
(fluid meniscus) makes with the solid surface, denoted
by the subscript s (c.f. Fig. 1). The contact angle,
hereafter denoted simply as 0, is clearly a property
characterizing the three-phase line of contact. Measurements of this property typically display hysteresis
effects(12). It should also be pointed out that a more
rigorous version of Equation (2) will include a term
accounting for the line-tension in the three-phase line
of contact°l). This "line property" is the analogue of
the interfacial tension for surfaces.
The contact angle provides the only direct and unambiguous specification of the so-called "wettability"
property characteristic of a given oil-water-reservoir
rock system. If the angle (J is small, say less than 40
degrees, the system is usually said to be water-wet.
Similarly, an oil-wet system is characterized by an
angle 0 greater than 140 degrees. For angles of intermediate magnitude the wettability is best described as
intermediate.
The measurement of the interfacial properties defined by Equations (1) and (2) under the temperature and pressure conditions typical of petroleum reservoirs presents a number of difficulties. Even for
systems of relatively simple composition, such data
are only gradually being reported in the literature.
Recent studies of this type which should be noted are
those of Jennings(13, 14) and McCaffery OS-l7). An earlier
study by Hocott ( 8 ) reported surface and interfacial
tension data for several field systems. Measurements
of contact angles for crude oil - reservoir water systems on flat, polished surfaces of quartz and calcite
have heen carried out by Treiber, Archer and
Owens
This work indicates that many petroleum
reservoirs may be intermediate in wettability type, a
conclusion which is in disagreement with the commonly held belief that water-wet conditions prevail in
nearly all cases.
(9
).
APPLICATION TO POROUS SOLIDS
The application of Equations (1) and (2) to determine the configurations of the fluid-fluid interfaces
Technology, October-December, 1974, Montreal
/ /
/
/
/
/
/
/
/ ~
/
/
/
/
/ / /
/
/SOLI D PHASE /
/
/ / / /
'//
/
/
// / / /
/
~
~
Yow
(1)
Here, P denotes pressure in a fluid phase, ')I the interfacial tension and J the curvature of the fluidfluid interface (c.f. Fig. 1). The subscripts 0 and w
refer to oil and water, and the double subscript ow
refers to the corresponding interface. The thermodynamic derivation of Equation (1) provides a context within which the effects of temperature, pressure
and interfacial curvature on the interfacial tension
can be interrelated Oo ,121, These considerations also
apply, of course, to interfaces for which one of the
contiguous fluids is a gas (denoted by the subscript g).
In this case, the quantities ')IgO and ')Igw are referred to
as surface tensions.
The second hydrostatic equation is the Young
equation,
/
/
/
Yos
WATER
OIL
PHASE
PHASE
P
w
/
Po
radius of
cu rvatu re
FIGURE 1 Hydrostatic equilibrium
in contact with a solid phase.
two fluid phases
within porous solids presents difficulties not readily
apparent from the usual textbook treatments of the
problem. Because the average pore size is assumed to
be small, the pressures in the two fluid phases may
be taken as constant over distances comparable to the
dimensions of an average pore. Also, the interfacial
tension may be taken as independent of the magnitude
of the curvature OO ). According to Equation (1), then,
the interfacial configurations will correspond to surfaces of constant curvature. Even under this simplifi~ation, the algebraic expression for the curvature, J,
IS not a simple one. The expression includes terms
which involve the various first and second partial
derivatives of the function representing the shape of
the interface. In fact, the expression corresponds to
a second-order non-linear differential equation for
which general solutions in terms of known functions
are not available. The problem is generally circumvented by the use of highly simplified geometrical
models to represent the shape of the actual pores
involved. The necessary boundary conditions are then
provided by the assumed configuration of the pore
walls and by the contact angle, 0, defined by Equation
(2) .
For example, in the case of a porous solid formed
by a random packing of uniform spherical particles,
the pore openings may be approximated by the inner
surface of a torus(20, 21). In such a model, the radius of
the torus ring corresponds to the radius of the
particles forming the packing. The equatorial distance
from the axis of symmetry to the ring surface is
then adjusted to provide pore openings of the appropriate sizes.
A further problem arises in connection with the
stability of the fluid-fluid interfacial configuration
which results from the solution of the differential
equation represented by Equation (1). For typical
pore models, such as packings of spherical particles
of uniform size, it is easily shown(22,23) that a limited
range of such configurations are stable (c.f. Fig. 2).
This fact accounts for the hysteresis in the relationship between the capillary pressure, Pc = Po - Pw,
and the saturation of the aqueous phase, expressed as
Sw = Vw/(Vo + V w), where V is the volume of a
particular fluid in a test sample. (Under the implicit
assumption that 0 = 0, the quantity Sw is often referred to as the wetting-phase saturation.) The oilphase saturation, So, is similarly defined, so that
Sw + S, = 1.
55
WETIING
PHASE
NON - WEnI NG
PHASE
UNSTABLE
INTERFACE
CONFI GURATIONS
STABLE
~INTERFACE
CONFIGURATIONS
pIe, if the model is a random packing of uniform
spherical particles, R may be taken to be the particle
radius.
Experimental systems approximating this model
indicate that Hdr == 2.70 and H;mb == 1.75. The model
calculations(21) also show that, for drainage conditions,
Zdr (6) > 1 if 6 > o. . . . . . . . . . . . . . . . . . . . . . . . . . . .. (53)
and for imbibition conditions,
Zimb (6) < 1 if 6> 0
(5b)
Under the definitions adopted for Hdr and H;mb, it is
clear that both Zdr (In and Zimb (e) approach unity as
the contact angle, e, vanishes.
FIGURE 2 -
Stable and unstable interface configurations.
CAPILLARY PRESSURE VERSUS
FLUID SATURATION RELATIONSHIPS
Consider a sample of reservoir rock or similar
porous solid which is fully saturated with an aqueous
phase, and assume that the condition e = 0 holds.
The quasi-static displacement of this phase by a nonwetting phase is then defined by the set of increasing values of the capillary pressure, Pc, and a
corresponding set of decreasing values of the wettingphase saturation, Sw (c.f. Fig. 3a). The relationship
defined by such data is called the drainage capillary
pressure curve. The pressure-versus-saturation relationship for the reverse process, in which the wetting
phase displaces the non-wetting phase, is known as
the imbibition capillary pressure curve. Because, for a
typical pore configuration, the imbibition capillary
pressure is of the order of half of the drainage pressure, the curves will display considerable hysteresis'24, 33).
Excellent examples of such curves, with well-defined
interior scanning loops, have been reported by Morrow
and Harris'34l, Topp and Miller'3.) and Bomba(36). A
review emphasizing the fundamental physical principles underlying such hysteresis behaviour has been
given by Morrow (26 ). An earlier, but illuminating,
account of the manner in which interface stability
conditions give rise to rheons or Haines jumps, on the
microscopic or pore-size level, and to hysteresis effects on the macroscopic level is due to Miller and
Miller
As rheons are associated with both branches of the
capillary
pressure curve, it may be convenient to disdJow/dVw < 0, for 6 < 40
(3a)
tinguish between the two resulting types of rheons.
dJow/dVo < 0, for 6 > 140
(3b) On the drainage branch, the rheons correspond to a
As mentioned previously, solutions to Equation (1) process in which the wetting-phase saturation is
can only be obtained for highly simplified models of decreasing; hence such rheons may be referred to as
the pore geometry thought to be characteristic of xerons. Similarly, rheons on the imbibition branch corporous solids C2·,21). Regardless of the particular geo- respond to an increasing wetting-phase saturation
metrical forms which may be adopted as the basis and may be referred to as hygrons.
of such a model, the meniscus configurations corresA characteristic feature of the drainage capillary
ponding to the limiting conditions for stability will pressure curve for two nearly incompressible fluids,
be given by Equations (3). The maximum and mini- such as oil and water, is the minimum value of the
mum curvatures so obtained are referred to as drain- wetting-phase saturation. It appears that this reage and imbibition curvatures, J dr and J;mb, respect- sidual saturation, SWi, is reached asymptotically as the
ively. The effect of varying contact angle can then be capillary pressure is increased indefinitely. In the
expressed(21) in terms of a cos e factor and normalized case of a typical unconsolidated pack of closely sized
function, Z (e), as follows:
sand particles, the wetting-phase fluid corresponding
Jdr = 2 H dr Zdr (6) cos 6/R.
(4a) to the minimum saturation is largely in configurations
know as pendular rings. These are individually isolated
J;mb = 2 H;mb Zimb (6) cos 6/R... . . ... .
. . . . . . .. (4b)
segments of fluid surrounding the particle-to-particle
Here, Hdr and H;mb are the normalized values of the contact points. As the particle size distribution is made
two limiting curvatures for the case of zero contact broader, or possibly bimodal in character, it is possible
angle. The normalization factor is 2/R, where R is to develop local, small-scale packing heterogeneities.
a characteristic distance for the pore model. For exam- If this occurs, the magnitude of the residual wetting-
If the limits of stability are exceeded for a given
pore, the system will respond by a sudden and localized flow in which one fluid is displaced from the pore
by the other fluid. Such microscopic flow processes
were first discussed by Haines(24) and are usually referred to in the field of soil physics as "Haines
jumps"(""). The alternative designation of "rheon"
has been suggested(12J, and this term is used by Morrow'26\ although in a more restricted sense than
originally proposed. Rheons can be easily observed
visually by allowing air to displace a liquid phase (or
vice versa) from the pore space formed by packing
small glass beads in a glass capillary tube. A study of
the pressure oscillations associated with a displacement experiment of this type has been reported by
Crawford and Hoover C27l •
The stability limits applicable to interfacial configurations with axial symmetry have recently been
investigated by Dyson and co-workers,28-3.). This work
has shown that the conjugate-point criterion of the
calculus of variations leads to explicit solutions for the
limiting interfacial configurations which are observed
experimentally. A thermodynamic approach to this
problem, which is applicable to more general interfacial shapes, has been initiated by Everett and
Haynes'31,32). An intuitive treatment(l2), also thermodynamic in nature, suggests that the appropriate condition for configurational stability can be expressed
as
(25
)•
0
•••••••••
. . ..
0
••••••••. • • • . • • • • • • • • ••
56
The Journal of Canadian Petroleum
NARROW PORE
SIZE 01 ST'N
WIOE PORE
SIZE OIST'N
1
ti
u
u
0-
c...
u..J'
u..J-
a:
""
:::J
:::J
Vl
Vl
Vl
Vl
u..J
L.W
""
"'-
""c...
>:s""
ORA INAGE
>-
:s""
-'
DRAINAGE
-'
c...
«
u
0-
«
'-'
IMBIBITION
IMBIBITION
0.0
1.0
0.0
1.0
WE;TiNG PHASE SATURATION, Sw
WETTING PHASE SATURATION, Sw
Capillary pressure versus wettingFIGURE 3a phase saturation narrow pore size distribution.
FIGURE 3b Capillary pressure versus wettingphase saturation wide pore size distribution.
phase saturation for a drainage process increases, as
Morrow(31,38) has recently shown (c.f. Fig. 3b). The
additional fluid is now trapped in such a way that
small clusters of pores remain entirely filled. The
relationship between the residual wetting-phase saturation, as measured in the laboratory, and the water
saturation in a petroleum reservoir at the time of
discovery (the connate water saturation) is discussed
by Morrow(39).
saturation. For typical sandstone rocks, this residual
saturation is of the order of 30 to 40 per cent. Because
such rocks are usually considered to be water-wet,
this residual saturation, denoted as Sor, corresponds
to residual oil. It also defines what is termed the
microscopic displacement efficiency,
In a manner somewhat analogous to the drainage
case, the imbibition capillary pressure curve is characterized by a minimum value of the non-wetting-phase
saturation. The residual non-wetting fluid is, however,
trapped in configurations which are quite different
from those of the residual wetting fluid under
drainage conditions. In an imbibition process, the nonwetting fluid is trapped in such a way as to nearly
fill individual pores, or small clusters of neighbouring
pores. This trapping occurs because any given pore
will have only a limited number of openings leading
to other pores. There is then a finite probability that
any given pore will be by-passed"·). Evidence has been
reported(4l) which indicates that this probability increases as the larger pores of the pore-size distribution are subjected to displacement; i.e., as the
capillary pressure decreases.
As was pointed out some time ago by Frisch and
Hammersley(42\ a rather sophisticated theory for
treating the statistical aspects of the trapping process
has been developed. This theory, known as the theory
of percolation processes, is closely related to the wellknown problem of treating a random, self-avoiding
"walk" on an idealized lattice system. It is apparent
from a recent review(43) that a direct application of
this theory to the prediction of residual non-wettingphase saturations for porous solids has not yet been
developed.
Experimental results for various porous systems
clearly show that the broader the pore-size distribution, the larger is the residual non-wetting-phase
If, as may well be the case, many oil-water-reservoir
rock systems are not completely water-wet, it may be
possible to alter the wettability in such a way as to
reduce the value of Sor and hence recover additional
oil. In this connection, Morrow and co-workers(44, 45) are
currently studying the dependence of both the drainage and imbibition capillary pressure versus saturation
relationships on the wettability of the system, as defined by the contact angle. The model pore studies(2l),
mentioned above, suggest that if the angle is larger
than 45 degrees, a wetting phase, which has been displaced through the application of a positive capillary
pressure, will not be able to reimbibe. This prediction
has been borne out, at least qualitatively, by the recently reported experimental results"4,45), as well as
by unpublished work by Sutula and Wilson"6).
Technology, October-December, 1974, Montreal
E
_
m-
1 - Sor - Swi ...........................
I-S w i
(6)
Low Interfacial Tensions
For Improved Recovery
SIMULTANEOUS FLOW OF IMMISCIBLE FLUIDS
When oil is displaced from a homogeneous porous
solid by a waterflood process, both fluid phases undergo hydrodynamic transport through the pore space.
Only when, in a given region of the porous solid, the
oil-phase saturation is reduced to its residual value
does the oil phase cease to flow. Under simultaneous
two-phase flow conditions, the relationships between
the normalized coefficients of permeability for the
two phases and the water saturation are known as
relative permeability curves(47).
57
Many fundamental studies of these relationships
have been reported in the literature. It is found that
for so-called steady-state conditions (saturations not
changing with time), such curves are closely related
to the corresponding capillary pressure curves. Thus,
hysteresis behaviour, as defined by differing curves
for drainage and imbibition conditions, is observed'4S,49). Also, the final saturations reached for
both drainage (oil displacing water) and imbibition
(water displacing oil) are those which are defined
by the capillary pressure curves. In particular, it is
found that the relative permeability relationships are
independent of the flow rates employed and of the
viscosities of the two fluids'50, 5l). This evidence suffices to prove that, on a microscopic level, the commingled fluids are substantially in mutual equilibrium
as far as capillary pressure is concerned'52,53).
Studies of actual displacement processes, under nonsteady-state conditions, are also revealing"O). Perkins(54) has shown that when unconsolidated sand packs
are waterflooded, the residual oil saturation is independent of flood rate. Loomis and Crowell(55) found
that for gas-oil systems under drainage conditions
in homogeneous rocks, the relative permeability relationships are in good agreement with the relationships
measured by steady-state methods. In the case of
water-oil systems, however, these workers found that
steady-state and non-steady-state methods were not
in agreement, unless oil-wet conditions were maintained. Because for water displacing oil an oil-wet
condition gives rise to a drainage process, it is seen
that only for the non-steady-state imbibition process
are departures from local capillary equilibrium significant. This strongly suggests the possibility that this
type of behaviour is due to dynamic contact-angle
effects. However, it is not known at present how information on the flow-rate dependence of the contact
angle is actually to be applied in this context.
CRITICAL RANGE OF THE CAPILLARY NUMBER
It has been stated in the previous section that for
a completely water-wet system the residual oil saturation is independent of the flood rate. This implies
that, under ordinary flooding conditions, capillary
forces dominate the macroscopic displacement process
and that the microscopic distribution of the oil and
water phases is determined by the conditions for hydrostatic equilibrium. Clearly, however, if the flood
rate is made sufficiently high, this situation will no
longer be maintained. Viscous forces will begin to
have an effect on the magnitude of the residual oil;
Le., on the microscopic displacement efficiency as defined by Equation (6).
In order to assess the transition between a displacement process dominated by capillary forces and one
dominated by viscous forces, it is convenient to consider the dependence of the microscopic displacement
efficiency, Em, on a suitable dimensionless parameter,
such as the "capillary number", defined as
to the magnitude of the capillary forces'6o, 61). For
ordinary waterflooding conditions, N ca is of the order
of 10-6 (8).
Laboratory studies of the relationship between the
capillary number and the microscopic displacement
efficiency have usually been carried out by means
of "tertiary" waterfloods. Such an experiment is preceded by a conventional waterflood. The resulting
value of the residual oil saturation is that which corresponds to secondary recovery under field conditions.
The tertiary flood then involves greatly increasing
the flow rate, Uw, greatly decreasing the interfacial
tension, yow, or both. An increased wetting-phase vi.3cosity, p.,w, may also be used. As the capillary number
is substantially increased, eventually some additional
oil is recovered. This increased recovery is attributed
to the tertiary flood.
For packs of very coarse sand, Leverett(47) reported
data which showed indications of slightly improved
recovery efficiency for capillary numbers as low as
10-5. These measurements were not extended to higher
values of N ca . Studies on several sandstone samples
were carried out by Moore and Slobod"9), who found
the critical value of N ca for the initial reduction in
residual oil saturation to be about an order of magnitude higher; i.e., about 10-4. Results reported by
Taber'62,63) suggest that this lower critical value of
N ca is of the order of 5 x 10-" whereas values reported
by Lefebre du Prey"6, 57) and by Foster'S) are again
about 10-4. Variations in the lower critical value of
N ca are to be expected, as various reservoir rocks will
differ widely with respect to pore structure and pore
size distribution. Also, it is by no means certain that
all of the reported investigations were carried out
under the same contact-angle conditions.
Evidence for the magnitude of the upper critical
value of N Ca, which corresponds to complete oil
recovery, has not been so extensively reported in the
literature. It appears that in order to reduce the value
of the residual oil saturation by a factor of about
one-half, it is necessary to increase the capillary number by a factor of 10'62, 63) to 100 (56,57,59). Work reported by Foster'S) indicates that increasing N Ca by
a factor of 200-300 will result in microscopic displacement efficiencies which approach 100 per cent (c.f.
Fig. 4). Thus, the upper critical value of the capillary
number is of the order of 10-2 to 10-'. In a recent study
by Dullien and co-workers(64), the upper critical value
of N ca was not established. Although water-wet conditions apparently were maintained in these experiments, the techniques used to establish that such
conditions existed are not specified.
1.0
PORE SIZE DI ST'N
~E
NARROW
~
-'
CL
V>
..------------------:77",.....-----,
0.5
AVERAGE
WIDE
C
N Ca
=
............................
(7)
cC
u
~
Here, J1-w and U w are the water phase viscosity and
flow rate per unit cross-sectional area, respectively,
and qJ is the porosity of the solid. This number, or
its equivalent, has been used by a number of
authors'S, 47, 56-63). Physically, it represents the non-dimensional ratio of the magnitude of the viscous forces
58
WATER, WET
SYSTEMS
0.0
10'4
10,2
CAP ILlARY NUMBER. NCA
FIGURE 4 Correlation of microscopic displacement efficiency with capillary number.
The Journal of Canadian Petroleum
MOBILIZATION OF RESIDUAL OIL
The correlation between the displacement efficiency
and the capillary number, as obtained from tertiary
waterflood experiments, strongly suggests that the
process of mobilizing residual oil depends on a competition between viscous and capillary forces. A
description of the mechanism by which this competition is brought into play would obviously be instructive. Although various authors have attempted
to develop such mechanistic descriptions, these efforts
are widely scattered in the literature and have not
been subjected to critical examination. Clearly, a test
of any such description is whether it is consistent
with the observed critical range of the capillary number, N ca.
As pointed out by Taber(62l, the classical "Jamin
effect", discovered over 100 years ago, provides the
basic concept required for the development of a
mechanistic interpretation of the process of mobilizing
residual oil. An early attempt in this direction is due
to Gardescu(65). This work is concerned, however, with
two distinct effects, both referred to as the Jamin
effect. As pointed out by Smith and Crane(66), these
two effects are associated with (a) contact-angle hysteresis, as observed in cylindrical capillary tubes, and
(b) capillary-pressure hysteresis, as observed in noncylindrical tubes, i.e. tubes in which a series of constrictions are introduced.
The resulting confusion in terminology seems to
have been compounded by an insufficient appreciation of the actual basis of capillary pressure hysteresis effects(59J. As discussed above, this hysteresis
is a consequence of the stability conditions applicable
to the configuration of any fluid-fluid interface which
is bounded by contact with a solid phase(21.23J. These
stability conditions are given by Equations (3). Thus,
the model calculations summarized in Equations (4)
and (5) should provide a more satisfactory basis for
developing the desired mechanistic interpretation of
the critical range of N Ca than does the approach
followed by Gardescu(65J.
Figure 5 is a schematic representation of an isolated
segment of non-wetting-phase fluid trapped in the
interstices of a porous solid by an invading wetting
phase. The residual oil saturation (water-wet condition) is the sum of such trapped volumes of oil, divided
by the total pore volume. Microscopic studies indicate
that these oil "ganglia" usually extend over no more
than about 10 neighbouring pores. If now the surrounding water phase is caused to flow at an extremely
high rate, such an isolated segment will also be caused
to flow. However, this flow will be accompanied by at
least two individual displacement events, or rheons.
One of these will be of the drainage type, in which
oil moves out of the original volume occupied by the
ga~glion. The other will be of the imbibition type, in
WhICh water moves into the original volume. The
drainage rheon (xeron) occurs at the downstream or
low-pressure side of the ganglion; the imbibition rheon
(hygron) occurs at the upstream or high-pressure side.
This picture of the oil mobilization process thus
requires that a critical pressure gradient be exceeded
in the surrounding water phase. For the simplified
pore-shape model discussed above(21J, this critical
gradient is given by
( 6 P w /6 L).
en' =
Yow
Ode - Jimb)
2N R
.. (8)
Here, Jdr and J imb are the drainage and imbibition curvatures given by Equations (4a) and (4b), Ragain
Technology, October-December, 1974, Montreal
~
DIRECTION OF
WATER FLOW
>
"'L
FIGURE 5 -
Configuration of trapped oil ganglion.
denotes the radius of the uniform spherical particles
forming the porous solid, and N is the ratio of the
length of the mobilized ganglion, 6 L, to the particle
diameter. The quantity N thus measures the minimum
number of pores comprising a particular ganglion;
i.e., the number of pores encountered if the ganglion
is traversed in the direction of the surrounding waterphase flow.
In order to relate the critical pressure gradient given
by Equation (8) to the corresponding capillary number, (NCa)cd" it is only necessary to introduce Darcy's
law. Equation (7) can then be written as
(Nca)"it
k,w K(6"PI:) ... .. .. .. .. .. ..... ...
=
'PlOW
(9)
edt
L.:::..
where K is the value of the single-phase permeability
and k rw is the relative permeability to water. As some
authors report the results of tertiary waterflood
experiments in terms of the quantity (6Pw/6L)critlyow,
Equation (9) can be used to compute the corresponding values of the capillary number. In evaluating the
range of capillary numbers used in the work reported
by Taber(62,63), for instance, it is necessary to employ
this relationship.
The permeability, K, may also be related to the
capillary pressure for the case of zero contact angle
through a non-dimensional quantity, the Leverett
number(33), defined as
N Lc
Jdr vK/ cp
=
(6
=
0). . . . . . . . . . . . . . . . . . . . . ..
(10)
If now Equations (9) and (10) are combined with
Equations (4) and (8), the following expression is
obtained for the case of ()
(N ) . _
Ca
en'
-
= 0,
k nv (NLY G
4 N H dr
(11)
where
G = 1 -
(Himb/Hdr)
·
· (12)
In order to test Equation (11), the following average values of the various parameters may be used:
k rw = 0.3, N Le = 1/\1'5, H dr = 2.70, G = 0.35 and
N = 2. These values are typical for unconsolidated
packs of uniform glass beads or nearly spherical sand
grains. Using these values, the computed value of
(Nca)cd' is approximately 10-3. Because this value falls
within the critical range of capillary numbers observed
experimentally, the mechanism for oil mobilization
which has been described is reasonable and can be
accepted as a working hypothesis.
59
Further consideration of Equation (11) also suggests that the observed critical range of capillary
numbers (10-4 to 3 X 10-2 ) is associated with the expected ranges for the parameters appearing in this
equation. In particular, the lower critical value of
N ca is seen to be related to those ganglia, formed
during the original trapping processes under ordinary
waterflood (imbibition) conditions, which have the
largest values of the ratio N/(N Le )2. In a similar
manner, the upper critical value of N Ca will correspond
to the mobilization of oil trapped in single pores
(N = 1) of the smallest sizes within the pore-size
distribution. Thus, the proposed mechanism easily
accounts for a rather broad critical range of capillary
numbers.
It should be noted that, according to Equation (11),
neither the size of the spherical particles forming
the porous solid nor the permeability of the solid
directly influences the critical value of the capillary
number. Therefore, even though reservoir rocks will
differ widely with respect to these parameters, it is
not expected that such variables will contribute
significantly to the observed range of critical capillary
numbers. On the other hand, variations in the width
of the pore-size distribution and in the shapes of the
pores could, as already indicated, contribute to variations in the critical capillary number.
It is clear that the picture of the oil mobilization
process which leads to Equation (11) is based on the
same concepts which are used to explain the occurrence
of hysteresis in the capillary pressure versus fluid
saturation relationship. Thus, the pressure difference
in the wetting phase which appears in Equation (8)
is directly related to the difference between the drainage and imbibition interfacial curvatures discussed in
a previous section. It should be remarked in this connection that Dullien(67) has suggested a somewhat
similar expression. However, the relationship which
is proposed appears to be derived solely by considering
the magnitudes of the equivalent radii of a void and of
its pore entry. Thus, the mechanism of the oil mobilization process envisaged by Dullien does not explicitly,
at least, involve the condition for the configurational
stability of fluid interfaces. This condition, as has
been pointed out above, is responsible for the occurrence of hysteresis in the capillary pressure versus
saturation relationship.
The model calculations(2l) on which Equation (11)
is based can also be used to predict the effect of
contact angle. If a normalized function expressing this
effect is defined by the relationship
W(6) = {Nca (6)/(Nca )6=O}e,il
.
..
(13)
then the model leading to Equations (4) will yield the
following result,
W(6) = {Zd,(6) -
(1 - G)Zimb(6) I (cos 6/G) . . . . . . . . ..
(14)
If now Equations (5) are taken into account, it is
seen that the effect of increasing 0 on Zd' and Zimb
may be such that W(O) will actually increase. This is
in contrast to the effect predicted by taking W (0) to
be identical with cos O. It is found, in fact, that W (0)
may be as large as 2.2 when 0 is about 45 degrees.
Thus, the optimum wettability condition for tertiary
waterflooding is predicted to be one of complete
water-wetness (0
0).
Having provided a satisfactory mechanistic basis
for the observed range of critical capillary numbers,
the problem of achieving improved oil recovery by
=
60
altering the value of the interfacial tension, yow, may
be approached with added confidence as to the
magnitude of the required change. Thus, if N ca is to
be increased by a factor which is of the order of 10~,
the interfacial tension must be decreased by a similar
factor. In other words, the interfacial tension must be
reduced to what may be called an ultra-low value.
Such a value is of the order of 10-3 to 10-4 dynes/em.
Earlier estimates'6,,69) of the required level suggested
that 0.05 dynes/em was an appropriate target for
the purposes of tertiary oil recovery by interfacial
tension alteration. These estimates, however, were
based on limited evidence concerning the relationship
between the capillary number and displacement efficiency.
SURFACE CHEMISTRY OF
LOW-TENSION SYSTEMS
The achievement of ultra-low values of the interfacial tension between oil and water phases represents
a challenge of considerable magnitude. The literature
dealing with the fundamental aspects of the adsorption
of surface-active chemical species at the oil-water
interface offers limited guidance in this respect.
The early work of Harkins and Zollman'70), who reported a single instance of a tension as low as 2 x 10-3
dynes/em, appears never to have been followed up by
any systematic study of the composition variables
suggested by this observation. Presumably, it has
been believed that any such system will inevitably undergo spontaneous or self-emulsification. If this type
of behaviour were of universal occurrence, the study
of such interfacial states by conventional techniques
based on thermodynamics would, of course, be impossible.
In a pioneering study of the interfacial behaviour of
crude oil-water systems, Reisberg and Doscher(7l)
attributed the observation of an ultra-low value for
yow by Harkins and Zollman to the in-situ formation
of a highly surface-active component. Using a sample
of crude oil from Ventura, California, and the pendantdrop technique for measuring interfacial tension,
Reisberg and Doscher found that a tension as low as
10-3 dyne/em could be observed when the concentration
of NaOH in the aqueous phase was about 0.1 N. The
tensions for such systems, however, were rather unstable, so that equilibrium values of yow were not
obtained. Jennings(72) also used the pendant-drop
method and reported an ultra-low value of yow for a
crude oil-surfactant solution system.
More recently, studies carried out with aqueous
solutions of the petroleum sulphonate class of surfactants C7 - 9,73) have also demonstrated that ultra-low
values of yow can be achieved. It has been found that
the average surfactant molecular weight, the nature
of the surfactant molecular weight distribution and
the electrolyte concentration of the aqueous phase
are significant variables. In fact, a rather close
specification of each of these parameters is essential.
The interfacial compositions of such systems will be,
of course, of considerable complexity. It appears,
however, that the tendency for spontaneous emulsification is greatly reduced, if not eliminated, in
these cases. The conditions under which instabilities
of this type could arise are also complex. These conditions are the subject of an analysis recently reported
by Miller and Scriven'74'.
Although it seems quite likely that the basic mechanism involved in low-tension waterflooding for terThe Journal of Canadian Petroleum
tiary oil recovery is as described above, the design
of a practical field process must take into account a
number of other significant factors. The various requirements involved have been surveyed by Ahearn C75 J,
who includes references to the patent literature. Excessive adsorption of the surfactant components on
the reservoir rock surfaces must be avoided, as was
emphasized a decade ago by Wyllie'7.) and even earlier
by Muskat(3). The latter author also foresaw the need
for using a polymeric material as a thickener in the
aqueous drive fluid. By this means, the viscosity of
the· displacing fluid can be made larger than that of
the displaced fluid, and the necessary conditions for
hydrodynamic stability can be satisfied. Practical
techniques for meeting these and other critical requirements are discussed in recently reported work'7.".
relative permeability
single-phase permeability
distance in direction of macroscopic flow
number of pores in trapped oil ganglion
capillary numher, non-dimensional
Leverett number, non-dimensional
fluid phase pressure
capillary pressure
grain or particle radius
saturation, fraction of void volume occupied by a fluid
fluid phase flow rate per unit cross-stlctional area
fluid phase volume
function expressing effect of contact angle on N Ca
function expressing effect of contact angle on ratio,
J/cos6
kr
K
L
N
N Ca
N Le
P
Pc
R
S
U
V
W(6)
Z(6)
Greek Letters
interfacial or surface tension
contact angle
fluid phase viscosity
porosity, fractional void volume
"'(
6
IJ.
'!'
Summary and Conclusions
1. The distribution of commingled immiscible fluid
phases, e.g., oil and water, within the interstices of a
homogeneous porous solid, e.g., reservoir rock, is determined by capillary forces. When the conditions for
the configurational stability of a particular fluid-fluid
interface within an individual pore are not satisfied,
a sudden and rapid burst of flow occurs. The ensemble
of these microscopic flow events, or rheons, constitutes
the macroscopic displacement process.
2. This mechanism accounts for hysteresis in the
capillary pressure and relative permeability versus
saturation relationships; i.e., the differences between
the drainage and imbibition processes. It also accounts for the existence of residual saturations under
both types of displacement process; Le., the minimum
wetting-phase saturation for drainage and the minimum non-wetting-phase saturation for imbibition.
3. If a lower critical value of a non-dimensional quantity, the capillary number, is exceeded, the residual
non-wetting-phase saturation can be reduced. When
the upper critical value of the capillary number is
reached, the residual non-wetting-phase saturation
is eliminated. This range of critical capillary numbers
can be predicted from simplified pore-shape models
and a consideration of the conditions for the configurational stability of fluid interfaces.
4. In order to achieve a value of the capillary number
sufficiently high that the residual oil saturation can
be eliminated, an ultra-low value of the oil-water interfacial tension is required. Such a value is of the
order of 10-3 to 10-4 dynes/cm, and it has been recently
shown that aqueous solutions of petroleum sulphonates
can provide tensions in this range. This finding forms
the basis of several new processes for tertiary waterflooding. Model calculations indicate that a condition
of complete water-wetness should provide the optimum
wettability for oil recovery by this means.
Acknowledgment
Appreciation is expressed to Mobil Research and
Development Corporation for permission to publish
this paper.
Notation
Em
G
H
J
microscopic displacement efficiency
factor defined by Equation (12)
mean curvature of fluid-fluid interface, non-dimensional, 6 = 0 case
mean curvature of fluid-fluid interface
Technology, October-December, 1974, Montreal
Subscripts
go
gw
=
=
=
=
=
=
w.
dr
imb
wi
=
=
=
Or'
edt
=
oil phase
water phase
gas-oil interface
gas-water interface
oil-water interface
oil-solid interface
water-solid interface
oil-water-solid line of contact
drainage branch of hysteresis curve
imbibition branch of hysteresis curve
residual wetting phase, water-wet condition
residual non-wetting phase, water-wet condition
critical value of the capillary number
Literature Cited
(1) Elkins, L. E., World Oil, June, 1971, p. 69.
(2) Arnold, C. W., AIChE Symp. Ser. No. 127, 69, 25
(1973).
(3) Muskat, M., Ind. Eng. Chem., 45, 1401 (1953).
(4) Craig, D. R, and Bray, J. A., Eighth World Petro
Congr. Proc., 3, 275 (1971).
(5) Gogarty, W. B., and Tosch, W. C., Trans. AIME, 243,
1407 (1968).
(6) Davis, J. A., and Jones, S. C., Trans. AIME, 243,
1415 (1968).
(7) Hill, H. J., Reisberg, J., and Stegemeier, G. L., J.
Petro Tech., February, 1973, p. 18'6.
(8) Foster, W. R, J. Petro Tech., February, 1973, p. 205.
(9) Gale, W. W., and Sandvik, E. I., Soc. Petro Eng. J.,
13,191 (1973).
(10) Melrose, J. C., Ind. Eng. Chem., 60, No.3, 53 (1968).
(11) Pujado, P. R, and Scriven, L. E., J. Colloid Interfa,ce Sci., 40, 82 (1972).
(12) Melrose, J. C., Can. J. Chem. Eng., 48,638 (1970).
(13) Jennings, H. Y., Jr., J. Colloid Interface Sci., 24, 323
(1967).
(14) Jennings, H. Y., Jr., and Newman, G. H., Soc. Petro
Eng. J., 11,171 (1971).
(15) McCaffery, F. G., and Mungan, N., J. Canadian Petro
Te,ch., July-Sept., 1970, p. 185.
(16) McCaffery, F. G., J. Canadian Petro Tech., JulySept., 1972, p. 26.
(17) McCaffery, F. G., and Cram, P. J., Paper No. 56,
Div. ColI. and Surf. Chern., 161st National Meeting
of the ACS, Los Angeles, California, March 29April 2, 1971.
(18) Hocott, C. R., Trans. AIME, 132, 184 (1939).
(19) Treiber, L. E., Archer, D. L., and Owens, W. W.,
Soc. Petro Eng. J., 12, 531 (1972).
(20) Purcell, W. R, Trans. AIME, 189,369 (1950).
(21) Melrose, J. C., Soc. Petro Eng. J., 5, 259 (1965).
(22) Smith, W.O., Foote, P. D., and Busang, P. F.,
Physics, 1, 18 (1931).
(23) Smith, W.O., Physics, 4, 184 (1933).
(24) Haines, W. B., J. Agri. Sci., 20, 97 (1930).
(25) Miller, E. E., and Miller, R. D., J. Appl. Phys., 27,
324 (1956).
(26) Morrow, N. R, Ind. Eng. Chem., 62, No. 6,32 (1970).
(27) Crawford, F. W., and Hoover, G. M., J. Geophys. Res.,
7'1, 2911 (1966).
(28) ErIe, M. A., Dyson, D. C., and Morrow, N. R, AIChE
J., 17, 115 (1971).
61
(29) Erle, M. A., Gillette, R. D., and Dyson, D. C., Chem.
Eng., J., 1, 97 (1970).
(30) Gillette, R. D., and Dyson, D. C., Chem. Eng. J., 2,
44 (1971).
(31) Everett, D. H., and Haynes, J. M., J. Colloid Interface Sci., 38, 125 (1972).
(32) Everett, D. H., and Haynes, J. M., Zeit. physik. Chem.
(N.F.), 82,36 (1972).
(33) Leverett, M. C., Trans. AIME, 142, 152 (1941).
(34) Morrow, N. R, and Harris, C. C., Soc. Petro Eng. J.,
5, 15 (1965).
(35) Topp, G. C., and Miller, E. E., Soil Sci. So,c. Amer.
Proc., 30,156 (1966).
(36) Bomba, S. J., Ph.D. thesis, Dniv. of Wisconsin, 1967.
(Order No. 68-7089, Dniv. Microfilms, Inc., Ann
Arbor, Mich.)
(37) Morrow, N. R., Chem. Eng. Sci., 25, 1799 (1970).
(38) Morrow, N. R., Am. Assoc. Petro Geol. Bull., 55, 514
(1971).
(39) Morrow, N. R, J. Canadian Petro Tech., Jan.-March,
1971, p. 38.
(40) Fatt, 1., Trans. AIME, 207, 144 (1956).
(41) Raimondi, P., and Torcaso, M. A., Soc. Petro Eng. J.,
4, 49 (1964).
(42) Frisch, H. L., and Hammersley, J. M., J. Soc. Indust.
Appl. Math., 11, 894 (1963).
(43) Shante, V. K. S., and Kirkpatrick, S., Advan. Phys.,
20, 32,5 (1971).
(44) Morrow, N. R., and Mungan, N., Revue IFP, 26, 629
(1971).
(45) Morrow, N. R., Cram, P. J., and McCaffery, F. G.,
Soc. Petro Eng. J., 13, 221 (1973).
(46) Sutula, C. L., and Wilson, J. E., Paper No. 30f, 53rd
National AIChE Meeting, Pittsburgh, Pa., May 1720, 1964.
(47) Leverett, M. C., Trans. AIME, 132, 149 (1939).
(48) Osoba, J. S., Richardson, J. G., Kerver, J. K., Hafford,
J. A., and Blair, R. M., Trans. AIME, 192, 47 (1951).
(49) Geffen, T. M., Owens, W. W., Parrish, D. R., and
Morse, R. A., Trans. AIME, 192, 9'9 (1951).
(50) Richardson, J. G., in "Handbook of Fluid Dynamics",
V. L. Streeter, ed., Sec. 16, McGraw-Hill, New York,
1961.
(51) Donaldson, E. C., Lorenz., P. B., and Thomas, RD.,
Paper No. 1562, 41st Annual Fall Meeting of the
SPE, Dallas, Texas, Oct. 2-5, 1966.
(52) Brown, H. W., Trans. AIME, 192, 67 (1951).
62
(53) Ehrlich, R, and Crane, F. E., Soc. Petro Eng. J., 9,
221 (1969).
(54) Perkins, F. M., Jr., Trans. AIME, 210, 409 (1957).
(55) ,Loomis, A. G., and Crowell, D. C., Bureau of Mines
Bulletin 599, D.S. Govt. Print. Off., Washington,
1962.
(56) Lefebre du Prey, E. J., Compte-rendu du IIIO Colloque de l'ARTFP, Ed. TECHNIP, Paris, 1969, p.
251.
(57) Lefebre du Prey, E. J., Soc. Petro Eng. J., 13, 39
(1973).
(58) Brownell, L. E., and Katz, D. L., Chem. Eng. Progr.,
43, 601 (1947).
(59) Moore, T. F., and Slobod, R L., Producers Monthly,
August, 1956, p. 20.
(60) Saffman, P. G., and Taylor, Sir Geoffrey, Proc. Roy.
Soc., A245, 312 (1958).
(61) Lard€!, Mme, Briant, J., Labrid, J., and Marle, C.,
Revue IFP, 20, 25,3 (1965).
(62) Taber, J. J., Soc. Petro Eng. J., 9, 3 (1969).
(63) Taber, J. J., Kirby, J. C., and Schroeder, F. D.,
AIChE Symp. Ser. No. 127, 69, 53 (1973).
,
(64) Dullien, F. A. L., Dhawan, G. K., Gurak, N., and
Babjak, L, Soc. Petro Eng. J., 12, 289 (1972).
(65) Gardescu, 1. 1., Trans. AIME, 86, 351 (1930).
(66) Smith, W.O., and Crane, M. D., J. Am. Chem. Soc.,
52, 1345 (1930).
(67) Dullien, F. A. L., J. Petro Tech., January, 1969, p. 14.
(68) Moore, T. F., and Blum, H. A., Oil and Gas J., December 8, 1952, p. 108.
(69) Wagner, O. R., and Leach, R. 0., Soc. Petro Engl'. J.,
6, 335 (1966).
(70) Harkins, W. D., and Zollman, H., J. Am. Chem. Soc.,
48,69 (1926).
(71) Reisberg, J., and Doscher, T. M., Producers Monthly,
November, 1956, p. 43.
(72) Jennings, H. Y., Jr., Rev. Sci. Instrum., 28, 774
(1957).
(73) Brandner, C. F., and Dunlap, Peggy M., Paper No.
9, Div. ColI. and Surf. Chern., 165th National Meeting
of the ACS, Dallas, Texas, April 9-13, 1973.
(74) Miller, C. A., and Scriven, L. E., J. Colloid Interface
Sci., 33,360 (1970).
(75) Ahearn, G. P., J. Am. Oil Chemist's Soc., 46, 542A
(1969).
(76) Wylie, M. R. J., J. Petro Tech., June, 1962, p. 583.
The Journal of Canadian Petroleum